D3 Examples - Haroopad 0.13.0 issue with inline math

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<h3 id="the-lorenz-equations">The Lorenz Equations</h3><p><span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-1-Frame"><nobr><span class="math" id="MathJax-Span-1" style="width: 9.017em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.377em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(-0.22em 1000em 3.982em -0.304em); top: -2.131em; left: 0em;"><span class="mrow" id="MathJax-Span-2"><span class="mtable" id="MathJax-Span-3" style="padding-right: 0.167em; padding-left: 0.167em;"><span style="display: inline-block; position: relative; width: 7.029em; height: 0px;"><span style="position: absolute; clip: rect(2.34em 1000em 6.247em -0.471em); top: -4.59em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.631em; height: 0px;"><span style="position: absolute; clip: rect(3.127em 1000em 4.164em -0.457em); top: -5.376em; right: 0em;"><span class="mtd" id="MathJax-Span-4"><span class="mrow" id="MathJax-Span-5"><span class="texatom" id="MathJax-Span-6"><span class="mrow" id="MathJax-Span-7"><span class="munderover" id="MathJax-Span-8"><span style="display: inline-block; position: relative; width: 0.602em; height: 0px;"><span style="position: absolute; clip: rect(1.908em 1000em 2.689em -0.457em); top: -2.514em; left: 0em;"><span class="mi" id="MathJax-Span-9" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; clip: rect(1.899em 1000em 2.347em -0.302em); top: -2.762em; left: 0.055em;"><span class="mo" id="MathJax-Span-10" style="font-family: MathJax_Main;">˙</span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; clip: rect(3.127em 1000em 4.358em -0.471em); top: -3.928em; right: 0em;"><span class="mtd" id="MathJax-Span-20"><span class="mrow" id="MathJax-Span-21"><span class="texatom" id="MathJax-Span-22"><span class="mrow" id="MathJax-Span-23"><span class="munderover" id="MathJax-Span-24"><span style="display: inline-block; position: relative; width: 0.631em; height: 0px;"><span style="position: absolute; clip: rect(1.908em 1000em 2.883em -0.471em); top: -2.514em; left: 0em;"><span class="mi" id="MathJax-Span-25" style="font-family: MathJax_Math; font-style: italic;">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.006em;"></span></span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; clip: rect(1.899em 1000em 2.347em -0.302em); top: -2.762em; left: 0.085em;"><span class="mo" id="MathJax-Span-26" style="font-family: MathJax_Main;">˙</span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; clip: rect(3.127em 1000em 4.164em -0.429em); top: -2.507em; right: 0em;"><span class="mtd" id="MathJax-Span-37"><span class="mrow" id="MathJax-Span-38"><span class="texatom" id="MathJax-Span-39"><span class="mrow" id="MathJax-Span-40"><span class="munderover" id="MathJax-Span-41"><span style="display: inline-block; position: relative; width: 0.629em; height: 0px;"><span style="position: absolute; clip: rect(1.908em 1000em 2.689em -0.457em); top: -2.514em; left: 0.027em;"><span class="mi" id="MathJax-Span-42" style="font-family: MathJax_Math; font-style: italic;">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; clip: rect(1.899em 1000em 2.347em -0.302em); top: -2.762em; left: 0.083em;"><span class="mo" id="MathJax-Span-43" style="font-family: MathJax_Main;">˙</span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.59em;"></span></span><span style="position: absolute; clip: rect(2.508em 1000em 6.66em -0.436em); top: -4.809em; left: 0.909em;"><span style="display: inline-block; position: relative; width: 6.12em; height: 0px;"><span style="position: absolute; clip: rect(1.818em 1000em 3.146em -0.436em); top: -4.12em; left: 0em;"><span class="mtd" id="MathJax-Span-11"><span class="mrow" id="MathJax-Span-12"><span class="mo" id="MathJax-Span-13" style="font-family: MathJax_Main;">=</span><span class="mi" id="MathJax-Span-14" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span class="mo" id="MathJax-Span-15" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-16" style="font-family: MathJax_Math; font-style: italic;">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.006em;"></span></span><span class="mo" id="MathJax-Span-17" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mi" id="MathJax-Span-18" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">x</span><span class="mo" id="MathJax-Span-19" style="font-family: MathJax_Main;">)</span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(2.126em 1000em 3.112em -0.436em); top: -2.671em; left: 0em;"><span class="mtd" id="MathJax-Span-27"><span class="mrow" id="MathJax-Span-28"><span class="mo" id="MathJax-Span-29" style="font-family: MathJax_Main;">=</span><span class="mi" id="MathJax-Span-30" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">ρ</span><span class="mi" id="MathJax-Span-31" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-32" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mi" id="MathJax-Span-33" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.006em;"></span></span><span class="mo" id="MathJax-Span-34" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mi" id="MathJax-Span-35" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">x</span><span class="mi" id="MathJax-Span-36" style="font-family: MathJax_Math; font-style: italic;">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.863em 1000em 3.101em -0.436em); top: -1.25em; left: 0em;"><span class="mtd" id="MathJax-Span-44"><span class="mrow" id="MathJax-Span-45"><span class="mo" id="MathJax-Span-46" style="font-family: MathJax_Main;">=</span><span class="mo" id="MathJax-Span-47" style="font-family: MathJax_Main; padding-left: 0.278em;">−</span><span class="mi" id="MathJax-Span-48" style="font-family: MathJax_Math; font-style: italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.007em;"></span></span><span class="mi" id="MathJax-Span-49" style="font-family: MathJax_Math; font-style: italic;">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-50" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-51" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">x</span><span class="mi" id="MathJax-Span-52" style="font-family: MathJax_Math; font-style: italic;">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.006em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.809em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.131em;"></span></span></span><span style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 4.86em; vertical-align: -2.125em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-1">
\begin{aligned}
\dot{x} & = \sigma(y-x) \\
\dot{y} & = \rho x - y - xz \\
\dot{z} & = -\beta z + xy
\end{aligned}</script><p></p><h3 id="the-cauchy-schwarz-inequality">The Cauchy-Schwarz Inequality</h3><p><span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-2-Frame"><nobr><span class="math" id="MathJax-Span-53" style="width: 18.683em; display: inline-block;"><span style="display: inline-block; position: relative; width: 15.301em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.479em 1000em 6.004em -0.255em); top: -4.59em; left: 0em;"><span class="mrow" id="MathJax-Span-54"><span class="msubsup" id="MathJax-Span-55"><span style="display: inline-block; position: relative; width: 5.594em; height: 0px;"><span style="position: absolute; clip: rect(2.677em 1000em 6.004em -0.255em); top: -4.59em; left: 0em;"><span class="mrow" id="MathJax-Span-56"><span class="mo" id="MathJax-Span-57" style="vertical-align: 0em;"><span style="font-family: MathJax_Size4;">(</span></span><span class="munderover" id="MathJax-Span-58"><span style="display: inline-block; position: relative; width: 1.475em; height: 0px;"><span style="position: absolute; clip: rect(2.657em 1000em 4.384em -0.437em); top: -3.77em; left: 0em;"><span class="mo" id="MathJax-Span-59" style="font-family: MathJax_Size2; vertical-align: 0em;">∑</span><span style="display: inline-block; width: 0px; height: 3.77em;"></span></span><span style="position: absolute; clip: rect(1.64em 1000em 2.567em -0.458em); top: -1.188em; left: 0.082em;"><span class="texatom" id="MathJax-Span-60"><span class="mrow" id="MathJax-Span-61"><span class="mi" id="MathJax-Span-62" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-63" style="font-size: 70.7%; font-family: MathJax_Main;">=</span><span class="mn" id="MathJax-Span-64" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span><span style="position: absolute; clip: rect(1.664em 1000em 2.412em -0.477em); top: -3.39em; left: 0.519em;"><span class="mi" id="MathJax-Span-65" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 2.24em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-66" style="padding-left: 0.167em;"><span style="display: inline-block; position: relative; width: 1.004em; height: 0px;"><span style="position: absolute; clip: rect(1.909em 1000em 2.688em -0.459em); top: -2.514em; left: 0em;"><span class="mi" id="MathJax-Span-67" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; top: -2.09em; left: 0.546em;"><span class="mi" id="MathJax-Span-68" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span style="display: inline-block; width: 0px; height: 2.24em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-69"><span style="display: inline-block; position: relative; width: 0.895em; height: 0px;"><span style="position: absolute; clip: rect(1.656em 1000em 2.689em -0.452em); top: -2.514em; left: 0em;"><span class="mi" id="MathJax-Span-70" style="font-family: MathJax_Math; font-style: italic;">b</span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; top: -2.09em; left: 0.437em;"><span class="mi" id="MathJax-Span-71" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span style="display: inline-block; width: 0px; height: 2.24em;"></span></span></span></span><span class="mo" id="MathJax-Span-72" style="vertical-align: 0em;"><span style="font-family: MathJax_Size4;">)</span></span></span><span style="display: inline-block; width: 0px; height: 4.59em;"></span></span><span style="position: absolute; top: -3.772em; left: 5.137em;"><span class="mn" id="MathJax-Span-73" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span></span></span><span class="mo" id="MathJax-Span-74" style="font-family: MathJax_Main; padding-left: 0.278em;">≤</span><span class="mrow" id="MathJax-Span-75" style="padding-left: 0.278em;"><span class="mo" id="MathJax-Span-76" style="vertical-align: 0em;"><span style="font-family: MathJax_Size4;">(</span></span><span class="munderover" id="MathJax-Span-77"><span style="display: inline-block; position: relative; width: 1.475em; height: 0px;"><span style="position: absolute; clip: rect(2.657em 1000em 4.384em -0.437em); top: -3.77em; left: 0em;"><span class="mo" id="MathJax-Span-78" style="font-family: MathJax_Size2; vertical-align: 0em;">∑</span><span style="display: inline-block; width: 0px; height: 3.77em;"></span></span><span style="position: absolute; clip: rect(1.64em 1000em 2.567em -0.458em); top: -1.188em; left: 0.082em;"><span class="texatom" id="MathJax-Span-79"><span class="mrow" id="MathJax-Span-80"><span class="mi" id="MathJax-Span-81" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-82" style="font-size: 70.7%; font-family: MathJax_Main;">=</span><span class="mn" id="MathJax-Span-83" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span><span style="position: absolute; clip: rect(1.664em 1000em 2.412em -0.477em); top: -3.39em; left: 0.519em;"><span class="mi" id="MathJax-Span-84" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 2.24em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-85" style="padding-left: 0.167em;"><span style="display: inline-block; position: relative; width: 1.004em; height: 0px;"><span style="position: absolute; clip: rect(1.909em 1000em 2.688em -0.459em); top: -2.514em; left: 0em;"><span class="mi" id="MathJax-Span-86" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; clip: rect(1.66em 1000em 2.459em -0.456em); top: -2.64em; left: 0.546em;"><span class="mn" id="MathJax-Span-87" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span><span style="position: absolute; clip: rect(1.586em 1000em 2.412em -0.458em); top: -1.914em; left: 0.546em;"><span class="mi" id="MathJax-Span-88" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span style="display: inline-block; width: 0px; height: 2.24em;"></span></span></span></span><span class="mo" id="MathJax-Span-89" style="vertical-align: 0em;"><span style="font-family: MathJax_Size4;">)</span></span></span><span class="mrow" id="MathJax-Span-90"><span class="mo" id="MathJax-Span-91" style="vertical-align: 0em;"><span style="font-family: MathJax_Size4;">(</span></span><span class="munderover" id="MathJax-Span-92"><span style="display: inline-block; position: relative; width: 1.475em; height: 0px;"><span style="position: absolute; clip: rect(2.657em 1000em 4.384em -0.437em); top: -3.77em; left: 0em;"><span class="mo" id="MathJax-Span-93" style="font-family: MathJax_Size2; vertical-align: 0em;">∑</span><span style="display: inline-block; width: 0px; height: 3.77em;"></span></span><span style="position: absolute; clip: rect(1.64em 1000em 2.567em -0.458em); top: -1.188em; left: 0.082em;"><span class="texatom" id="MathJax-Span-94"><span class="mrow" id="MathJax-Span-95"><span class="mi" id="MathJax-Span-96" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-97" style="font-size: 70.7%; font-family: MathJax_Main;">=</span><span class="mn" id="MathJax-Span-98" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span><span style="position: absolute; clip: rect(1.664em 1000em 2.412em -0.477em); top: -3.39em; left: 0.519em;"><span class="mi" id="MathJax-Span-99" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 2.24em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-100" style="padding-left: 0.167em;"><span style="display: inline-block; position: relative; width: 0.895em; height: 0px;"><span style="position: absolute; clip: rect(1.656em 1000em 2.689em -0.452em); top: -2.514em; left: 0em;"><span class="mi" id="MathJax-Span-101" style="font-family: MathJax_Math; font-style: italic;">b</span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; clip: rect(1.66em 1000em 2.459em -0.456em); top: -2.716em; left: 0.437em;"><span class="mn" id="MathJax-Span-102" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span><span style="position: absolute; clip: rect(1.586em 1000em 2.412em -0.458em); top: -1.991em; left: 0.437em;"><span class="mi" id="MathJax-Span-103" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span style="display: inline-block; width: 0px; height: 2.24em;"></span></span></span></span><span class="mo" id="MathJax-Span-104" style="vertical-align: 0em;"><span style="font-family: MathJax_Size4;">)</span></span></span></span><span style="display: inline-block; width: 0px; height: 4.59em;"></span></span></span><span style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 4.034em; vertical-align: -1.591em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-2">
 \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)</script><p></p><h3 id="a-cross-product-formula">A Cross Product Formula</h3><p><span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-3-Frame"><nobr><span class="math" id="MathJax-Span-105" style="width: 15.017em; display: inline-block;"><span style="display: inline-block; position: relative; width: 12.295em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.264em 1000em 8.602em -0.467em); top: -5.683em; left: 0em;"><span class="mrow" id="MathJax-Span-106"><span class="msubsup" id="MathJax-Span-107"><span style="display: inline-block; position: relative; width: 1.332em; height: 0px;"><span style="position: absolute; clip: rect(1.937em 1000em 2.958em -0.467em); top: -2.787em; left: 0em;"><span class="texatom" id="MathJax-Span-108"><span class="mrow" id="MathJax-Span-109"><span class="mi" id="MathJax-Span-110" style="font-family: MathJax_Main; font-weight: bold;">V</span></span></span><span style="display: inline-block; width: 0px; height: 2.787em;"></span></span><span style="position: absolute; top: -2.145em; left: 0.874em;"><span class="mn" id="MathJax-Span-111" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span></span></span><span class="mo" id="MathJax-Span-112" style="font-family: MathJax_Main; padding-left: 0.222em;">×</span><span class="msubsup" id="MathJax-Span-113" style="padding-left: 0.222em;"><span style="display: inline-block; position: relative; width: 1.332em; height: 0px;"><span style="position: absolute; clip: rect(1.937em 1000em 2.958em -0.467em); top: -2.787em; left: 0em;"><span class="texatom" id="MathJax-Span-114"><span class="mrow" id="MathJax-Span-115"><span class="mi" id="MathJax-Span-116" style="font-family: MathJax_Main; font-weight: bold;">V</span></span></span><span style="display: inline-block; width: 0px; height: 2.787em;"></span></span><span style="position: absolute; top: -2.145em; left: 0.874em;"><span class="mn" id="MathJax-Span-117" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span></span></span><span class="mo" id="MathJax-Span-118" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mrow" id="MathJax-Span-119" style="padding-left: 0.278em;"><span class="mo" id="MathJax-Span-120" style="vertical-align: 3.255em;"><span style="display: inline-block; position: relative; width: 0.278em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.239em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: 1.772em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.404em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -1.569em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -0.733em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: 0.102em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: 0.937em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span></span><span class="mtable" id="MathJax-Span-121" style="padding-right: 0.167em; padding-left: 0.167em;"><span style="display: inline-block; position: relative; width: 6.163em; height: 0px;"><span style="position: absolute; clip: rect(3.571em 1000em 9.804em -0.452em); top: -6.885em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.781em; height: 0px;"><span style="position: absolute; clip: rect(1.928em 1000em 2.951em -0.452em); top: -5.242em; left: 50%; margin-left: -0.164em;"><span class="mtd" id="MathJax-Span-122"><span class="mrow" id="MathJax-Span-123"><span class="texatom" id="MathJax-Span-124"><span class="mrow" id="MathJax-Span-125"><span class="mi" id="MathJax-Span-126" style="font-family: MathJax_Main; font-weight: bold;">i</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.787em;"></span></span><span style="position: absolute; clip: rect(2.434em 1000em 4.867em -0.372em); top: -4.453em; left: 50%; margin-left: -0.89em;"><span class="mtd" id="MathJax-Span-137"><span class="mrow" id="MathJax-Span-138"><span class="mfrac" id="MathJax-Span-139" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 1.541em; height: 0px;"><span style="position: absolute; clip: rect(1.853em 1000em 2.918em -0.45em); top: -3.409em; left: 50%; margin-left: -0.71em;"><span class="mrow" id="MathJax-Span-140"><span class="mi" id="MathJax-Span-141" style="font-family: MathJax_Main;">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span class="mi" id="MathJax-Span-142" style="font-family: MathJax_Math; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.853em 1000em 2.918em -0.45em); top: -2.041em; left: 50%; margin-left: -0.574em;"><span class="mrow" id="MathJax-Span-143"><span class="mi" id="MathJax-Span-144" style="font-family: MathJax_Main;">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span class="mi" id="MathJax-Span-145" style="font-family: MathJax_Math; font-style: italic;">u</span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.313em; left: 0em;"><span style="border-left-width: 1.541em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; clip: rect(2.434em 1000em 4.867em -0.372em); top: -1.948em; left: 50%; margin-left: -0.89em;"><span class="mtd" id="MathJax-Span-158"><span class="mrow" id="MathJax-Span-159"><span class="mfrac" id="MathJax-Span-160" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 1.541em; height: 0px;"><span style="position: absolute; clip: rect(1.853em 1000em 2.918em -0.45em); top: -3.409em; left: 50%; margin-left: -0.71em;"><span class="mrow" id="MathJax-Span-161"><span class="mi" id="MathJax-Span-162" style="font-family: MathJax_Main;">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span class="mi" id="MathJax-Span-163" style="font-family: MathJax_Math; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.853em 1000em 2.918em -0.45em); top: -2.041em; left: 50%; margin-left: -0.546em;"><span class="mrow" id="MathJax-Span-164"><span class="mi" id="MathJax-Span-165" style="font-family: MathJax_Main;">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span class="mi" id="MathJax-Span-166" style="font-family: MathJax_Math; font-style: italic;">v</span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.313em; left: 0em;"><span style="border-left-width: 1.541em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span><span style="display: inline-block; width: 0px; height: 6.885em;"></span></span><span style="position: absolute; clip: rect(3.571em 1000em 9.804em -0.563em); top: -6.885em; left: 2.781em;"><span style="display: inline-block; position: relative; width: 1.726em; height: 0px;"><span style="position: absolute; clip: rect(1.928em 1000em 3.151em -0.563em); top: -5.242em; left: 50%; margin-left: -0.191em;"><span class="mtd" id="MathJax-Span-127"><span class="mrow" id="MathJax-Span-128"><span class="texatom" id="MathJax-Span-129"><span class="mrow" id="MathJax-Span-130"><span class="mi" id="MathJax-Span-131" style="font-family: MathJax_Main; font-weight: bold;">j</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.787em;"></span></span><span style="position: absolute; clip: rect(2.434em 1000em 4.867em -0.372em); top: -4.453em; left: 50%; margin-left: -0.863em;"><span class="mtd" id="MathJax-Span-146"><span class="mrow" id="MathJax-Span-147"><span class="mfrac" id="MathJax-Span-148" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 1.486em; height: 0px;"><span style="position: absolute; clip: rect(1.853em 1000em 2.918em -0.45em); top: -3.409em; left: 50%; margin-left: -0.683em;"><span class="mrow" id="MathJax-Span-149"><span class="mi" id="MathJax-Span-150" style="font-family: MathJax_Main;">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span class="mi" id="MathJax-Span-151" style="font-family: MathJax_Math; font-style: italic;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.182em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.853em 1000em 2.918em -0.45em); top: -2.041em; left: 50%; margin-left: -0.574em;"><span class="mrow" id="MathJax-Span-152"><span class="mi" id="MathJax-Span-153" style="font-family: MathJax_Main;">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span class="mi" id="MathJax-Span-154" style="font-family: MathJax_Math; font-style: italic;">u</span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.313em; left: 0em;"><span style="border-left-width: 1.486em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; clip: rect(2.434em 1000em 4.867em -0.372em); top: -1.948em; left: 50%; margin-left: -0.863em;"><span class="mtd" id="MathJax-Span-167"><span class="mrow" id="MathJax-Span-168"><span class="mfrac" id="MathJax-Span-169" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 1.486em; height: 0px;"><span style="position: absolute; clip: rect(1.853em 1000em 2.918em -0.45em); top: -3.409em; left: 50%; margin-left: -0.683em;"><span class="mrow" id="MathJax-Span-170"><span class="mi" id="MathJax-Span-171" style="font-family: MathJax_Main;">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span class="mi" id="MathJax-Span-172" style="font-family: MathJax_Math; font-style: italic;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.182em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.853em 1000em 2.918em -0.45em); top: -2.041em; left: 50%; margin-left: -0.546em;"><span class="mrow" id="MathJax-Span-173"><span class="mi" id="MathJax-Span-174" style="font-family: MathJax_Main;">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span class="mi" id="MathJax-Span-175" style="font-family: MathJax_Math; font-style: italic;">v</span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.313em; left: 0em;"><span style="border-left-width: 1.486em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span><span style="display: inline-block; width: 0px; height: 6.885em;"></span></span><span style="position: absolute; clip: rect(2.916em 1000em 8.457em -0.463em); top: -6.23em; left: 5.507em;"><span style="display: inline-block; position: relative; width: 0.656em; height: 0px;"><span style="position: absolute; clip: rect(1.929em 1000em 2.951em -0.463em); top: -5.242em; left: 50%; margin-left: -0.328em;"><span class="mtd" id="MathJax-Span-132"><span class="mrow" id="MathJax-Span-133"><span class="texatom" id="MathJax-Span-134"><span class="mrow" id="MathJax-Span-135"><span class="mi" id="MathJax-Span-136" style="font-family: MathJax_Main; font-weight: bold;">k</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.787em;"></span></span><span style="position: absolute; clip: rect(1.902em 1000em 2.918em -0.453em); top: -3.196em; left: 50%; margin-left: -0.273em;"><span class="mtd" id="MathJax-Span-155"><span class="mrow" id="MathJax-Span-156"><span class="mn" id="MathJax-Span-157" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.902em 1000em 2.918em -0.453em); top: -0.691em; left: 50%; margin-left: -0.273em;"><span class="mtd" id="MathJax-Span-176"><span class="mrow" id="MathJax-Span-177"><span class="mn" id="MathJax-Span-178" style="font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span></span><span style="display: inline-block; width: 0px; height: 6.23em;"></span></span></span></span><span class="mo" id="MathJax-Span-179" style="vertical-align: 3.255em;"><span style="display: inline-block; position: relative; width: 0.278em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.239em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: 1.772em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.404em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -1.569em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -0.733em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: 0.102em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: 0.937em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 5.683em;"></span></span></span><span style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 7.466em; vertical-align: -3.428em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-3">
\mathbf{V}_1 \times \mathbf{V}_2 =  \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial X}{\partial u} &  \frac{\partial Y}{\partial u} & 0 \\
\frac{\partial X}{\partial v} &  \frac{\partial Y}{\partial v} & 0
\end{vmatrix}</script><p></p><h3 id="the-probability-of-getting-k-heads-when-flipping-n-coins-is">The probability of getting <p><span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-17-Frame"><nobr><span class="math" id="MathJax-Span-821" style="width: 0.683em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.546em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.656em 1000em 2.689em -0.444em); top: -2.514em; left: 0em;"><span class="mrow" id="MathJax-Span-822"><span class="mi" id="MathJax-Span-823" style="font-family: MathJax_Math; font-style: italic;">k</span></span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span></span><span style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 0.993em; vertical-align: -0.08em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-17">k</script><p></p> heads when flipping <p><span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-18-Frame"><nobr><span class="math" id="MathJax-Span-824" style="width: 0.75em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.601em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.908em 1000em 2.689em -0.471em); top: -2.514em; left: 0em;"><span class="mrow" id="MathJax-Span-825"><span class="mi" id="MathJax-Span-826" style="font-family: MathJax_Math; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span></span><span style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 0.686em; vertical-align: -0.08em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-18">n</script><p></p> coins is</h3><p><span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-4-Frame"><nobr><span class="math" id="MathJax-Span-180" style="width: 13.683em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.202em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.321em 1000em 5.048em -0.459em); top: -3.934em; left: 0em;"><span class="mrow" id="MathJax-Span-181"><span class="mi" id="MathJax-Span-182" style="font-family: MathJax_Math; font-style: italic;">P<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.109em;"></span></span><span class="mo" id="MathJax-Span-183" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-184" style="font-family: MathJax_Math; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.026em;"></span></span><span class="mo" id="MathJax-Span-185" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-186" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="texatom" id="MathJax-Span-187" style="padding-left: 0.278em;"><span class="mrow" id="MathJax-Span-188"><span class="mrow" id="MathJax-Span-189"><span class="mo" id="MathJax-Span-190" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">(</span></span><span class="mfrac" id="MathJax-Span-191"><span style="display: inline-block; position: relative; width: 0.601em; height: 0px;"><span style="position: absolute; clip: rect(1.908em 1000em 2.689em -0.471em); top: -3.19em; left: 50%; margin-left: -0.301em;"><span class="mi" id="MathJax-Span-192" style="font-family: MathJax_Math; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; clip: rect(1.656em 1000em 2.689em -0.444em); top: -1.828em; left: 50%; margin-left: -0.273em;"><span class="mi" id="MathJax-Span-193" style="font-family: MathJax_Math; font-style: italic;">k</span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span></span></span><span class="mo" id="MathJax-Span-194" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">)</span></span></span></span></span><span class="msubsup" id="MathJax-Span-195"><span style="display: inline-block; position: relative; width: 1.004em; height: 0px;"><span style="position: absolute; clip: rect(1.908em 1000em 2.872em -0.531em); top: -2.514em; left: 0em;"><span class="mi" id="MathJax-Span-196" style="font-family: MathJax_Math; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; top: -2.653em; left: 0.546em;"><span class="mi" id="MathJax-Span-197" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span style="display: inline-block; width: 0px; height: 2.24em;"></span></span></span></span><span class="mo" id="MathJax-Span-198" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-199" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-200" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mi" id="MathJax-Span-201" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">p</span><span class="msubsup" id="MathJax-Span-202"><span style="display: inline-block; position: relative; width: 1.878em; height: 0px;"><span style="position: absolute; clip: rect(1.818em 1000em 3.146em -0.437em); top: -2.732em; left: 0em;"><span class="mo" id="MathJax-Span-203" style="font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; top: -2.772em; left: 0.437em;"><span class="texatom" id="MathJax-Span-204"><span class="mrow" id="MathJax-Span-205"><span class="mi" id="MathJax-Span-206" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">n</span><span class="mo" id="MathJax-Span-207" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-208" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span></span></span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.934em;"></span></span></span><span style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 3.06em; vertical-align: -1.225em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-4">
P(E)   = {n \choose k} p^k (1-p)^{ n-k}</script><p></p><h3 id="an-identity-of-ramanujan">An Identity of Ramanujan</h3><p><span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-5-Frame"><nobr><span class="math" id="MathJax-Span-209" style="width: 20.617em; display: inline-block;"><span style="display: inline-block; position: relative; width: 16.885em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.058em 1000em 5.738em -0.372em); top: -2.732em; left: 0em;"><span class="mrow" id="MathJax-Span-210"><span class="mfrac" id="MathJax-Span-211" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 7.716em; height: 0px;"><span style="position: absolute; clip: rect(1.902em 1000em 2.896em -0.409em); top: -3.409em; left: 50%; margin-left: -0.273em;"><span class="mn" id="MathJax-Span-212" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(2.457em 1000em 4.584em -0.312em); top: -2.644em; left: 50%; margin-left: -3.798em;"><span class="mrow" id="MathJax-Span-213"><span class="texatom" id="MathJax-Span-214"><span class="mrow" id="MathJax-Span-215"><span class="mo" id="MathJax-Span-216" style="vertical-align: 0em;"><span style="font-family: MathJax_Size2;">(</span></span></span></span><span class="msqrt" id="MathJax-Span-217"><span style="display: inline-block; position: relative; width: 2.976em; height: 0px;"><span style="position: absolute; clip: rect(1.51em 1000em 2.883em -0.449em); top: -2.514em; left: 0.984em;"><span class="mrow" id="MathJax-Span-218"><span class="mi" id="MathJax-Span-219" style="font-family: MathJax_Math; font-style: italic;">ϕ</span><span class="msqrt" id="MathJax-Span-220"><span style="display: inline-block; position: relative; width: 1.366em; height: 0px;"><span style="position: absolute; clip: rect(1.902em 1000em 2.918em -0.442em); top: -2.732em; left: 0.82em;"><span class="mrow" id="MathJax-Span-221"><span class="mn" id="MathJax-Span-222" style="font-family: MathJax_Main;">5</span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(0.929em 1000em 1.325em -0.492em); top: -1.933em; left: 0.82em;"><span style="border-left-width: 0.546em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: -0.068em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span><span style="position: absolute; clip: rect(3.025em 1000em 4.353em -0.42em); top: -4.02em; left: 0em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; clip: rect(3.555em 1000em 4.023em -0.492em); top: -4.732em; left: 0.984em;"><span style="display: inline-block; position: relative; width: 1.992em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.989em; left: -0.084em;">−<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -3.989em; left: 1.298em;">−<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.989em; left: 0.36em;">−<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.989em; left: 0.829em;">−<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; clip: rect(2.975em 1000em 4.503em -0.381em); top: -4.141em; left: 0em;"><span style="font-family: MathJax_Size1;">√</span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span></span><span class="mo" id="MathJax-Span-223" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mi" id="MathJax-Span-224" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">ϕ</span><span class="texatom" id="MathJax-Span-225"><span class="mrow" id="MathJax-Span-226"><span class="mo" id="MathJax-Span-227" style="vertical-align: 0em;"><span style="font-family: MathJax_Size2;">)</span></span></span></span><span class="msubsup" id="MathJax-Span-228"><span style="display: inline-block; position: relative; width: 1.605em; height: 0px;"><span style="position: absolute; clip: rect(1.908em 1000em 2.689em -0.453em); top: -2.514em; left: 0em;"><span class="mi" id="MathJax-Span-229" style="font-family: MathJax_Math; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; top: -2.678em; left: 0.492em;"><span class="texatom" id="MathJax-Span-230"><span class="mrow" id="MathJax-Span-231"><span class="mfrac" id="MathJax-Span-232" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 0.393em; height: 0px;"><span style="position: absolute; clip: rect(1.689em 1000em 2.35em -0.467em); top: -2.525em; left: 50%; margin-left: -0.137em;"><span class="mn" id="MathJax-Span-233" style="font-size: 50%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 2.186em;"></span></span><span style="position: absolute; clip: rect(1.689em 1000em 2.361em -0.467em); top: -1.867em; left: 50%; margin-left: -0.137em;"><span class="mn" id="MathJax-Span-234" style="font-size: 50%; font-family: MathJax_Main;">5</span><span style="display: inline-block; width: 0px; height: 2.186em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.24em; left: 0em;"><span style="border-left-width: 0.393em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span><span class="mi" id="MathJax-Span-235" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.24em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.77em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.313em; left: 0em;"><span style="border-left-width: 7.716em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span><span class="mo" id="MathJax-Span-236" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-237" style="font-family: MathJax_Main; padding-left: 0.278em;">1</span><span class="mo" id="MathJax-Span-238" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mfrac" id="MathJax-Span-239" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 5.694em; height: 0px;"><span style="position: absolute; clip: rect(2.991em 1000em 4.164em -0.453em); top: -4.666em; left: 50%; margin-left: -0.939em;"><span class="msubsup" id="MathJax-Span-240"><span style="display: inline-block; position: relative; width: 1.878em; height: 0px;"><span style="position: absolute; clip: rect(1.908em 1000em 2.689em -0.453em); top: -2.514em; left: 0em;"><span class="mi" id="MathJax-Span-241" style="font-family: MathJax_Math; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; top: -2.658em; left: 0.492em;"><span class="texatom" id="MathJax-Span-242"><span class="mrow" id="MathJax-Span-243"><span class="mo" id="MathJax-Span-244" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-245" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span class="mi" id="MathJax-Span-246" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; clip: rect(1.605em 1000em 4.798em -0.409em); top: -1.793em; left: 50%; margin-left: -2.787em;"><span class="mrow" id="MathJax-Span-247"><span class="mn" id="MathJax-Span-248" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-249" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mfrac" id="MathJax-Span-250" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 3.617em; height: 0px;"><span style="position: absolute; clip: rect(3.282em 1000em 4.161em -0.464em); top: -4.409em; left: 50%; margin-left: -0.682em;"><span class="msubsup" id="MathJax-Span-251"><span style="display: inline-block; position: relative; width: 1.365em; height: 0px;"><span style="position: absolute; clip: rect(1.764em 1000em 2.412em -0.464em); top: -2.24em; left: 0em;"><span class="mi" id="MathJax-Span-252" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 2.24em;"></span></span><span style="position: absolute; top: -2.39em; left: 0.383em;"><span class="texatom" id="MathJax-Span-253"><span class="mrow" id="MathJax-Span-254"><span class="mo" id="MathJax-Span-255" style="font-size: 50%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-256" style="font-size: 50%; font-family: MathJax_Main;">4</span><span class="mi" id="MathJax-Span-257" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.186em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; clip: rect(1.309em 1000em 3.627em -0.433em); top: -1.56em; left: 50%; margin-left: -1.749em;"><span class="mrow" id="MathJax-Span-258"><span class="mn" id="MathJax-Span-259" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-260" style="font-size: 70.7%; font-family: MathJax_Main;">+</span><span class="mfrac" id="MathJax-Span-261" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 2.36em; height: 0px;"><span style="position: absolute; clip: rect(3.348em 1000em 4.159em -0.472em); top: -4.334em; left: 50%; margin-left: -0.62em;"><span class="msubsup" id="MathJax-Span-262"><span style="display: inline-block; position: relative; width: 1.24em; height: 0px;"><span style="position: absolute; clip: rect(1.801em 1000em 2.355em -0.472em); top: -2.186em; left: 0em;"><span class="mi" id="MathJax-Span-263" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 2.186em;"></span></span><span style="position: absolute; top: -2.33em; left: 0.273em;"><span class="texatom" id="MathJax-Span-264"><span class="mrow" id="MathJax-Span-265"><span class="mo" id="MathJax-Span-266" style="font-size: 50%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-267" style="font-size: 50%; font-family: MathJax_Main;">6</span><span class="mi" id="MathJax-Span-268" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.186em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; clip: rect(1.251em 1000em 2.761em -0.45em); top: -1.43em; left: 50%; margin-left: -1.12em;"><span class="mrow" id="MathJax-Span-269"><span class="mn" id="MathJax-Span-270" style="font-size: 50%; font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-271" style="font-size: 50%; font-family: MathJax_Main;">+</span><span class="mfrac" id="MathJax-Span-272" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 1.377em; height: 0px;"><span style="position: absolute; clip: rect(3.348em 1000em 4.159em -0.472em); top: -4.282em; left: 50%; margin-left: -0.62em;"><span class="msubsup" id="MathJax-Span-273"><span style="display: inline-block; position: relative; width: 1.24em; height: 0px;"><span style="position: absolute; clip: rect(1.801em 1000em 2.355em -0.472em); top: -2.186em; left: 0em;"><span class="mi" id="MathJax-Span-274" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 2.186em;"></span></span><span style="position: absolute; top: -2.33em; left: 0.273em;"><span class="texatom" id="MathJax-Span-275"><span class="mrow" id="MathJax-Span-276"><span class="mo" id="MathJax-Span-277" style="font-size: 50%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-278" style="font-size: 50%; font-family: MathJax_Main;">8</span><span class="mi" id="MathJax-Span-279" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.186em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; clip: rect(1.689em 1000em 2.391em -0.45em); top: -1.815em; left: 50%; margin-left: -0.628em;"><span class="mrow" id="MathJax-Span-280"><span class="mn" id="MathJax-Span-281" style="font-size: 50%; font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-282" style="font-size: 50%; font-family: MathJax_Main;">+</span><span class="mo" id="MathJax-Span-283" style="font-size: 50%; font-family: MathJax_Main;">…</span></span><span style="display: inline-block; width: 0px; height: 2.186em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.188em; left: 0em;"><span style="border-left-width: 1.377em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.186em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.24em; left: 0em;"><span style="border-left-width: 2.36em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.313em; left: 0em;"><span style="border-left-width: 3.617em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.313em; left: 0em;"><span style="border-left-width: 5.694em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span></span><span style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 5.443em; vertical-align: -3.534em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-5">
\frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} =
1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}
{1+\frac{e^{-8\pi}} {1+\ldots} } } }</script><p></p><h3 id="a-rogers-ramanujan-identity">A Rogers-Ramanujan Identity</h3><p><span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true"><span class="MathJax" id="MathJax-Element-6-Frame"><nobr><span class="math" id="MathJax-Span-284" style="width: 100%; display: inline-block; min-width: 22.417em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(5.156em 1000em 11.306em -0.446em); top: -6.831em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-285"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.315em 1000em 5.139em -0.409em); top: -3.989em; left: 50%; margin-left: -8.256em;"><span class="mn" id="MathJax-Span-286" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-287" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mfrac" id="MathJax-Span-288" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 3.071em; height: 0px;"><span style="position: absolute; clip: rect(2.991em 1000em 4.347em -0.459em); top: -4.666em; left: 50%; margin-left: -0.502em;"><span class="msubsup" id="MathJax-Span-289"><span style="display: inline-block; position: relative; width: 1.004em; height: 0px;"><span style="position: absolute; clip: rect(1.908em 1000em 2.872em -0.459em); top: -2.514em; left: 0em;"><span class="mi" id="MathJax-Span-290" style="font-family: MathJax_Math; font-style: italic;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; top: -2.658em; left: 0.546em;"><span class="mn" id="MathJax-Span-291" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; clip: rect(1.818em 1000em 3.146em -0.398em); top: -2.006em; left: 50%; margin-left: -1.475em;"><span class="mrow" id="MathJax-Span-292"><span class="mo" id="MathJax-Span-293" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-294" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-295" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mi" id="MathJax-Span-296" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span class="mo" id="MathJax-Span-297" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.313em; left: 0em;"><span style="border-left-width: 3.071em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span><span class="mo" id="MathJax-Span-298" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mfrac" id="MathJax-Span-299" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 6.568em; height: 0px;"><span style="position: absolute; clip: rect(2.991em 1000em 4.347em -0.459em); top: -4.666em; left: 50%; margin-left: -0.502em;"><span class="msubsup" id="MathJax-Span-300"><span style="display: inline-block; position: relative; width: 1.004em; height: 0px;"><span style="position: absolute; clip: rect(1.908em 1000em 2.872em -0.459em); top: -2.514em; left: 0em;"><span class="mi" id="MathJax-Span-301" style="font-family: MathJax_Math; font-style: italic;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; top: -2.658em; left: 0.546em;"><span class="mn" id="MathJax-Span-302" style="font-size: 70.7%; font-family: MathJax_Main;">6</span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; clip: rect(1.808em 1000em 3.146em -0.398em); top: -1.996em; left: 50%; margin-left: -3.224em;"><span class="mrow" id="MathJax-Span-303"><span class="mo" id="MathJax-Span-304" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-305" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-306" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mi" id="MathJax-Span-307" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span class="mo" id="MathJax-Span-308" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-309" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-310" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-311" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="msubsup" id="MathJax-Span-312" style="padding-left: 0.222em;"><span style="display: inline-block; position: relative; width: 1.004em; height: 0px;"><span style="position: absolute; clip: rect(1.908em 1000em 2.872em -0.459em); top: -2.514em; left: 0em;"><span class="mi" id="MathJax-Span-313" style="font-family: MathJax_Math; font-style: italic;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; top: -2.584em; left: 0.546em;"><span class="mn" id="MathJax-Span-314" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span></span></span><span class="mo" id="MathJax-Span-315" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.313em; left: 0em;"><span style="border-left-width: 6.568em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span><span class="mo" id="MathJax-Span-316" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mo" id="MathJax-Span-317" style="font-family: MathJax_Main; padding-left: 0.222em;">⋯</span><span class="mo" id="MathJax-Span-318" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; clip: rect(2.263em 1000em 5.485em -0.446em); top: -1.01em; left: 50%; margin-left: -9.185em;"><span class="munderover" id="MathJax-Span-319" style=""><span style="display: inline-block; position: relative; width: 1.311em; height: 0px;"><span style="position: absolute; clip: rect(2.657em 1000em 4.384em -0.436em); top: -3.77em; left: 0em;"><span class="mo" id="MathJax-Span-320" style="font-family: MathJax_Size2; vertical-align: 0em;">∏</span><span style="display: inline-block; width: 0px; height: 3.77em;"></span></span><span style="position: absolute; clip: rect(1.66em 1000em 2.703em -0.5em); top: -1.207em; left: 0.055em;"><span class="texatom" id="MathJax-Span-321"><span class="mrow" id="MathJax-Span-322"><span class="mi" id="MathJax-Span-323" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">j</span><span class="mo" id="MathJax-Span-324" style="font-size: 70.7%; font-family: MathJax_Main;">=</span><span class="mn" id="MathJax-Span-325" style="font-size: 70.7%; font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span><span style="position: absolute; clip: rect(1.719em 1000em 2.467em -0.453em); top: -3.445em; left: 0.301em;"><span class="texatom" id="MathJax-Span-326"><span class="mrow" id="MathJax-Span-327"><span class="mi" id="MathJax-Span-328" style="font-size: 70.7%; font-family: MathJax_Main;">∞</span></span></span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span></span></span><span class="mfrac" id="MathJax-Span-329" style="padding-left: 0.287em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 9.519em; height: 0px;"><span style="position: absolute; clip: rect(1.902em 1000em 2.896em -0.409em); top: -3.409em; left: 50%; margin-left: -0.273em;"><span class="mn" id="MathJax-Span-330" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.808em 1000em 3.146em -0.398em); top: -1.996em; left: 50%; margin-left: -4.699em;"><span class="mrow" id="MathJax-Span-331"><span class="mo" id="MathJax-Span-332" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-333" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-334" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="msubsup" id="MathJax-Span-335" style="padding-left: 0.222em;"><span style="display: inline-block; position: relative; width: 2.206em; height: 0px;"><span style="position: absolute; clip: rect(1.908em 1000em 2.872em -0.459em); top: -2.514em; left: 0em;"><span class="mi" id="MathJax-Span-336" style="font-family: MathJax_Math; font-style: italic;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; top: -2.584em; left: 0.546em;"><span class="texatom" id="MathJax-Span-337"><span class="mrow" id="MathJax-Span-338"><span class="mn" id="MathJax-Span-339" style="font-size: 70.7%; font-family: MathJax_Main;">5</span><span class="mi" id="MathJax-Span-340" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">j</span><span class="mo" id="MathJax-Span-341" style="font-size: 70.7%; font-family: MathJax_Main;">+</span><span class="mn" id="MathJax-Span-342" style="font-size: 70.7%; font-family: MathJax_Main;">2</span></span></span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span></span></span><span class="mo" id="MathJax-Span-343" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-344" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-345" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-346" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="msubsup" id="MathJax-Span-347" style="padding-left: 0.222em;"><span style="display: inline-block; position: relative; width: 2.206em; height: 0px;"><span style="position: absolute; clip: rect(1.908em 1000em 2.872em -0.459em); top: -2.514em; left: 0em;"><span class="mi" id="MathJax-Span-348" style="font-family: MathJax_Math; font-style: italic;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; top: -2.584em; left: 0.546em;"><span class="texatom" id="MathJax-Span-349"><span class="mrow" id="MathJax-Span-350"><span class="mn" id="MathJax-Span-351" style="font-size: 70.7%; font-family: MathJax_Main;">5</span><span class="mi" id="MathJax-Span-352" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">j</span><span class="mo" id="MathJax-Span-353" style="font-size: 70.7%; font-family: MathJax_Main;">+</span><span class="mn" id="MathJax-Span-354" style="font-size: 70.7%; font-family: MathJax_Main;">3</span></span></span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span></span></span><span class="mo" id="MathJax-Span-355" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.313em; left: 0em;"><span style="border-left-width: 9.519em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span><span class="mo" id="MathJax-Span-356" style="font-family: MathJax_Main; padding-left: 0.167em;">,</span><span class="mspace" id="MathJax-Span-357" style="height: 0em; vertical-align: 0em; width: 1em; display: inline-block; overflow: hidden;"></span><span class="mspace" id="MathJax-Span-358" style="height: 0em; vertical-align: 0em; width: 1em; display: inline-block; overflow: hidden;"></span><span class="mtext" id="MathJax-Span-359" style="font-family: MathJax_Main; padding-left: 0.167em;">for&nbsp;</span><span class="texatom" id="MathJax-Span-360"><span class="mrow" id="MathJax-Span-361"><span class="texatom" id="MathJax-Span-362"><span class="mrow" id="MathJax-Span-363"><span class="mo" id="MathJax-Span-364" style="font-family: MathJax_Main;">|</span></span></span><span class="mi" id="MathJax-Span-365" style="font-family: MathJax_Math; font-style: italic;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span class="texatom" id="MathJax-Span-366"><span class="mrow" id="MathJax-Span-367"><span class="mo" id="MathJax-Span-368" style="font-family: MathJax_Main;">|</span></span></span><span class="mtext" id="MathJax-Span-369" style="font-family: MathJax_Main; color: red;">\&lt;</span><span class="mn" id="MathJax-Span-370" style="font-family: MathJax_Main;">1</span></span></span><span class="mo" id="MathJax-Span-371" style="font-family: MathJax_Main;">.</span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 6.831em;"></span></span></span><span style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 7.236em; vertical-align: -5.327em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-6">
1 +  \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =
\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},
\quad\quad \text{for $|q|\<1$}.</script><p></p><h3 id="maxwell-s-equations">Maxwell’s Equations</h3><p><span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-10-Frame"><nobr><span class="math" id="MathJax-Span-630" style="width: 13.55em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.093em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(-2.666em 1000em 6.428em -0.279em); top: -2.131em; left: 0em;"><span class="mrow" id="MathJax-Span-631"><span class="mtable" id="MathJax-Span-632" style="padding-right: 0.167em; padding-left: 0.167em;"><span style="display: inline-block; position: relative; width: 10.715em; height: 0px;"><span style="position: absolute; clip: rect(4.82em 1000em 13.748em -0.446em); top: -9.617em; left: 0em;"><span style="display: inline-block; position: relative; width: 7.377em; height: 0px;"><span style="position: absolute; clip: rect(0.877em 1000em 3.61em -0.446em); top: -5.674em; right: 0em;"><span class="mtd" id="MathJax-Span-633"><span class="mrow" id="MathJax-Span-634"><span class="mi" id="MathJax-Span-635" style="font-family: MathJax_Main;">∇</span><span class="mo" id="MathJax-Span-636" style="font-family: MathJax_Main; padding-left: 0.222em;">×</span><span class="texatom" id="MathJax-Span-637" style="padding-left: 0.222em;"><span class="mrow" id="MathJax-Span-638"><span class="munderover" id="MathJax-Span-639"><span style="display: inline-block; position: relative; width: 0.82em; height: 0px;"><span style="position: absolute; clip: rect(1.937em 1000em 2.951em -0.453em); top: -2.787em; left: 0em;"><span class="texatom" id="MathJax-Span-640"><span class="mrow" id="MathJax-Span-641"><span class="mi" id="MathJax-Span-642" style="font-family: MathJax_Main; font-weight: bold;">B</span></span></span><span style="display: inline-block; width: 0px; height: 2.787em;"></span></span><span style="position: absolute; top: -4.296em; left: 0.189em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-643" style="font-family: MathJax_Main;">⃗&nbsp;<span style="height: 0em; vertical-align: 0em; margin-left: -0.273em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-644" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mspace" id="MathJax-Span-645" style="height: 0em; vertical-align: 0em; width: 0.167em; display: inline-block; overflow: hidden;"></span><span class="mfrac" id="MathJax-Span-646" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 0.666em; height: 0px;"><span style="position: absolute; clip: rect(1.902em 1000em 2.896em -0.409em); top: -3.409em; left: 50%; margin-left: -0.273em;"><span class="mn" id="MathJax-Span-647" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.908em 1000em 2.689em -0.458em); top: -1.828em; left: 50%; margin-left: -0.219em;"><span class="mi" id="MathJax-Span-648" style="font-family: MathJax_Math; font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.313em; left: 0em;"><span style="border-left-width: 0.666em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span><span class="mspace" id="MathJax-Span-649" style="height: 0em; vertical-align: 0em; width: 0.167em; display: inline-block; overflow: hidden;"></span><span class="mfrac" id="MathJax-Span-650" style="padding-left: 0.287em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 1.486em; height: 0px;"><span style="position: absolute; clip: rect(1.554em 1000em 2.918em -0.45em); top: -3.409em; left: 50%; margin-left: -0.683em;"><span class="mrow" id="MathJax-Span-651"><span class="mi" id="MathJax-Span-652" style="font-family: MathJax_Main;">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span class="texatom" id="MathJax-Span-653"><span class="mrow" id="MathJax-Span-654"><span class="munderover" id="MathJax-Span-655"><span style="display: inline-block; position: relative; width: 0.765em; height: 0px;"><span style="position: absolute; clip: rect(1.943em 1000em 2.951em -0.453em); top: -2.787em; left: 0em;"><span class="texatom" id="MathJax-Span-656"><span class="mrow" id="MathJax-Span-657"><span class="mi" id="MathJax-Span-658" style="font-family: MathJax_Main; font-weight: bold;">E</span></span></span><span style="display: inline-block; width: 0px; height: 2.787em;"></span></span><span style="position: absolute; top: -4.29em; left: 0.162em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-659" style="font-family: MathJax_Main;">⃗&nbsp;<span style="height: 0em; vertical-align: 0em; margin-left: -0.273em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.853em 1000em 2.918em -0.45em); top: -2.041em; left: 50%; margin-left: -0.464em;"><span class="mrow" id="MathJax-Span-660"><span class="mi" id="MathJax-Span-661" style="font-family: MathJax_Main;">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span class="mi" id="MathJax-Span-662" style="font-family: MathJax_Math; font-style: italic;">t</span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.313em; left: 0em;"><span style="border-left-width: 1.486em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.554em 1000em 2.929em -0.446em); top: -3.446em; right: 0em;"><span class="mtd" id="MathJax-Span-678"><span class="mrow" id="MathJax-Span-679"><span class="mi" id="MathJax-Span-680" style="font-family: MathJax_Main;">∇</span><span class="mo" id="MathJax-Span-681" style="font-family: MathJax_Main; padding-left: 0.222em;">⋅</span><span class="texatom" id="MathJax-Span-682" style="padding-left: 0.222em;"><span class="mrow" id="MathJax-Span-683"><span class="munderover" id="MathJax-Span-684"><span style="display: inline-block; position: relative; width: 0.765em; height: 0px;"><span style="position: absolute; clip: rect(1.943em 1000em 2.951em -0.453em); top: -2.787em; left: 0em;"><span class="texatom" id="MathJax-Span-685"><span class="mrow" id="MathJax-Span-686"><span class="mi" id="MathJax-Span-687" style="font-family: MathJax_Main; font-weight: bold;">E</span></span></span><span style="display: inline-block; width: 0px; height: 2.787em;"></span></span><span style="position: absolute; top: -4.29em; left: 0.162em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-688" style="font-family: MathJax_Main;">⃗&nbsp;<span style="height: 0em; vertical-align: 0em; margin-left: -0.273em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(0.871em 1000em 3.61em -0.446em); top: -1.033em; right: 0em;"><span class="mtd" id="MathJax-Span-695"><span class="mrow" id="MathJax-Span-696"><span class="mi" id="MathJax-Span-697" style="font-family: MathJax_Main;">∇</span><span class="mo" id="MathJax-Span-698" style="font-family: MathJax_Main; padding-left: 0.222em;">×</span><span class="texatom" id="MathJax-Span-699" style="padding-left: 0.222em;"><span class="mrow" id="MathJax-Span-700"><span class="munderover" id="MathJax-Span-701"><span style="display: inline-block; position: relative; width: 0.765em; height: 0px;"><span style="position: absolute; clip: rect(1.943em 1000em 2.951em -0.453em); top: -2.787em; left: 0em;"><span class="texatom" id="MathJax-Span-702"><span class="mrow" id="MathJax-Span-703"><span class="mi" id="MathJax-Span-704" style="font-family: MathJax_Main; font-weight: bold;">E</span></span></span><span style="display: inline-block; width: 0px; height: 2.787em;"></span></span><span style="position: absolute; top: -4.29em; left: 0.162em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-705" style="font-family: MathJax_Main;">⃗&nbsp;<span style="height: 0em; vertical-align: 0em; margin-left: -0.273em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span></span></span></span><span class="mspace" id="MathJax-Span-706" style="height: 0em; vertical-align: 0em; width: 0.167em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-707" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mspace" id="MathJax-Span-708" style="height: 0em; vertical-align: 0em; width: 0.167em; display: inline-block; overflow: hidden;"></span><span class="mfrac" id="MathJax-Span-709" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 0.666em; height: 0px;"><span style="position: absolute; clip: rect(1.902em 1000em 2.896em -0.409em); top: -3.409em; left: 50%; margin-left: -0.273em;"><span class="mn" id="MathJax-Span-710" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.908em 1000em 2.689em -0.458em); top: -1.828em; left: 50%; margin-left: -0.219em;"><span class="mi" id="MathJax-Span-711" style="font-family: MathJax_Math; font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.313em; left: 0em;"><span style="border-left-width: 0.666em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span><span class="mspace" id="MathJax-Span-712" style="height: 0em; vertical-align: 0em; width: 0.167em; display: inline-block; overflow: hidden;"></span><span class="mfrac" id="MathJax-Span-713" style="padding-left: 0.287em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 1.541em; height: 0px;"><span style="position: absolute; clip: rect(1.548em 1000em 2.918em -0.45em); top: -3.409em; left: 50%; margin-left: -0.71em;"><span class="mrow" id="MathJax-Span-714"><span class="mi" id="MathJax-Span-715" style="font-family: MathJax_Main;">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span class="texatom" id="MathJax-Span-716"><span class="mrow" id="MathJax-Span-717"><span class="munderover" id="MathJax-Span-718"><span style="display: inline-block; position: relative; width: 0.82em; height: 0px;"><span style="position: absolute; clip: rect(1.937em 1000em 2.951em -0.453em); top: -2.787em; left: 0em;"><span class="texatom" id="MathJax-Span-719"><span class="mrow" id="MathJax-Span-720"><span class="mi" id="MathJax-Span-721" style="font-family: MathJax_Main; font-weight: bold;">B</span></span></span><span style="display: inline-block; width: 0px; height: 2.787em;"></span></span><span style="position: absolute; top: -4.296em; left: 0.189em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-722" style="font-family: MathJax_Main;">⃗&nbsp;<span style="height: 0em; vertical-align: 0em; margin-left: -0.273em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.853em 1000em 2.918em -0.45em); top: -2.041em; left: 50%; margin-left: -0.464em;"><span class="mrow" id="MathJax-Span-723"><span class="mi" id="MathJax-Span-724" style="font-family: MathJax_Main;">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span class="mi" id="MathJax-Span-725" style="font-family: MathJax_Math; font-style: italic;">t</span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.313em; left: 0em;"><span style="border-left-width: 1.541em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.548em 1000em 2.929em -0.446em); top: 1.201em; right: 0em;"><span class="mtd" id="MathJax-Span-736"><span class="mrow" id="MathJax-Span-737"><span class="mi" id="MathJax-Span-738" style="font-family: MathJax_Main;">∇</span><span class="mo" id="MathJax-Span-739" style="font-family: MathJax_Main; padding-left: 0.222em;">⋅</span><span class="texatom" id="MathJax-Span-740" style="padding-left: 0.222em;"><span class="mrow" id="MathJax-Span-741"><span class="munderover" id="MathJax-Span-742"><span style="display: inline-block; position: relative; width: 0.82em; height: 0px;"><span style="position: absolute; clip: rect(1.937em 1000em 2.951em -0.453em); top: -2.787em; left: 0em;"><span class="texatom" id="MathJax-Span-743"><span class="mrow" id="MathJax-Span-744"><span class="mi" id="MathJax-Span-745" style="font-family: MathJax_Main; font-weight: bold;">B</span></span></span><span style="display: inline-block; width: 0px; height: 2.787em;"></span></span><span style="position: absolute; top: -4.296em; left: 0.189em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-746" style="font-family: MathJax_Main;">⃗&nbsp;<span style="height: 0em; vertical-align: 0em; margin-left: -0.273em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span></span><span style="display: inline-block; width: 0px; height: 9.617em;"></span></span><span style="position: absolute; clip: rect(4.775em 1000em 13.354em -0.436em); top: -9.235em; left: 7.655em;"><span style="display: inline-block; position: relative; width: 3.06em; height: 0px;"><span style="position: absolute; clip: rect(1.215em 1000em 3.593em -0.436em); top: -5.674em; left: 0em;"><span class="mtd" id="MathJax-Span-663"><span class="mrow" id="MathJax-Span-664"><span class="mo" id="MathJax-Span-665" style="font-family: MathJax_Main;">=</span><span class="mfrac" id="MathJax-Span-666" style="padding-left: 0.398em; padding-right: 0.12em;"><span style="display: inline-block; position: relative; width: 1.213em; height: 0px;"><span style="position: absolute; clip: rect(1.891em 1000em 2.907em -0.464em); top: -3.409em; left: 50%; margin-left: -0.546em;"><span class="mrow" id="MathJax-Span-667"><span class="mn" id="MathJax-Span-668" style="font-family: MathJax_Main;">4</span><span class="mi" id="MathJax-Span-669" style="font-family: MathJax_Math; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.908em 1000em 2.689em -0.458em); top: -1.828em; left: 50%; margin-left: -0.219em;"><span class="mi" id="MathJax-Span-670" style="font-family: MathJax_Math; font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 2.514em;"></span></span><span style="position: absolute; clip: rect(0.861em 1000em 1.257em -0.492em); top: -1.313em; left: 0em;"><span style="border-left-width: 1.213em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.093em;"></span></span></span></span><span class="texatom" id="MathJax-Span-671" style="padding-left: 0.167em;"><span class="mrow" id="MathJax-Span-672"><span class="munderover" id="MathJax-Span-673"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px;"><span style="position: absolute; clip: rect(1.928em 1000em 3.151em -0.563em); top: -2.787em; left: 0em;"><span class="texatom" id="MathJax-Span-674"><span class="mrow" id="MathJax-Span-675"><span class="mi" id="MathJax-Span-676" style="font-family: MathJax_Main; font-weight: bold;">j</span></span></span><span style="display: inline-block; width: 0px; height: 2.787em;"></span></span><span style="position: absolute; top: -4.305em; left: -0.03em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-677" style="font-family: MathJax_Main;">⃗&nbsp;<span style="height: 0em; vertical-align: 0em; margin-left: -0.273em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.891em 1000em 3.112em -0.436em); top: -3.446em; left: 0em;"><span class="mtd" id="MathJax-Span-689"><span class="mrow" id="MathJax-Span-690"><span class="mo" id="MathJax-Span-691" style="font-family: MathJax_Main;">=</span><span class="mn" id="MathJax-Span-692" style="font-family: MathJax_Main; padding-left: 0.278em;">4</span><span class="mi" id="MathJax-Span-693" style="font-family: MathJax_Math; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-694" style="font-family: MathJax_Math; font-style: italic;">ρ</span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.58em 1000em 2.906em -0.436em); top: -1.033em; left: 0em;"><span class="mtd" id="MathJax-Span-726"><span class="mrow" id="MathJax-Span-727"><span class="mo" id="MathJax-Span-728" style="font-family: MathJax_Main;">=</span><span class="texatom" id="MathJax-Span-729" style="padding-left: 0.278em;"><span class="mrow" id="MathJax-Span-730"><span class="munderover" id="MathJax-Span-731"><span style="display: inline-block; position: relative; width: 0.601em; height: 0px;"><span style="position: absolute; clip: rect(1.969em 1000em 2.961em -0.447em); top: -2.787em; left: 0em;"><span class="texatom" id="MathJax-Span-732"><span class="mrow" id="MathJax-Span-733"><span class="mn" id="MathJax-Span-734" style="font-family: MathJax_Main; font-weight: bold;">0</span></span></span><span style="display: inline-block; width: 0px; height: 2.787em;"></span></span><span style="position: absolute; top: -4.264em; left: 0.08em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-735" style="font-family: MathJax_Main;">⃗&nbsp;<span style="height: 0em; vertical-align: 0em; margin-left: -0.273em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(1.902em 1000em 2.918em -0.436em); top: 1.201em; left: 0em;"><span class="mtd" id="MathJax-Span-747"><span class="mrow" id="MathJax-Span-748"><span class="mo" id="MathJax-Span-749" style="font-family: MathJax_Main;">=</span><span class="mn" id="MathJax-Span-750" style="font-family: MathJax_Main; padding-left: 0.278em;">0</span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span></span><span style="display: inline-block; width: 0px; height: 9.235em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.131em;"></span></span></span><span style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 10.829em; vertical-align: -5.109em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-10">\begin{aligned}
\nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\   \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\
\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\
\nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}</script><p></p><p>Finally, while display equations look good for a page of samples, the ability to mix math and text in a paragraph is also important. This expression </p><p><span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-20-Frame"><nobr><span class="math" id="MathJax-Span-843" style="width: 10.283em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.415em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.62em 1000em 3.146em -0.42em); top: -2.732em; left: 0em;"><span class="mrow" id="MathJax-Span-844"><span class="msqrt" id="MathJax-Span-845"><span style="display: inline-block; position: relative; width: 3.632em; height: 0px;"><span style="position: absolute; clip: rect(1.902em 1000em 2.918em -0.45em); top: -2.732em; left: 0.82em;"><span class="mrow" id="MathJax-Span-846"><span class="mn" id="MathJax-Span-847" style="font-family: MathJax_Main;">3</span><span class="mi" id="MathJax-Span-848" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-849" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-850" style="font-family: MathJax_Main; padding-left: 0.222em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; clip: rect(3.555em 1000em 4.023em -0.492em); top: -4.526em; left: 0.82em;"><span style="display: inline-block; position: relative; width: 2.812em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.989em; left: -0.084em;">−<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -3.989em; left: 2.118em;">−<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.989em; left: 0.448em;">−<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.989em; left: 1.004em;">−<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.989em; left: 1.561em;">−<span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span><span style="position: absolute; clip: rect(3.025em 1000em 4.353em -0.42em); top: -3.987em; left: 0em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; width: 0px; height: 3.989em;"></span></span></span></span><span class="mo" id="MathJax-Span-851" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mo" id="MathJax-Span-852" style="font-family: MathJax_Main; padding-left: 0.222em;">(</span><span class="mn" id="MathJax-Span-853" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-854" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-855" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">x</span><span class="msubsup" id="MathJax-Span-856"><span style="display: inline-block; position: relative; width: 0.895em; height: 0px;"><span style="position: absolute; clip: rect(1.818em 1000em 3.146em -0.437em); top: -2.732em; left: 0em;"><span class="mo" id="MathJax-Span-857" style="font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span><span style="position: absolute; top: -2.772em; left: 0.437em;"><span class="mn" id="MathJax-Span-858" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 2.295em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.732em;"></span></span></span><span style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.595em; vertical-align: -0.372em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-20">\sqrt{3x-1}+(1+x)^2</script><p></p><p></p>
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