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dwinter/gradient functions

Created January 28, 2015 21:12
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\sum _{k=0}^{\infty } \frac{u^k e^{-\frac{(w-a k)^2}{2 k V}-u}}{\sqrt{2 \pi } k! \sqrt{k V}} p A ÐiR àØS À+% °Fd �8% ��d 1
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gradient functions
<p><span class="MathJax_Preview"></span><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-1456" style="width: 13.316em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.964em; height: 0px; font-size: 111%;"><span style="position: absolute; clip: rect(-0.148em 1000.003em 3.606em -0.398em); top: -2.25em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1457"><span class="munderover" id="MathJax-Span-1458"><span style="display: inline-block; position: relative; width: 1.454em; height: 0px;"><span style="position: absolute; clip: rect(2.655em 1000.003em 4.357em -0.398em); top: -3.751em; left: 0.003em;"><span class="mo" id="MathJax-Span-1459" style="font-family: MathJax_Size2; vertical-align: 0.003em;">∑</span><span style="display: inline-block; width: 0px; height: 3.756em;"></span></span><span style="position: absolute; clip: rect(1.754em 1000.003em 2.655em -0.398em); top: -1.299em; left: 0.053em;"><span class="texatom" id="MathJax-Span-1460"><span class="mrow" id="MathJax-Span-1461"><span class="mi" id="MathJax-Span-1462" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-1463" style="font-size: 70.7%; font-family: MathJax_Main;">=</span><span class="mn" id="MathJax-Span-1464" style="font-size: 70.7%; font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span><span style="position: absolute; clip: rect(1.854em 1000.003em 2.555em -0.398em); top: -3.551em; left: 0.353em;"><span class="texatom" id="MathJax-Span-1465"><span class="mrow" id="MathJax-Span-1466"><span class="mi" id="MathJax-Span-1467" style="font-size: 70.7%; font-family: MathJax_Main;">∞</span></span></span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span></span></span><span class="mfrac" id="MathJax-Span-1468" style="padding-left: 0.303em; padding-right: 0.103em;"><span style="display: inline-block; position: relative; width: 10.063em; height: 0px;"><span style="position: absolute; clip: rect(1.004em 1000.003em 3.106em -0.398em); top: -3.401em; left: 50%; margin-left: -4.952em;"><span class="mrow" id="MathJax-Span-1469"><span class="mi" id="MathJax-Span-1470" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="msubsup" id="MathJax-Span-1471"><span style="display: inline-block; position: relative; width: 1.054em; height: 0px;"><span style="position: absolute; clip: rect(1.904em 1000.003em 2.655em -0.448em); top: -2.5em; left: 0.003em;"><span class="mi" id="MathJax-Span-1472" style="font-family: MathJax_Math; font-style: italic;">u</span><span style="display: inline-block; width: 0px; height: 2.505em;"></span></span><span style="position: absolute; top: -2.7em; left: 0.603em;"><span class="mi" id="MathJax-Span-1473" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span style="display: inline-block; width: 0px; height: 2.355em;"></span></span></span></span><span class="mo" id="MathJax-Span-1474" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1475" style="font-family: MathJax_Math; font-style: italic;">w</span><span class="mo" id="MathJax-Span-1476" style="font-family: MathJax_Main; padding-left: 0.203em;">−</span><span class="mi" id="MathJax-Span-1477" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.203em;">a</span><span class="mi" id="MathJax-Span-1478" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-1479" style="font-family: MathJax_Main;">)</span><span class="msubsup" id="MathJax-Span-1480"><span style="display: inline-block; position: relative; width: 4.657em; height: 0px;"><span style="position: absolute; clip: rect(1.904em 1000.003em 2.655em -0.398em); top: -2.5em; left: 0.003em;"><span class="mi" id="MathJax-Span-1481" style="font-family: MathJax_Math; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 2.505em;"></span></span><span style="position: absolute; top: -3.001em; left: 0.503em;"><span class="texatom" id="MathJax-Span-1482"><span class="mrow" id="MathJax-Span-1483"><span class="mo" id="MathJax-Span-1484" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mfrac" id="MathJax-Span-1485" style="padding-left: 0.103em; padding-right: 0.103em;"><span style="display: inline-block; position: relative; width: 2.355em; height: 0px;"><span style="position: absolute; clip: rect(1.654em 1000.003em 2.555em -0.398em); top: -2.75em; left: 50%; margin-left: -0.998em;"><span class="mrow" id="MathJax-Span-1486"><span class="mo" id="MathJax-Span-1487" style="font-size: 50%; font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1488" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">w</span><span class="mo" id="MathJax-Span-1489" style="font-size: 50%; font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-1490" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">a</span><span class="mi" id="MathJax-Span-1491" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">k</span><span class="msubsup" id="MathJax-Span-1492"><span style="display: inline-block; position: relative; width: 0.503em; height: 0px;"><span style="position: absolute; clip: rect(1.804em 1000.003em 2.555em -0.398em); top: -2.3em; left: 0.003em;"><span class="mo" id="MathJax-Span-1493" style="font-size: 50%; font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span><span style="position: absolute; top: -2.5em; left: 0.203em;"><span class="mn" id="MathJax-Span-1494" style="font-size: 50%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span><span style="position: absolute; clip: rect(1.804em 1000.003em 2.555em -0.448em); top: -1.949em; left: 50%; margin-left: -1.099em;"><span class="mrow" id="MathJax-Span-1495"><span class="mn" id="MathJax-Span-1496" style="font-size: 50%; font-family: MathJax_Main;">2</span><span class="mo" id="MathJax-Span-1497" style="font-size: 50%; font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1498" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">k</span><span class="mi" id="MathJax-Span-1499" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.103em;"></span></span><span class="mo" id="MathJax-Span-1500" style="font-size: 50%; font-family: MathJax_Main;">+</span><span class="mtext" id="MathJax-Span-1501" style="font-size: 50%; font-family: MathJax_Main;">Ve</span><span class="mo" id="MathJax-Span-1502" style="font-size: 50%; font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span><span style="position: absolute; clip: rect(0.853em 1000.003em 1.204em -0.448em); top: -1.199em; left: 0.003em;"><span style="border-left-width: 2.355em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.054em;"></span></span></span></span><span class="mo" id="MathJax-Span-1503" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-1504" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">u</span></span></span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.705em;"></span></span><span style="position: absolute; clip: rect(1.554em 1000.003em 3.106em -0.398em); top: -1.749em; left: 50%; margin-left: -4.252em;"><span class="mrow" id="MathJax-Span-1505"><span class="msqrt" id="MathJax-Span-1506"><span style="display: inline-block; position: relative; width: 2.055em; height: 0px;"><span style="position: absolute; clip: rect(1.904em 1000.003em 2.855em -0.398em); top: -2.7em; left: 0.853em;"><span class="mrow" id="MathJax-Span-1507"><span class="mn" id="MathJax-Span-1508" style="font-family: MathJax_Main;">2</span><span class="mi" id="MathJax-Span-1509" 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: 2.705em;"></span></span><span style="position: absolute; clip: rect(3.606em 1000.003em 4.007em -0.448em); top: -4.602em; left: 0.853em;"><span style="display: inline-block; position: relative; width: 1.204em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -4.002em; left: -0.098em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -4.002em; left: 0.503em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; clip: rect(3.056em 1000.003em 4.357em -0.398em); top: -4.052em; left: 0.003em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span></span><span class="mi" id="MathJax-Span-1510" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-1511" style="font-family: MathJax_Main;">!</span><span class="mo" id="MathJax-Span-1512" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1513" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="mi" id="MathJax-Span-1514" style="font-family: MathJax_Math; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.203em;"></span></span><span class="mo" id="MathJax-Span-1515" style="font-family: MathJax_Main; padding-left: 0.203em;">+</span><span class="mtext" id="MathJax-Span-1516" style="font-family: MathJax_Main; padding-left: 0.203em;">Ve</span><span class="msubsup" id="MathJax-Span-1517"><span style="display: inline-block; position: relative; width: 1.554em; height: 0px;"><span style="position: absolute; clip: rect(1.804em 1000.003em 3.106em -0.398em); top: -2.7em; left: 0.003em;"><span class="mo" id="MathJax-Span-1518" style="font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 2.705em;"></span></span><span style="position: absolute; top: -2.9em; left: 0.403em;"><span class="texatom" id="MathJax-Span-1519"><span class="mrow" id="MathJax-Span-1520"><span class="mn" id="MathJax-Span-1521" style="font-size: 70.7%; font-family: MathJax_Main;">3</span><span class="texatom" id="MathJax-Span-1522"><span class="mrow" id="MathJax-Span-1523"><span class="mo" id="MathJax-Span-1524" style="font-size: 70.7%; font-family: MathJax_Main;">/</span></span></span><span class="mn" id="MathJax-Span-1525" style="font-size: 70.7%; font-family: MathJax_Main;">2</span></span></span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.705em;"></span></span><span style="position: absolute; clip: rect(0.853em 1000.003em 1.204em -0.448em); top: -1.249em; left: 0.003em;"><span style="border-left-width: 10.063em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.054em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.255em;"></span></span></span><span style="border-left-width: 0.003em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 4.003em; vertical-align: -1.386em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-18">
\sum_{k=0}^{\infty } \frac{k u^k (w-a k) e^{-\frac{(w-a k)^2}{2 (k
V+\text{Ve})}-u}}{\sqrt{2 \pi } k! (k V+\text{Ve})^{3/2}}
</script> <br>
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-19-Frame"><nobr><span class="math" id="MathJax-Span-1526" style="width: 27.18em; display: inline-block;"><span style="display: inline-block; position: relative; width: 24.477em; height: 0px; font-size: 111%;"><span style="position: absolute; clip: rect(2.355em 1000.003em 6.659em -0.398em); top: -4.752em; left: 0.003em;"><span class="mrow" id="MathJax-Span-1527"><span class="munderover" id="MathJax-Span-1528"><span style="display: inline-block; position: relative; width: 1.454em; height: 0px;"><span style="position: absolute; clip: rect(2.655em 1000.003em 4.357em -0.398em); top: -3.751em; left: 0.003em;"><span class="mo" id="MathJax-Span-1529" style="font-family: MathJax_Size2; vertical-align: 0.003em;">∑</span><span style="display: inline-block; width: 0px; height: 3.756em;"></span></span><span style="position: absolute; clip: rect(1.754em 1000.003em 2.655em -0.398em); top: -1.299em; left: 0.053em;"><span class="texatom" id="MathJax-Span-1530"><span class="mrow" id="MathJax-Span-1531"><span class="mi" id="MathJax-Span-1532" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-1533" style="font-size: 70.7%; font-family: MathJax_Main;">=</span><span class="mn" id="MathJax-Span-1534" style="font-size: 70.7%; font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span><span style="position: absolute; clip: rect(1.854em 1000.003em 2.555em -0.398em); top: -3.551em; left: 0.353em;"><span class="texatom" id="MathJax-Span-1535"><span class="mrow" id="MathJax-Span-1536"><span class="mi" id="MathJax-Span-1537" style="font-size: 70.7%; font-family: MathJax_Main;">∞</span></span></span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span></span></span><span class="mrow" id="MathJax-Span-1538"><span class="mo" id="MathJax-Span-1539" style="vertical-align: 2.255em;"><span style="display: inline-block; position: relative; width: 0.853em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.85em; left: 0.003em;">⎛<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: -0.598em; left: 0.003em;">⎝<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.699em; left: 0.003em;">⎜<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span></span><span class="mfrac" id="MathJax-Span-1540" style="padding-left: 0.103em; padding-right: 0.103em;"><span style="display: inline-block; position: relative; width: 10.563em; height: 0px;"><span style="position: absolute; clip: rect(1.004em 1000.003em 3.106em -0.398em); top: -3.401em; left: 50%; margin-left: -5.203em;"><span class="mrow" id="MathJax-Span-1541"><span class="mi" id="MathJax-Span-1542" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="msubsup" id="MathJax-Span-1543"><span style="display: inline-block; position: relative; width: 1.054em; height: 0px;"><span style="position: absolute; clip: rect(1.904em 1000.003em 2.655em -0.448em); top: -2.5em; left: 0.003em;"><span class="mi" id="MathJax-Span-1544" style="font-family: MathJax_Math; font-style: italic;">u</span><span style="display: inline-block; width: 0px; height: 2.505em;"></span></span><span style="position: absolute; top: -2.7em; left: 0.603em;"><span class="mi" id="MathJax-Span-1545" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span style="display: inline-block; width: 0px; height: 2.355em;"></span></span></span></span><span class="mo" id="MathJax-Span-1546" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1547" style="font-family: MathJax_Math; font-style: italic;">w</span><span class="mo" id="MathJax-Span-1548" style="font-family: MathJax_Main; padding-left: 0.203em;">−</span><span class="mi" id="MathJax-Span-1549" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.203em;">a</span><span class="mi" id="MathJax-Span-1550" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="msubsup" id="MathJax-Span-1551"><span style="display: inline-block; position: relative; width: 0.853em; height: 0px;"><span style="position: absolute; clip: rect(1.804em 1000.003em 3.106em -0.398em); top: -2.7em; left: 0.003em;"><span class="mo" id="MathJax-Span-1552" style="font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 2.705em;"></span></span><span style="position: absolute; top: -2.9em; left: 0.403em;"><span class="mn" id="MathJax-Span-1553" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-1554"><span style="display: inline-block; position: relative; width: 4.657em; height: 0px;"><span style="position: absolute; clip: rect(1.904em 1000.003em 2.655em -0.398em); top: -2.5em; left: 0.003em;"><span class="mi" id="MathJax-Span-1555" style="font-family: MathJax_Math; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 2.505em;"></span></span><span style="position: absolute; top: -3.001em; left: 0.503em;"><span class="texatom" id="MathJax-Span-1556"><span class="mrow" id="MathJax-Span-1557"><span class="mo" id="MathJax-Span-1558" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mfrac" id="MathJax-Span-1559" style="padding-left: 0.103em; padding-right: 0.103em;"><span style="display: inline-block; position: relative; width: 2.355em; height: 0px;"><span style="position: absolute; clip: rect(1.654em 1000.003em 2.555em -0.398em); top: -2.75em; left: 50%; margin-left: -0.998em;"><span class="mrow" id="MathJax-Span-1560"><span class="mo" id="MathJax-Span-1561" style="font-size: 50%; font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1562" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">w</span><span class="mo" id="MathJax-Span-1563" style="font-size: 50%; font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-1564" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">a</span><span class="mi" id="MathJax-Span-1565" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">k</span><span class="msubsup" id="MathJax-Span-1566"><span style="display: inline-block; position: relative; width: 0.503em; height: 0px;"><span style="position: absolute; clip: rect(1.804em 1000.003em 2.555em -0.398em); top: -2.3em; left: 0.003em;"><span class="mo" id="MathJax-Span-1567" style="font-size: 50%; font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span><span style="position: absolute; top: -2.5em; left: 0.203em;"><span class="mn" id="MathJax-Span-1568" style="font-size: 50%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span><span style="position: absolute; clip: rect(1.804em 1000.003em 2.555em -0.448em); top: -1.949em; left: 50%; margin-left: -1.099em;"><span class="mrow" id="MathJax-Span-1569"><span class="mn" id="MathJax-Span-1570" style="font-size: 50%; font-family: MathJax_Main;">2</span><span class="mo" id="MathJax-Span-1571" style="font-size: 50%; font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1572" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">k</span><span class="mi" id="MathJax-Span-1573" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.103em;"></span></span><span class="mo" id="MathJax-Span-1574" style="font-size: 50%; font-family: MathJax_Main;">+</span><span class="mtext" id="MathJax-Span-1575" style="font-size: 50%; font-family: MathJax_Main;">Ve</span><span class="mo" id="MathJax-Span-1576" style="font-size: 50%; font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span><span style="position: absolute; clip: rect(0.853em 1000.003em 1.204em -0.448em); top: -1.199em; left: 0.003em;"><span style="border-left-width: 2.355em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.054em;"></span></span></span></span><span class="mo" id="MathJax-Span-1577" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-1578" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">u</span></span></span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.705em;"></span></span><span style="position: absolute; clip: rect(1.554em 1000.003em 3.106em -0.398em); top: -1.749em; left: 50%; margin-left: -4.502em;"><span class="mrow" id="MathJax-Span-1579"><span class="mn" id="MathJax-Span-1580" style="font-family: MathJax_Main;">2</span><span class="msqrt" id="MathJax-Span-1581"><span style="display: inline-block; position: relative; width: 2.055em; height: 0px;"><span style="position: absolute; clip: rect(1.904em 1000.003em 2.855em -0.398em); top: -2.7em; left: 0.853em;"><span class="mrow" id="MathJax-Span-1582"><span class="mn" id="MathJax-Span-1583" style="font-family: MathJax_Main;">2</span><span class="mi" id="MathJax-Span-1584" 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: 2.705em;"></span></span><span style="position: absolute; clip: rect(3.606em 1000.003em 4.007em -0.448em); top: -4.602em; left: 0.853em;"><span style="display: inline-block; position: relative; width: 1.204em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -4.002em; left: -0.098em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -4.002em; left: 0.503em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; clip: rect(3.056em 1000.003em 4.357em -0.398em); top: -4.052em; left: 0.003em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span></span><span class="mi" id="MathJax-Span-1585" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-1586" style="font-family: MathJax_Main;">!</span><span class="mo" id="MathJax-Span-1587" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1588" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="mi" id="MathJax-Span-1589" style="font-family: MathJax_Math; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.203em;"></span></span><span class="mo" id="MathJax-Span-1590" style="font-family: MathJax_Main; padding-left: 0.203em;">+</span><span class="mtext" id="MathJax-Span-1591" style="font-family: MathJax_Main; padding-left: 0.203em;">Ve</span><span class="msubsup" id="MathJax-Span-1592"><span style="display: inline-block; position: relative; width: 1.554em; height: 0px;"><span style="position: absolute; clip: rect(1.804em 1000.003em 3.106em -0.398em); top: -2.7em; left: 0.003em;"><span class="mo" id="MathJax-Span-1593" style="font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 2.705em;"></span></span><span style="position: absolute; top: -2.9em; left: 0.403em;"><span class="texatom" id="MathJax-Span-1594"><span class="mrow" id="MathJax-Span-1595"><span class="mn" id="MathJax-Span-1596" style="font-size: 70.7%; font-family: MathJax_Main;">5</span><span class="texatom" id="MathJax-Span-1597"><span class="mrow" id="MathJax-Span-1598"><span class="mo" id="MathJax-Span-1599" style="font-size: 70.7%; font-family: MathJax_Main;">/</span></span></span><span class="mn" id="MathJax-Span-1600" style="font-size: 70.7%; font-family: MathJax_Main;">2</span></span></span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.705em;"></span></span><span style="position: absolute; clip: rect(0.853em 1000.003em 1.204em -0.448em); top: -1.249em; left: 0.003em;"><span style="border-left-width: 10.563em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.054em;"></span></span></span></span><span class="mo" id="MathJax-Span-1601" style="font-family: MathJax_Main; padding-left: 0.203em;">−</span><span class="mfrac" id="MathJax-Span-1602" style="padding-left: 0.353em; padding-right: 0.103em;"><span style="display: inline-block; position: relative; width: 9.112em; height: 0px;"><span style="position: absolute; clip: rect(0.803em 1000.003em 2.655em -0.398em); top: -3.201em; left: 50%; margin-left: -3.101em;"><span class="mrow" id="MathJax-Span-1603"><span class="mi" id="MathJax-Span-1604" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="msubsup" id="MathJax-Span-1605"><span style="display: inline-block; position: relative; width: 1.054em; height: 0px;"><span style="position: absolute; clip: rect(1.904em 1000.003em 2.655em -0.448em); top: -2.5em; left: 0.003em;"><span class="mi" id="MathJax-Span-1606" style="font-family: MathJax_Math; font-style: italic;">u</span><span style="display: inline-block; width: 0px; height: 2.505em;"></span></span><span style="position: absolute; top: -2.7em; left: 0.603em;"><span class="mi" id="MathJax-Span-1607" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span style="display: inline-block; width: 0px; height: 2.355em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-1608"><span style="display: inline-block; position: relative; width: 4.657em; height: 0px;"><span style="position: absolute; clip: rect(1.904em 1000.003em 2.655em -0.398em); top: -2.5em; left: 0.003em;"><span class="mi" id="MathJax-Span-1609" style="font-family: MathJax_Math; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 2.505em;"></span></span><span style="position: absolute; top: -3.001em; left: 0.503em;"><span class="texatom" id="MathJax-Span-1610"><span class="mrow" id="MathJax-Span-1611"><span class="mo" id="MathJax-Span-1612" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mfrac" id="MathJax-Span-1613" style="padding-left: 0.103em; padding-right: 0.103em;"><span style="display: inline-block; position: relative; width: 2.355em; height: 0px;"><span style="position: absolute; clip: rect(1.654em 1000.003em 2.555em -0.398em); top: -2.75em; left: 50%; margin-left: -0.998em;"><span class="mrow" id="MathJax-Span-1614"><span class="mo" id="MathJax-Span-1615" style="font-size: 50%; font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1616" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">w</span><span class="mo" id="MathJax-Span-1617" style="font-size: 50%; font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-1618" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">a</span><span class="mi" id="MathJax-Span-1619" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">k</span><span class="msubsup" id="MathJax-Span-1620"><span style="display: inline-block; position: relative; width: 0.503em; height: 0px;"><span style="position: absolute; clip: rect(1.804em 1000.003em 2.555em -0.398em); top: -2.3em; left: 0.003em;"><span class="mo" id="MathJax-Span-1621" style="font-size: 50%; font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span><span style="position: absolute; top: -2.5em; left: 0.203em;"><span class="mn" id="MathJax-Span-1622" style="font-size: 50%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span><span style="position: absolute; clip: rect(1.804em 1000.003em 2.555em -0.448em); top: -1.949em; left: 50%; margin-left: -1.099em;"><span class="mrow" id="MathJax-Span-1623"><span class="mn" id="MathJax-Span-1624" style="font-size: 50%; font-family: MathJax_Main;">2</span><span class="mo" id="MathJax-Span-1625" style="font-size: 50%; font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1626" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">k</span><span class="mi" id="MathJax-Span-1627" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.103em;"></span></span><span class="mo" id="MathJax-Span-1628" style="font-size: 50%; font-family: MathJax_Main;">+</span><span class="mtext" id="MathJax-Span-1629" style="font-size: 50%; font-family: MathJax_Main;">Ve</span><span class="mo" id="MathJax-Span-1630" style="font-size: 50%; font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span><span style="position: absolute; clip: rect(0.853em 1000.003em 1.204em -0.448em); top: -1.199em; left: 0.003em;"><span style="border-left-width: 2.355em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.054em;"></span></span></span></span><span class="mo" id="MathJax-Span-1631" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-1632" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">u</span></span></span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.505em;"></span></span><span style="position: absolute; clip: rect(1.554em 1000.003em 3.106em -0.398em); top: -1.749em; left: 50%; margin-left: -4.502em;"><span class="mrow" id="MathJax-Span-1633"><span class="mn" id="MathJax-Span-1634" style="font-family: MathJax_Main;">2</span><span class="msqrt" id="MathJax-Span-1635"><span style="display: inline-block; position: relative; width: 2.055em; height: 0px;"><span style="position: absolute; clip: rect(1.904em 1000.003em 2.855em -0.398em); top: -2.7em; left: 0.853em;"><span class="mrow" id="MathJax-Span-1636"><span class="mn" id="MathJax-Span-1637" style="font-family: MathJax_Main;">2</span><span class="mi" id="MathJax-Span-1638" 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: 2.705em;"></span></span><span style="position: absolute; clip: rect(3.606em 1000.003em 4.007em -0.448em); top: -4.602em; left: 0.853em;"><span style="display: inline-block; position: relative; width: 1.204em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -4.002em; left: -0.098em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -4.002em; left: 0.503em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; clip: rect(3.056em 1000.003em 4.357em -0.398em); top: -4.052em; left: 0.003em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span></span><span class="mi" id="MathJax-Span-1639" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-1640" style="font-family: MathJax_Main;">!</span><span class="mo" id="MathJax-Span-1641" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1642" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="mi" id="MathJax-Span-1643" style="font-family: MathJax_Math; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.203em;"></span></span><span class="mo" id="MathJax-Span-1644" style="font-family: MathJax_Main; padding-left: 0.203em;">+</span><span class="mtext" id="MathJax-Span-1645" style="font-family: MathJax_Main; padding-left: 0.203em;">Ve</span><span class="msubsup" id="MathJax-Span-1646"><span style="display: inline-block; position: relative; width: 1.554em; height: 0px;"><span style="position: absolute; clip: rect(1.804em 1000.003em 3.106em -0.398em); top: -2.7em; left: 0.003em;"><span class="mo" id="MathJax-Span-1647" style="font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 2.705em;"></span></span><span style="position: absolute; top: -2.9em; left: 0.403em;"><span class="texatom" id="MathJax-Span-1648"><span class="mrow" id="MathJax-Span-1649"><span class="mn" id="MathJax-Span-1650" style="font-size: 70.7%; font-family: MathJax_Main;">3</span><span class="texatom" id="MathJax-Span-1651"><span class="mrow" id="MathJax-Span-1652"><span class="mo" id="MathJax-Span-1653" style="font-size: 70.7%; font-family: MathJax_Main;">/</span></span></span><span class="mn" id="MathJax-Span-1654" style="font-size: 70.7%; font-family: MathJax_Main;">2</span></span></span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.705em;"></span></span><span style="position: absolute; clip: rect(0.853em 1000.003em 1.204em -0.448em); top: -1.249em; left: 0.003em;"><span style="border-left-width: 9.112em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.054em;"></span></span></span></span><span class="mo" id="MathJax-Span-1655" style="vertical-align: 2.255em;"><span style="display: inline-block; position: relative; width: 0.853em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.85em; left: 0.003em;">⎞<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: -0.598em; left: 0.003em;">⎠<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.699em; left: 0.003em;">⎟<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span></span></span><span class="texatom" id="MathJax-Span-1656"><span class="mrow" id="MathJax-Span-1657"><span class="mo" id="MathJax-Span-1658"><span style="font-family: STIXGeneral, 'Arial Unicode MS', serif; font-size: 90%; font-style: normal; font-weight: normal;"></span></span></span></span><span class="texatom" id="MathJax-Span-1659"><span class="mrow" id="MathJax-Span-1660"><span class="mo" id="MathJax-Span-1661"><span style="font-family: STIXGeneral, 'Arial Unicode MS', serif; font-size: 90%; font-style: normal; font-weight: normal;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.757em;"></span></span></span><span style="border-left-width: 0.003em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 4.614em; vertical-align: -1.997em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-19">
\sum _{k=0}^{\infty } \left(\frac{k u^k (w-a k)^2 e^{-\frac{(w-a k)^2}{2 (k V+\text{Ve})}-u}}{2 \sqrt{2 \pi } k! (k V+\text{Ve})^{5/2}}-\frac{k u^k e^{-\frac{(w-a k)^2}{2 (k V+\text{Ve})}-u}}{2 \sqrt{2 \pi } k! (k V+\text{Ve})^{3/2}}\right) </script></p>
<p><span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-22-Frame"><nobr><span class="math" id="MathJax-Span-2030" style="width: 22.125em; display: inline-block;"><span style="display: inline-block; position: relative; width: 19.922em; height: 0px; font-size: 111%;"><span style="position: absolute; clip: rect(2.355em 1000.003em 6.659em -0.398em); top: -4.752em; left: 0.003em;"><span class="mrow" id="MathJax-Span-2031"><span class="munderover" id="MathJax-Span-2032"><span style="display: inline-block; position: relative; width: 1.454em; height: 0px;"><span style="position: absolute; clip: rect(2.655em 1000.003em 4.357em -0.398em); top: -3.751em; left: 0.003em;"><span class="mo" id="MathJax-Span-2033" style="font-family: MathJax_Size2; vertical-align: 0.003em;">∑</span><span style="display: inline-block; width: 0px; height: 3.756em;"></span></span><span style="position: absolute; clip: rect(1.754em 1000.003em 2.655em -0.398em); top: -1.299em; left: 0.053em;"><span class="texatom" id="MathJax-Span-2034"><span class="mrow" id="MathJax-Span-2035"><span class="mi" id="MathJax-Span-2036" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-2037" style="font-size: 70.7%; font-family: MathJax_Main;">=</span><span class="mn" id="MathJax-Span-2038" style="font-size: 70.7%; font-family: MathJax_Main;">0</span></span></span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span><span style="position: absolute; clip: rect(1.854em 1000.003em 2.555em -0.398em); top: -3.551em; left: 0.353em;"><span class="texatom" id="MathJax-Span-2039"><span class="mrow" id="MathJax-Span-2040"><span class="mi" id="MathJax-Span-2041" style="font-size: 70.7%; font-family: MathJax_Main;">∞</span></span></span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span></span></span><span class="mrow" id="MathJax-Span-2042"><span class="mo" id="MathJax-Span-2043" style="vertical-align: 2.255em;"><span style="display: inline-block; position: relative; width: 0.853em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.85em; left: 0.003em;">⎛<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: -0.698em; left: 0.003em;">⎝<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.699em; left: 0.003em;">⎜<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span></span><span class="mfrac" id="MathJax-Span-2044" style="padding-left: 0.103em; padding-right: 0.103em;"><span style="display: inline-block; position: relative; width: 7.56em; height: 0px;"><span style="position: absolute; clip: rect(0.803em 1000.003em 2.655em -0.398em); top: -3.201em; left: 50%; margin-left: -3.551em;"><span class="mrow" id="MathJax-Span-2045"><span class="mi" id="MathJax-Span-2046" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="msubsup" id="MathJax-Span-2047"><span style="display: inline-block; position: relative; width: 1.954em; height: 0px;"><span style="position: absolute; clip: rect(1.904em 1000.003em 2.655em -0.448em); top: -2.5em; left: 0.003em;"><span class="mi" id="MathJax-Span-2048" style="font-family: MathJax_Math; font-style: italic;">u</span><span style="display: inline-block; width: 0px; height: 2.505em;"></span></span><span style="position: absolute; top: -2.75em; left: 0.603em;"><span class="texatom" id="MathJax-Span-2049"><span class="mrow" id="MathJax-Span-2050"><span class="mi" id="MathJax-Span-2051" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-2052" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-2053" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-2054"><span style="display: inline-block; position: relative; width: 4.657em; height: 0px;"><span style="position: absolute; clip: rect(1.904em 1000.003em 2.655em -0.398em); top: -2.5em; left: 0.003em;"><span class="mi" id="MathJax-Span-2055" style="font-family: MathJax_Math; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 2.505em;"></span></span><span style="position: absolute; top: -3.001em; left: 0.503em;"><span class="texatom" id="MathJax-Span-2056"><span class="mrow" id="MathJax-Span-2057"><span class="mo" id="MathJax-Span-2058" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mfrac" id="MathJax-Span-2059" style="padding-left: 0.103em; padding-right: 0.103em;"><span style="display: inline-block; position: relative; width: 2.355em; height: 0px;"><span style="position: absolute; clip: rect(1.654em 1000.003em 2.555em -0.398em); top: -2.75em; left: 50%; margin-left: -0.998em;"><span class="mrow" id="MathJax-Span-2060"><span class="mo" id="MathJax-Span-2061" style="font-size: 50%; font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-2062" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">w</span><span class="mo" id="MathJax-Span-2063" style="font-size: 50%; font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-2064" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">a</span><span class="mi" id="MathJax-Span-2065" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">k</span><span class="msubsup" id="MathJax-Span-2066"><span style="display: inline-block; position: relative; width: 0.503em; height: 0px;"><span style="position: absolute; clip: rect(1.804em 1000.003em 2.555em -0.398em); top: -2.3em; left: 0.003em;"><span class="mo" id="MathJax-Span-2067" style="font-size: 50%; font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span><span style="position: absolute; top: -2.5em; left: 0.203em;"><span class="mn" id="MathJax-Span-2068" style="font-size: 50%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span><span style="position: absolute; clip: rect(1.804em 1000.003em 2.555em -0.448em); top: -1.949em; left: 50%; margin-left: -1.099em;"><span class="mrow" id="MathJax-Span-2069"><span class="mn" id="MathJax-Span-2070" style="font-size: 50%; font-family: MathJax_Main;">2</span><span class="mo" id="MathJax-Span-2071" style="font-size: 50%; font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-2072" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">k</span><span class="mi" id="MathJax-Span-2073" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.103em;"></span></span><span class="mo" id="MathJax-Span-2074" style="font-size: 50%; font-family: MathJax_Main;">+</span><span class="mtext" id="MathJax-Span-2075" style="font-size: 50%; font-family: MathJax_Main;">Ve</span><span class="mo" id="MathJax-Span-2076" style="font-size: 50%; font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span><span style="position: absolute; clip: rect(0.853em 1000.003em 1.204em -0.448em); top: -1.199em; left: 0.003em;"><span style="border-left-width: 2.355em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.054em;"></span></span></span></span><span class="mo" id="MathJax-Span-2077" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-2078" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">u</span></span></span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.505em;"></span></span><span style="position: absolute; clip: rect(1.704em 1000.003em 3.056em -0.398em); top: -1.899em; left: 50%; margin-left: -3.701em;"><span class="mrow" id="MathJax-Span-2079"><span class="msqrt" id="MathJax-Span-2080"><span style="display: inline-block; position: relative; width: 2.055em; height: 0px;"><span style="position: absolute; clip: rect(1.904em 1000.003em 2.855em -0.398em); top: -2.7em; left: 0.853em;"><span class="mrow" id="MathJax-Span-2081"><span class="mn" id="MathJax-Span-2082" style="font-family: MathJax_Main;">2</span><span class="mi" id="MathJax-Span-2083" 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: 2.705em;"></span></span><span style="position: absolute; clip: rect(3.606em 1000.003em 4.007em -0.448em); top: -4.602em; left: 0.853em;"><span style="display: inline-block; position: relative; width: 1.204em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -4.002em; left: -0.098em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -4.002em; left: 0.503em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; clip: rect(3.056em 1000.003em 4.357em -0.398em); top: -4.052em; left: 0.003em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span></span><span class="mi" id="MathJax-Span-2084" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-2085" style="font-family: MathJax_Main;">!</span><span class="msqrt" id="MathJax-Span-2086"><span style="display: inline-block; position: relative; width: 4.557em; height: 0px;"><span style="position: absolute; clip: rect(1.854em 1000.003em 2.955em -0.398em); top: -2.7em; left: 0.853em;"><span class="mrow" id="MathJax-Span-2087"><span class="mi" id="MathJax-Span-2088" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="mi" id="MathJax-Span-2089" style="font-family: MathJax_Math; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.203em;"></span></span><span class="mo" id="MathJax-Span-2090" style="font-family: MathJax_Main; padding-left: 0.203em;">+</span><span class="mtext" id="MathJax-Span-2091" style="font-family: MathJax_Main; padding-left: 0.203em;">Ve</span></span><span style="display: inline-block; width: 0px; height: 2.705em;"></span></span><span style="position: absolute; clip: rect(3.606em 1000.003em 4.007em -0.448em); top: -4.552em; left: 0.853em;"><span style="display: inline-block; position: relative; width: 3.706em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -4.002em; left: -0.098em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -4.002em; left: 3.056em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -4.002em; left: 0.403em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -4.002em; left: 0.953em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -4.002em; left: 1.454em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -4.002em; left: 2.005em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -4.002em; left: 2.505em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; clip: rect(3.056em 1000.003em 4.357em -0.398em); top: -4.002em; left: 0.003em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.705em;"></span></span><span style="position: absolute; clip: rect(0.853em 1000.003em 1.204em -0.448em); top: -1.249em; left: 0.003em;"><span style="border-left-width: 7.56em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.054em;"></span></span></span></span><span class="mo" id="MathJax-Span-2092" style="font-family: MathJax_Main; padding-left: 0.203em;">−</span><span class="mfrac" id="MathJax-Span-2093" style="padding-left: 0.353em; padding-right: 0.103em;"><span style="display: inline-block; position: relative; width: 7.56em; height: 0px;"><span style="position: absolute; clip: rect(2.305em 1000.003em 4.157em -0.448em); top: -4.702em; left: 50%; margin-left: -2.85em;"><span class="mrow" id="MathJax-Span-2094"><span class="msubsup" id="MathJax-Span-2095"><span style="display: inline-block; position: relative; width: 1.054em; height: 0px;"><span style="position: absolute; clip: rect(1.904em 1000.003em 2.655em -0.448em); top: -2.5em; left: 0.003em;"><span class="mi" id="MathJax-Span-2096" style="font-family: MathJax_Math; font-style: italic;">u</span><span style="display: inline-block; width: 0px; height: 2.505em;"></span></span><span style="position: absolute; top: -2.7em; left: 0.603em;"><span class="mi" id="MathJax-Span-2097" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">k</span><span style="display: inline-block; width: 0px; height: 2.355em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-2098"><span style="display: inline-block; position: relative; width: 4.657em; height: 0px;"><span style="position: absolute; clip: rect(1.904em 1000.003em 2.655em -0.398em); top: -2.5em; left: 0.003em;"><span class="mi" id="MathJax-Span-2099" style="font-family: MathJax_Math; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 2.505em;"></span></span><span style="position: absolute; top: -3.001em; left: 0.503em;"><span class="texatom" id="MathJax-Span-2100"><span class="mrow" id="MathJax-Span-2101"><span class="mo" id="MathJax-Span-2102" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mfrac" id="MathJax-Span-2103" style="padding-left: 0.103em; padding-right: 0.103em;"><span style="display: inline-block; position: relative; width: 2.355em; height: 0px;"><span style="position: absolute; clip: rect(1.654em 1000.003em 2.555em -0.398em); top: -2.75em; left: 50%; margin-left: -0.998em;"><span class="mrow" id="MathJax-Span-2104"><span class="mo" id="MathJax-Span-2105" style="font-size: 50%; font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-2106" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">w</span><span class="mo" id="MathJax-Span-2107" style="font-size: 50%; font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-2108" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">a</span><span class="mi" id="MathJax-Span-2109" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">k</span><span class="msubsup" id="MathJax-Span-2110"><span style="display: inline-block; position: relative; width: 0.503em; height: 0px;"><span style="position: absolute; clip: rect(1.804em 1000.003em 2.555em -0.398em); top: -2.3em; left: 0.003em;"><span class="mo" id="MathJax-Span-2111" style="font-size: 50%; font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span><span style="position: absolute; top: -2.5em; left: 0.203em;"><span class="mn" id="MathJax-Span-2112" style="font-size: 50%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span><span style="position: absolute; clip: rect(1.804em 1000.003em 2.555em -0.448em); top: -1.949em; left: 50%; margin-left: -1.099em;"><span class="mrow" id="MathJax-Span-2113"><span class="mn" id="MathJax-Span-2114" style="font-size: 50%; font-family: MathJax_Main;">2</span><span class="mo" id="MathJax-Span-2115" style="font-size: 50%; font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-2116" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">k</span><span class="mi" id="MathJax-Span-2117" style="font-size: 50%; font-family: MathJax_Math; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.103em;"></span></span><span class="mo" id="MathJax-Span-2118" style="font-size: 50%; font-family: MathJax_Main;">+</span><span class="mtext" id="MathJax-Span-2119" style="font-size: 50%; font-family: MathJax_Main;">Ve</span><span class="mo" id="MathJax-Span-2120" style="font-size: 50%; font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.305em;"></span></span><span style="position: absolute; clip: rect(0.853em 1000.003em 1.204em -0.448em); top: -1.199em; left: 0.003em;"><span style="border-left-width: 2.355em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.054em;"></span></span></span></span><span class="mo" id="MathJax-Span-2121" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-2122" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">u</span></span></span><span style="display: inline-block; width: 0px; height: 2.405em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; clip: rect(1.704em 1000.003em 3.056em -0.398em); top: -1.899em; left: 50%; margin-left: -3.701em;"><span class="mrow" id="MathJax-Span-2123"><span class="msqrt" id="MathJax-Span-2124"><span style="display: inline-block; position: relative; width: 2.055em; height: 0px;"><span style="position: absolute; clip: rect(1.904em 1000.003em 2.855em -0.398em); top: -2.7em; left: 0.853em;"><span class="mrow" id="MathJax-Span-2125"><span class="mn" id="MathJax-Span-2126" style="font-family: MathJax_Main;">2</span><span class="mi" id="MathJax-Span-2127" 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: 2.705em;"></span></span><span style="position: absolute; clip: rect(3.606em 1000.003em 4.007em -0.448em); top: -4.602em; left: 0.853em;"><span style="display: inline-block; position: relative; width: 1.204em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -4.002em; left: -0.098em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -4.002em; left: 0.503em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; clip: rect(3.056em 1000.003em 4.357em -0.398em); top: -4.052em; left: 0.003em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span></span><span class="mi" id="MathJax-Span-2128" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-2129" style="font-family: MathJax_Main;">!</span><span class="msqrt" id="MathJax-Span-2130"><span style="display: inline-block; position: relative; width: 4.557em; height: 0px;"><span style="position: absolute; clip: rect(1.854em 1000.003em 2.955em -0.398em); top: -2.7em; left: 0.853em;"><span class="mrow" id="MathJax-Span-2131"><span class="mi" id="MathJax-Span-2132" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="mi" id="MathJax-Span-2133" style="font-family: MathJax_Math; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.203em;"></span></span><span class="mo" id="MathJax-Span-2134" style="font-family: MathJax_Main; padding-left: 0.203em;">+</span><span class="mtext" id="MathJax-Span-2135" style="font-family: MathJax_Main; padding-left: 0.203em;">Ve</span></span><span style="display: inline-block; width: 0px; height: 2.705em;"></span></span><span style="position: absolute; clip: rect(3.606em 1000.003em 4.007em -0.448em); top: -4.552em; left: 0.853em;"><span style="display: inline-block; position: relative; width: 3.706em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -4.002em; left: -0.098em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -4.002em; left: 3.056em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -4.002em; left: 0.403em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -4.002em; left: 0.953em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -4.002em; left: 1.454em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -4.002em; left: 2.005em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -4.002em; left: 2.505em;">−<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; clip: rect(3.056em 1000.003em 4.357em -0.398em); top: -4.002em; left: 0.003em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.705em;"></span></span><span style="position: absolute; clip: rect(0.853em 1000.003em 1.204em -0.448em); top: -1.249em; left: 0.003em;"><span style="border-left-width: 7.56em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.054em;"></span></span></span></span><span class="mo" id="MathJax-Span-2136" style="vertical-align: 2.255em;"><span style="display: inline-block; position: relative; width: 0.853em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.85em; left: 0.003em;">⎞<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: -0.698em; left: 0.003em;">⎠<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.699em; left: 0.003em;">⎟<span style="display: inline-block; width: 0px; height: 4.007em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.757em;"></span></span></span><span style="border-left-width: 0.003em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 4.503em; vertical-align: -1.997em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-22">
\sum _{k=0}^{\infty } \left(\frac{k u^{k-1} e^{-\frac{(w-a k)^2}{2 (k V+\text{Ve})}-u}}{\sqrt{2 \pi } k! \sqrt{k V+\text{Ve}}}-\frac{u^k e^{-\frac{(w-a k)^2}{2 (k V+\text{Ve})}-u}}{\sqrt{2 \pi } k! \sqrt{k V+\text{Ve}}}\right)
</script></p>
<blockquote>
<p>Written with <a href="https://stackedit.io/">StackEdit</a>.</p>
</blockquote>
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