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On integrals of biquadratic gaussians
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| <h1 class="line" data-line="0">On integrals of biquadratic gaussians</h1> | |
| <p class="line" data-line="2">This short note aims to prove that</p> | |
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| <path id="s30g0-23" d="m1.833126-3.283686c.00797-.03188 .01594-.079701 .01594-.119552c0-.00797 0-.111582-.127522-.111582s-.908593 .071731-1.044085 .087671c-.079701 .01594-.111582 .079701-.111582 .151432c0 .111582 .111582 .111582 .207223 .111582c.175342 0 .382565 0 .382565 .111582c0 .047821-.087671 .406476-.143462 .605729c-.03188 .151432-.167372 .6934-.199253 .804981l-.286924 1.147696c-.03188 .127522-.087671 .358655-.087671 .382565c0 .111582 .119552 .111582 .151432 .111582h.119552c.74122-.119552 1.641843-.502117 2.422914-1.195517c1.091905-.972354 1.267248-2.048319 1.267248-2.072229c0-.167372-.127522-.247073-.263014-.247073c-.151432 0-.286924 .087671-.350685 .302864c-.526027 1.825156-1.880946 2.646077-2.693898 2.893151l.74122-2.964882z"></path> | |
| </defs> | |
| <g id="s30page1"> | |
| <!--start 1872.019782 1491.397646 --> | |
| <use x="1872.019782" y="1491.397646" xlink:href="#s30g1-73"></use> | |
| <use x="1876.81525" y="1493.033996" xlink:href="#s30g0-23"></use> | |
| <use x="1881.997264" y="1491.397646" xlink:href="#s30g2-40"></use> | |
| <use x="1886.239702" y="1491.397646" xlink:href="#s30g1-120"></use> | |
| <use x="1892.474556" y="1491.397646" xlink:href="#s30g2-41"></use> | |
| </g> | |
| <script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("3.70313|30|14.25|"+window.location,"*");</script> | |
| </svg> is the <a href="https://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html">modified Bessel function of the first kind</a>. | |
| The result for the positive case appears (in slightly different form) in | |
| "Table of Integrals, Series, and Products" by Gradshteyn and Ryzhik | |
| (3.323 in 7th ed.), but the negative case doesn’t seem to be widely known. | |
| Note that the second form has removable singularity at <svg class="svg-preview" id="s2tex_4" style="vertical-align:-0.46763pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="43" height="13" viewBox="1872.02 1482.848 25.704 7.771"><defs><path id="s31g1-48" d="M5.018-3.49c0-.874-.054-1.746-.436-2.554C4.08-7.09 3.185-7.265 2.727-7.265c-.654 0-1.45.283-1.898 1.298C.48-5.215.425-4.364.425-3.491c0 .818.044 1.8.491 2.63A2 2 0 0 0 2.716.24c.59 0 1.419-.23 1.899-1.265.349-.753.403-1.604.403-2.466M2.716 0c-.425 0-1.069-.273-1.265-1.32-.12-.655-.12-1.658-.12-2.302 0-.698 0-1.418.087-2.007.207-1.298 1.026-1.396 1.298-1.396.36 0 1.08.196 1.288 1.276.109.61.109 1.44.109 2.127 0 .818 0 1.56-.12 2.258C3.829-.327 3.207 0 2.716 0"></path><path id="s31g1-61" d="M7.495-3.567c.163 0 .37 0 .37-.218s-.207-.219-.36-.219H.971c-.153 0-.36 0-.36.219s.207.218.37.218zm.01 2.116c.153 0 .36 0 .36-.218s-.207-.218-.37-.218H.982c-.164 0-.371 0-.371.218s.207.218.36.218z"></path><path id="s31g0-97" d="M4.07-4.124c-.197-.403-.514-.698-1.005-.698C1.79-4.822.436-3.218.436-1.625.436-.6 1.036.12 1.887.12c.218 0 .764-.044 1.418-.818.088.458.47.818.993.818.382 0 .633-.25.807-.6.186-.393.328-1.058.328-1.08 0-.11-.098-.11-.131-.11-.11 0-.12.045-.153.197-.185.71-.382 1.353-.829 1.353-.295 0-.327-.284-.327-.502 0-.24.022-.327.142-.807.12-.458.141-.567.24-.982l.392-1.527c.077-.306.077-.327.077-.371 0-.186-.131-.295-.317-.295-.262 0-.425.24-.458.48m-.71 2.826c-.055.196-.055.218-.218.403-.48.6-.927.775-1.233.775-.545 0-.698-.6-.698-1.025 0-.546.349-1.888.6-2.39.338-.643.829-1.047 1.265-1.047.71 0 .862.895.862.96s-.022.131-.033.186z"></path></defs><g id="s31page1"><use xlink:href="#s31g0-97" x="1872.02" y="1490.246"></use><use xlink:href="#s31g1-61" x="1880.816" y="1490.246"></use><use xlink:href="#s31g1-48" x="1892.332" y="1490.246"></use></g><script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("0.46763|32.25|9.75|"+window.location,"*");</script></svg>.</p> | |
| <p class="line" data-line="15">Let us define</p> | |
| <p class="line" data-line="17" align="center"><svg class="svg-preview" id="s2tex_5" style="vertical-align:-12.77481pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="228" height="45" viewBox="1872.019782 1483.007447 136.288917 26.899128"> | |
| <defs> | |
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| <path id="s27g4-50" d="m3.21594-1.117808h-.221171c-.011955 .083686-.071731 .478207-.161395 .54396c-.041843 .035866-.526027 .035866-.609714 .035866h-1.117808l.765131-.621669c.203238-.161395 .735243-.54396 .920548-.723288c.179328-.179328 .424408-.484184 .424408-.908593c0-.747198-.675467-1.183562-1.476463-1.183562c-.771108 0-1.309091 .508095-1.309091 1.069988c0 .304857 .257036 .340722 .32279 .340722c.14944 0 .32279-.107597 .32279-.32279c0-.131507-.077709-.32279-.340722-.32279c.137484-.304857 .502117-.532005 .91457-.532005c.627646 0 .962391 .466252 .962391 .950436c0 .424408-.280946 .860772-.699377 1.243337l-1.416687 1.297136c-.059776 .059776-.065753 .065753-.065753 .251059h2.600249l.185305-1.117808z"></path> | |
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| <use x="1888.183861" y="1487.068041" xlink:href="#s32g4-52"></use> | |
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| <use x="1905.605801" y="1494.3851" xlink:href="#s32g1-6"></use> | |
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| <script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("3.58938|70.5|18|"+window.location,"*");</script> | |
| </svg>. The plot is</p> | |
| <p class="line" data-line="21"><img src="https://gist.github.com/user-attachments/assets/c83887d8-def5-4c1f-9013-d247e792ebb0" alt="Plot of exp(-w^2) * F(w)"></p> | |
| <p class="line" data-line="23">Differentiating by <svg class="svg-preview" id="s2tex_7" style="vertical-align:-0.42534pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="13" height="9" viewBox="1872.02 1482.872 7.771 5.38"><defs><path id="s33g0-119" d="M5.04-3.698c.055-.24.164-.644.164-.71 0-.196-.153-.294-.317-.294a.45.45 0 0 0-.403.306c-.033.076-.546 2.16-.622 2.443a3 3 0 0 0-.098.72c0 .12 0 .142.01.197-.25.578-.578.916-1.003.916-.873 0-.873-.807-.873-.993 0-.349.055-.774.567-2.116.12-.327.186-.48.186-.698a.87.87 0 0 0-.895-.895C.72-4.822.316-3.24.316-3.142c0 .11.11.11.131.11.11 0 .12-.023.175-.197.294-1.015.72-1.353 1.102-1.353.098 0 .272.011.272.36 0 .273-.12.578-.196.764-.48 1.287-.61 1.789-.61 2.203C1.19-.262 1.92.12 2.737.12c.186 0 .698 0 1.135-.764.283.699 1.036.764 1.363.764.819 0 1.299-.687 1.582-1.342.371-.85.72-2.313.72-2.836 0-.6-.294-.764-.48-.764-.273 0-.545.284-.545.524 0 .142.065.207.163.294.12.12.393.404.393.928 0 .37-.316 1.44-.6 1.996-.284.578-.665.96-1.2.96-.513 0-.807-.327-.807-.95 0-.305.076-.643.12-.795z"></path></defs><use xlink:href="#s33g0-119" id="s33page1" x="1872.02" y="1487.913"></use><script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("0.42534|9.75|6.75|"+window.location,"*");</script></svg>, we obtain</p> | |
| <p class="line" data-line="25" align="center"><svg class="svg-preview" id="s2tex_8" style="vertical-align:-12.77481pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="282" height="45" viewBox="1872.019782 1483.007447 168.56787 26.899128"> | |
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| <p class="line" data-line="39">Therefore, the function satisfies <a href="https://mathworld.wolfram.com/HermiteDifferentialEquation.html">Hermite differential equation</a>:</p> | |
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| <p class="line" data-line="50">With this, it is straightforward to verify that</p> | |
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| </g> | |
| <script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("13.09209|312.75|35.25|"+window.location,"*");</script> | |
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| <p class="line" data-line="55">satisfies both the equation and the initial conditions: the Hermite | |
| polynomial always satisfies the corresponding equation, and the initial | |
| conditions can be confirmed by evaluating its representation in terms of | |
| <a href="https://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheSecondKind.html">confluent hypergeometric function of the second kind</a> | |
| at <svg class="svg-preview" id="s2tex_19" style="vertical-align:-0.46763pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="47" height="13" viewBox="1872.019782 1482.847969 28.094645 7.770859"> | |
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| <script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("0.46763|35.25|9.75|"+window.location,"*");</script> | |
| </svg> using its Taylor series:</p> | |
| <p class="line" data-line="61" align="center"><svg class="svg-preview" id="s2tex_20" style="vertical-align:-15.21844pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="465" height="50" viewBox="1872.019782 1482.961484 277.957659 29.88792"> | |
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| <p class="line" data-line="69">Note that the "Hermite polynomial" of a fractional order is, of course, | |
| not a polynomial.</p> | |
| <p class="line" data-line="72">This could reasonably conclude the proof, but we also want to express | |
| the solution in terms of familiar (and more commonly implemented in | |
| numerical software) modified Bessel functions.</p> | |
| <p class="line" data-line="76">First, we re-express the formula using only modified Bessel functions | |
| of positive order, using <a href="https://dlmf.nist.gov/10.27">reflection formulae</a>:</p> | |
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| <p class="line" data-line="95">Note: these expressions can be useful in their own right, since | |
| e.g. some numerical software only provides the implementations | |
| for non-negative orders.</p> | |
| <p class="line" data-line="99">We immediately notice that left and right limits agree, since | |
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| <p class="line" data-line="102">Denoting</p> | |
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| <p class="line" data-line="108">It is straightforward to verify that</p> | |
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| <p class="line" data-line="112">Similarly (though with a bit more effort), we can show that</p> | |
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| </svg>, | |
| and finding that they agree.</p> | |
| <p class="line" data-line="119">With additional drudgework we obtain (omitting for brevity the argument to Bessel functions): | |
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| <use x="2091.097297" y="1516.771275" xlink:href="#s11g8-103"></use> | |
| <use x="2096.551862" y="1516.771275" xlink:href="#s11g8-110"></use> | |
| <use x="2102.612457" y="1516.771275" xlink:href="#s11g8-40"></use> | |
| <use x="2106.854895" y="1516.771275" xlink:href="#s11g5-119"></use> | |
| <use x="2114.958449" y="1516.771275" xlink:href="#s11g8-41"></use> | |
| <use x="2121.019036" y="1504.662098" xlink:href="#s11g0-16"></use> | |
| <use x="2127.5342" y="1516.771275" xlink:href="#s11g8-40"></use> | |
| <use x="2131.776638" y="1516.771275" xlink:href="#s11g5-73"></use> | |
| <use x="2137.767621" y="1516.865921" xlink:href="#s11g6-57"></use> | |
| <rect x="2137.767621" y="1517.952499" height=".358656" width="3.652912"></rect> | |
| <use x="2137.767621" y="1523.249956" xlink:href="#s11g6-52"></use> | |
| <use x="2145.538376" y="1516.771275" xlink:href="#s11g2-0"></use> | |
| <use x="2156.44745" y="1516.771275" xlink:href="#s11g8-50"></use> | |
| <use x="2161.902015" y="1516.771275" xlink:href="#s11g5-73"></use> | |
| <use x="2167.892997" y="1516.865921" xlink:href="#s11g6-49"></use> | |
| <rect x="2167.892997" y="1517.952499" height=".358656" width="3.652912"></rect> | |
| <use x="2167.892997" y="1523.249956" xlink:href="#s11g6-52"></use> | |
| <use x="2175.663753" y="1516.771275" xlink:href="#s11g8-43"></use> | |
| <use x="2186.572827" y="1516.771275" xlink:href="#s11g5-73"></use> | |
| <use x="2191.368296" y="1520.124363" xlink:href="#s11g1-0"></use> | |
| <use x="2199.150316" y="1516.865921" xlink:href="#s11g6-55"></use> | |
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| <use x="2199.150316" y="1523.249956" xlink:href="#s11g6-52"></use> | |
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| <use x="2216.842867" y="1512.812457" xlink:href="#s11g7-52"></use> | |
| <use x="2223.999379" y="1516.771275" xlink:href="#s11g8-43"></use> | |
| <use x="2234.908453" y="1516.771275" xlink:href="#s11g8-40"></use> | |
| <use x="2239.150891" y="1516.771275" xlink:href="#s11g8-52"></use> | |
| <use x="2244.605456" y="1516.771275" xlink:href="#s11g5-73"></use> | |
| <use x="2250.596438" y="1516.865921" xlink:href="#s11g6-53"></use> | |
| <rect x="2250.596438" y="1517.952499" height=".358656" width="3.652912"></rect> | |
| <use x="2250.596438" y="1523.249956" xlink:href="#s11g6-52"></use> | |
| <use x="2258.367194" y="1516.771275" xlink:href="#s11g8-43"></use> | |
| <use x="2269.276268" y="1516.771275" xlink:href="#s11g8-52"></use> | |
| <use x="2274.730833" y="1516.771275" xlink:href="#s11g5-73"></use> | |
| <use x="2279.526302" y="1520.124363" xlink:href="#s11g1-0"></use> | |
| <use x="2287.308322" y="1516.865921" xlink:href="#s11g6-51"></use> | |
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| <use x="2287.308322" y="1523.249956" xlink:href="#s11g6-52"></use> | |
| <use x="2292.65488" y="1516.771275" xlink:href="#s11g8-41"></use> | |
| <use x="2296.897318" y="1516.771275" xlink:href="#s11g5-119"></use> | |
| <use x="2305.000872" y="1512.812457" xlink:href="#s11g7-50"></use> | |
| <use x="2312.157385" y="1516.771275" xlink:href="#s11g2-0"></use> | |
| <use x="2323.066459" y="1516.771275" xlink:href="#s11g5-73"></use> | |
| <use x="2329.057441" y="1516.865921" xlink:href="#s11g6-49"></use> | |
| <rect x="2329.057441" y="1517.952499" height=".358656" width="3.652912"></rect> | |
| <use x="2329.057441" y="1523.249956" xlink:href="#s11g6-52"></use> | |
| <use x="2334.403999" y="1504.662098" xlink:href="#s11g0-17"></use> | |
| <use x="2340.919078" y="1506.266327" xlink:href="#s11g5-58"></use> | |
| </g> | |
| <script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("22.36908|589.5|51.75|"+window.location,"*");</script> | |
| </svg></p> | |
| <p class="line" data-line="142">Using the <a href="https://dlmf.nist.gov/10.29">recurrence relations</a> | |
| it is then straightforward to verify that both top-level parenthesized | |
| terms are <svg class="svg-preview" id="s2tex_35" style="vertical-align:-0.46763pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="9" height="13" viewBox="1872.02 1482.848 5.38 7.771"><defs><path id="s56g0-48" d="M5.018-3.49c0-.874-.054-1.746-.436-2.554C4.08-7.09 3.185-7.265 2.727-7.265c-.654 0-1.45.283-1.898 1.298C.48-5.215.425-4.364.425-3.491c0 .818.044 1.8.491 2.63A2 2 0 0 0 2.716.24c.59 0 1.419-.23 1.899-1.265.349-.753.403-1.604.403-2.466M2.716 0c-.425 0-1.069-.273-1.265-1.32-.12-.655-.12-1.658-.12-2.302 0-.698 0-1.418.087-2.007.207-1.298 1.026-1.396 1.298-1.396.36 0 1.08.196 1.288 1.276.109.61.109 1.44.109 2.127 0 .818 0 1.56-.12 2.258C3.829-.327 3.207 0 2.716 0"></path></defs><use xlink:href="#s56g0-48" id="s56page1" x="1872.02" y="1490.246"></use><script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("0.46763|6.75|9.75|"+window.location,"*");</script></svg>.</p> | |
| <p class="line" data-line="146">We have thus shown that <svg class="svg-preview" id="s2tex_36" style="vertical-align:-3.70313pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="41" height="19" viewBox="1872.019782 1482.991669 24.508095 11.35741"> | |
| <defs> | |
| <path id="s57g1-40" d="m3.610908 2.618181c0-.032727 0-.054545-.185454-.24c-1.363636-1.374545-1.712727-3.436363-1.712727-5.105453c0-1.898181 .414545-3.796363 1.756363-5.159999c.141818-.130909 .141818-.152727 .141818-.185454c0-.076364-.043636-.109091-.109091-.109091c-.109091 0-1.090909 .741818-1.734545 2.127272c-.556363 1.2-.687273 2.410908-.687273 3.327272c0 .850909 .12 2.170908 .72 3.403635c.654545 1.341818 1.592727 2.050909 1.701818 2.050909c.065455 0 .109091-.032727 .109091-.109091z"></path> | |
| <path id="s57g1-41" d="m3.152726-2.727272c0-.850909-.12-2.170908-.72-3.403635c-.654545-1.341818-1.592727-2.050909-1.701818-2.050909c-.065455 0-.109091 .043636-.109091 .109091c0 .032727 0 .054545 .207273 .250909c1.069091 1.08 1.690909 2.814545 1.690909 5.094544c0 1.865454-.403636 3.785453-1.756363 5.159999c-.141818 .130909-.141818 .152727-.141818 .185454c0 .065455 .043636 .109091 .109091 .109091c.109091 0 1.090909-.741818 1.734545-2.127272c.556363-1.2 .687273-2.410908 .687273-3.327272z"></path> | |
| <path id="s57g0-71" d="m8.290907-7.581816c0-.032727-.021818-.109091-.12-.109091c-.032727 0-.043636 .010909-.163636 .130909l-.763636 .84c-.098182-.152727-.6-.970909-1.810909-.970909c-2.432727 0-4.887271 2.410908-4.887271 4.941817c0 1.734545 1.210909 2.98909 2.978181 2.98909c.48 0 .970909-.098182 1.363636-.261818c.545454-.218182 .752727-.447273 .949091-.665454c.098182 .272727 .381818 .676363 .490909 .676363c.054545 0 .076364-.032727 .076364-.043636c.021818-.021818 .130909-.436364 .185454-.665454l.207273-.84c.043636-.185454 .098182-.370909 .141818-.556363c.12-.490909 .130909-.512727 .752727-.523636c.054545 0 .174545-.010909 .174545-.218182c0-.076364-.054545-.12-.141818-.12c-.250909 0-.894545 .032727-1.145454 .032727c-.338182 0-1.189091-.032727-1.527272-.032727c-.098182 0-.229091 0-.229091 .218182c0 .12 .087273 .12 .327273 .12c.010909 0 .327273 0 .578182 .021818c.283636 .032727 .338182 .065455 .338182 .207273c0 .098182-.12 .589091-.229091 .992727c-.305454 1.2-1.723636 1.32-2.105454 1.32c-1.047272 0-2.192727-.621818-2.192727-2.290908c0-.338182 .109091-2.138181 1.254545-3.556363c.589091-.741818 1.647272-1.407272 2.727272-1.407272c1.112727 0 1.756363 .84 1.756363 2.105454c0 .436364-.032727 .447273-.032727 .556363s.12 .109091 .163636 .109091c.141818 0 .141818-.021818 .196364-.218182l.687273-2.781817z"></path> | |
| <path id="s57g0-119" d="m5.039999-3.698181c.054545-.24 .163636-.643636 .163636-.709091c0-.196364-.152727-.294545-.316364-.294545c-.130909 0-.327273 .087273-.403636 .305454c-.032727 .076364-.545454 2.159999-.621818 2.443636c-.076364 .327273-.098182 .523636-.098182 .72c0 .12 0 .141818 .010909 .196364c-.250909 .578182-.578182 .916363-1.003636 .916363c-.872727 0-.872727-.807273-.872727-.992727c0-.349091 .054545-.774545 .567273-2.116363c.12-.327273 .185454-.48 .185454-.698182c0-.490909-.36-.894545-.894545-.894545c-1.036363 0-1.44 1.581818-1.44 1.68c0 .109091 .109091 .109091 .130909 .109091c.109091 0 .12-.021818 .174545-.196364c.294545-1.014545 .72-1.352727 1.101818-1.352727c.098182 0 .272727 .010909 .272727 .36c0 .272727-.12 .578182-.196364 .763636c-.48 1.287272-.610909 1.78909-.610909 2.203636c0 .992727 .730909 1.374545 1.54909 1.374545c.185454 0 .698182 0 1.134545-.763636c.283636 .698182 1.036363 .763636 1.363636 .763636c.818182 0 1.298181-.687273 1.581818-1.341818c.370909-.850909 .72-2.312727 .72-2.836363c0-.6-.294545-.763636-.48-.763636c-.272727 0-.545454 .283636-.545454 .523636c0 .141818 .065455 .207273 .163636 .294545c.12 .12 .392727 .403636 .392727 .927272c0 .370909-.316364 1.44-.6 1.996363c-.283636 .578182-.665454 .96-1.2 .96c-.512727 0-.807273-.327273-.807273-.949091c0-.305454 .076364-.643636 .12-.796363l.458182-1.832727z"></path> | |
| </defs> | |
| <g id="s57page1"> | |
| <!--start 1872.019782 1491.397646 --> | |
| <use x="1872.019782" y="1491.397646" xlink:href="#s57g0-71"></use> | |
| <use x="1880.597043" y="1491.397646" xlink:href="#s57g1-40"></use> | |
| <use x="1884.839481" y="1491.397646" xlink:href="#s57g0-119"></use> | |
| <use x="1892.943035" y="1491.397646" xlink:href="#s57g1-41"></use> | |
| </g> | |
| <script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("3.70313|30.75|14.25|"+window.location,"*");</script> | |
| </svg> satisfies both the differential equation | |
| and the initial conditions, which concludes the proof.</p> | |
| <p class="line" data-line="149">It should probably be mentioned that several of the expressions were | |
| obtained using computer algebra software, with limited human | |
| oversight.</p> | |
| <p class="line" data-line="153">The numerical experiments found the expressions to agree with | |
| the numerical quadratures up to machine precision of float64.</p> | |
| <p class="line" data-line="156">Numerical considerations. Evaluating the integrals using formulae given | |
| here can cause several problems:</p> | |
| <ol> | |
| <li>For <svg class="svg-preview" id="s2tex_37" style="vertical-align:-0.46763pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="43" height="13" viewBox="1872.02 1482.848 25.704 7.771"><defs><path id="s31g1-48" d="M5.018-3.49c0-.874-.054-1.746-.436-2.554C4.08-7.09 3.185-7.265 2.727-7.265c-.654 0-1.45.283-1.898 1.298C.48-5.215.425-4.364.425-3.491c0 .818.044 1.8.491 2.63A2 2 0 0 0 2.716.24c.59 0 1.419-.23 1.899-1.265.349-.753.403-1.604.403-2.466M2.716 0c-.425 0-1.069-.273-1.265-1.32-.12-.655-.12-1.658-.12-2.302 0-.698 0-1.418.087-2.007.207-1.298 1.026-1.396 1.298-1.396.36 0 1.08.196 1.288 1.276.109.61.109 1.44.109 2.127 0 .818 0 1.56-.12 2.258C3.829-.327 3.207 0 2.716 0"></path><path id="s31g1-61" d="M7.495-3.567c.163 0 .37 0 .37-.218s-.207-.219-.36-.219H.971c-.153 0-.36 0-.36.219s.207.218.37.218zm.01 2.116c.153 0 .36 0 .36-.218s-.207-.218-.37-.218H.982c-.164 0-.371 0-.371.218s.207.218.36.218z"></path><path id="s31g0-97" d="M4.07-4.124c-.197-.403-.514-.698-1.005-.698C1.79-4.822.436-3.218.436-1.625.436-.6 1.036.12 1.887.12c.218 0 .764-.044 1.418-.818.088.458.47.818.993.818.382 0 .633-.25.807-.6.186-.393.328-1.058.328-1.08 0-.11-.098-.11-.131-.11-.11 0-.12.045-.153.197-.185.71-.382 1.353-.829 1.353-.295 0-.327-.284-.327-.502 0-.24.022-.327.142-.807.12-.458.141-.567.24-.982l.392-1.527c.077-.306.077-.327.077-.371 0-.186-.131-.295-.317-.295-.262 0-.425.24-.458.48m-.71 2.826c-.055.196-.055.218-.218.403-.48.6-.927.775-1.233.775-.545 0-.698-.6-.698-1.025 0-.546.349-1.888.6-2.39.338-.643.829-1.047 1.265-1.047.71 0 .862.895.862.96s-.022.131-.033.186z"></path></defs><g id="s31page1"><use xlink:href="#s31g0-97" x="1872.02" y="1490.246"></use><use xlink:href="#s31g1-61" x="1880.816" y="1490.246"></use><use xlink:href="#s31g1-48" x="1892.332" y="1490.246"></use></g><script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("0.46763|32.25|9.75|"+window.location,"*");</script></svg> the Bessel-based expression is indeterminate, <svg class="svg-preview" id="s2tex_38" style="vertical-align:-9.90726pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="10" height="38" viewBox="1873.215 1482.808 5.978 22.715"><defs><path id="s59g0-48" d="M5.018-3.49c0-.874-.054-1.746-.436-2.554C4.08-7.09 3.185-7.265 2.727-7.265c-.654 0-1.45.283-1.898 1.298C.48-5.215.425-4.364.425-3.491c0 .818.044 1.8.491 2.63A2 2 0 0 0 2.716.24c.59 0 1.419-.23 1.899-1.265.349-.753.403-1.604.403-2.466M2.716 0c-.425 0-1.069-.273-1.265-1.32-.12-.655-.12-1.658-.12-2.302 0-.698 0-1.418.087-2.007.207-1.298 1.026-1.396 1.298-1.396.36 0 1.08.196 1.288 1.276.109.61.109 1.44.109 2.127 0 .818 0 1.56-.12 2.258C3.829-.327 3.207 0 2.716 0"></path></defs><g id="s59page1"><use xlink:href="#s59g0-48" x="1873.215" y="1490.246"></use><path d="M1873.215 1494.681h5.455v.436h-5.455z"></path><use xlink:href="#s59g0-48" x="1873.215" y="1505.109"></use></g><script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("9.90726|7.5|28.5|"+window.location,"*");</script></svg>. Similarly, small <svg class="svg-preview" id="s2tex_39" style="vertical-align:-0.42534pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="10" height="9" viewBox="1872.02 1482.872 5.978 5.38"><defs><path id="s28g0-97" d="M4.07-4.124c-.197-.403-.514-.698-1.005-.698C1.79-4.822.436-3.218.436-1.625.436-.6 1.036.12 1.887.12c.218 0 .764-.044 1.418-.818.088.458.47.818.993.818.382 0 .633-.25.807-.6.186-.393.328-1.058.328-1.08 0-.11-.098-.11-.131-.11-.11 0-.12.045-.153.197-.185.71-.382 1.353-.829 1.353-.295 0-.327-.284-.327-.502 0-.24.022-.327.142-.807.12-.458.141-.567.24-.982l.392-1.527c.077-.306.077-.327.077-.371 0-.186-.131-.295-.317-.295-.262 0-.425.24-.458.48m-.71 2.826c-.055.196-.055.218-.218.403-.48.6-.927.775-1.233.775-.545 0-.698-.6-.698-1.025 0-.546.349-1.888.6-2.39.338-.643.829-1.047 1.265-1.047.71 0 .862.895.862.96s-.022.131-.033.186z"></path></defs><use xlink:href="#s28g0-97" id="s28page1" x="1872.02" y="1487.913"></use><script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("0.42534|7.5|6.75|"+window.location,"*");</script></svg> can be problematic (e.g. because <svg class="svg-preview" id="s2tex_40" style="vertical-align:-0.29955pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="17" height="17" viewBox="1872.02 1482.933 10.162 10.162"><defs><path id="s60g1-50" d="M2.248-1.626c.127-.12.462-.382.59-.494.494-.454.964-.893.964-1.618 0-.948-.797-1.562-1.794-1.562-.956 0-1.586.725-1.586 1.434 0 .391.311.447.423.447a.41.41 0 0 0 .414-.423c0-.414-.398-.414-.494-.414a1.23 1.23 0 0 1 1.156-.781c.741 0 1.124.63 1.124 1.299 0 .829-.582 1.435-1.523 2.399L.518-.303C.422-.215.422-.199.422 0h3.149l.23-1.427h-.246c-.024.16-.088.558-.184.71-.047.063-.653.063-.78.063H1.17z"></path><path id="s60g0-97" d="M4.07-4.124c-.197-.403-.514-.698-1.005-.698C1.79-4.822.436-3.218.436-1.625.436-.6 1.036.12 1.887.12c.218 0 .764-.044 1.418-.818.088.458.47.818.993.818.382 0 .633-.25.807-.6.186-.393.328-1.058.328-1.08 0-.11-.098-.11-.131-.11-.11 0-.12.045-.153.197-.185.71-.382 1.353-.829 1.353-.295 0-.327-.284-.327-.502 0-.24.022-.327.142-.807.12-.458.141-.567.24-.982l.392-1.527c.077-.306.077-.327.077-.371 0-.186-.131-.295-.317-.295-.262 0-.425.24-.458.48m-.71 2.826c-.055.196-.055.218-.218.403-.48.6-.927.775-1.233.775-.545 0-.698-.6-.698-1.025 0-.546.349-1.888.6-2.39.338-.643.829-1.047 1.265-1.047.71 0 .862.895.862.96s-.022.131-.033.186z"></path></defs><g id="s60page1"><use xlink:href="#s60g0-97" x="1872.02" y="1492.856"></use><use xlink:href="#s60g1-50" x="1877.786" y="1488.352"></use></g><script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("0.29955|12.75|12.75|"+window.location,"*");</script></svg> underflows).</li> | |
| <li>For large <svg class="svg-preview" id="s2tex_41" style="vertical-align:-3.70313pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="18" height="19" viewBox="1872.019782 1482.991669 10.759651 11.35741"> | |
| <defs> | |
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| <path id="s61g0-106" d="m1.734545-7.789089c0-.196364 0-.392727-.218182-.392727s-.218182 .196364-.218182 .392727v10.123634c0 .196364 0 .392727 .218182 .392727s.218182-.196364 .218182-.392727v-10.123634z"></path> | |
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| <script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("3.70313|13.5|14.25|"+window.location,"*");</script> | |
| </svg>, the expression is a product of factors tending to <svg class="svg-preview" id="s2tex_42" style="vertical-align:-0.46763pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="9" height="13" viewBox="1872.02 1482.848 5.38 7.771"><defs><path id="s56g0-48" d="M5.018-3.49c0-.874-.054-1.746-.436-2.554C4.08-7.09 3.185-7.265 2.727-7.265c-.654 0-1.45.283-1.898 1.298C.48-5.215.425-4.364.425-3.491c0 .818.044 1.8.491 2.63A2 2 0 0 0 2.716.24c.59 0 1.419-.23 1.899-1.265.349-.753.403-1.604.403-2.466M2.716 0c-.425 0-1.069-.273-1.265-1.32-.12-.655-.12-1.658-.12-2.302 0-.698 0-1.418.087-2.007.207-1.298 1.026-1.396 1.298-1.396.36 0 1.08.196 1.288 1.276.109.61.109 1.44.109 2.127 0 .818 0 1.56-.12 2.258C3.829-.327 3.207 0 2.716 0"></path></defs><use xlink:href="#s56g0-48" id="s56page1" x="1872.02" y="1490.246"></use><script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("0.46763|6.75|9.75|"+window.location,"*");</script></svg> and <svg class="svg-preview" id="s2tex_43" style="vertical-align:-0.05016pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="22" height="13" viewBox="1872.019782 1483.060698 13.150685 7.770859"> | |
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| <path id="s62g0-105" d="m1.930909-4.821817l-1.527272 .12v.338182c.709091 0 .807273 .065455 .807273 .6v2.934545c0 .490909-.12 .490909-.850909 .490909v.338182c.349091-.010909 .938182-.032727 1.2-.032727c.381818 0 .763636 .021818 1.134545 .032727v-.338182c-.72 0-.763636-.054545-.763636-.48v-4.003635zm.043636-1.898181c0-.349091-.272727-.578182-.578182-.578182c-.338182 0-.578182 .294545-.578182 .578182c0 .294545 .24 .578182 .578182 .578182c.305454 0 .578182-.229091 .578182-.578182z"></path> | |
| <path id="s62g0-110" d="m1.2-3.752726v2.923636c0 .490909-.12 .490909-.850909 .490909v.338182c.381818-.010909 .938182-.032727 1.232727-.032727c.283636 0 .850909 .021818 1.221818 .032727v-.338182c-.730909 0-.850909 0-.850909-.490909v-2.007272c0-1.134545 .774545-1.745454 1.472727-1.745454c.687273 0 .807273 .589091 .807273 1.210909v2.541817c0 .490909-.12 .490909-.850909 .490909v.338182c.381818-.010909 .938182-.032727 1.232727-.032727c.283636 0 .850909 .021818 1.221818 .032727v-.338182c-.567273 0-.84 0-.850909-.327273v-2.083636c0-.938182 0-1.276363-.338182-1.66909c-.152727-.185454-.512727-.403636-1.145454-.403636c-.796363 0-1.309091 .469091-1.614545 1.145454v-1.145454l-1.538181 .12v.338182c.763636 0 .850909 .076364 .850909 .610909z"></path> | |
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| <script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("0.05016|16.5|9.75|"+window.location,"*");</script> | |
| </svg>, respectively.</li> | |
| <li>Even modest positive <svg class="svg-preview" id="s2tex_44" style="vertical-align:-0.42534pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="10" height="9" viewBox="1872.02 1482.872 5.978 5.38"><defs><path id="s28g0-97" d="M4.07-4.124c-.197-.403-.514-.698-1.005-.698C1.79-4.822.436-3.218.436-1.625.436-.6 1.036.12 1.887.12c.218 0 .764-.044 1.418-.818.088.458.47.818.993.818.382 0 .633-.25.807-.6.186-.393.328-1.058.328-1.08 0-.11-.098-.11-.131-.11-.11 0-.12.045-.153.197-.185.71-.382 1.353-.829 1.353-.295 0-.327-.284-.327-.502 0-.24.022-.327.142-.807.12-.458.141-.567.24-.982l.392-1.527c.077-.306.077-.327.077-.371 0-.186-.131-.295-.317-.295-.262 0-.425.24-.458.48m-.71 2.826c-.055.196-.055.218-.218.403-.48.6-.927.775-1.233.775-.545 0-.698-.6-.698-1.025 0-.546.349-1.888.6-2.39.338-.643.829-1.047 1.265-1.047.71 0 .862.895.862.96s-.022.131-.033.186z"></path></defs><use xlink:href="#s28g0-97" id="s28page1" x="1872.02" y="1487.913"></use><script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("0.42534|7.5|6.75|"+window.location,"*");</script></svg> may lead to subtraction of close quantities (<svg class="svg-preview" id="s2tex_45" style="vertical-align:-8.30655pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="52" height="25" viewBox="1872.019782 1483.074098 31.083437 14.94396"> | |
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| <path id="s63g3-41" d="m3.152726-2.727272c0-.850909-.12-2.170908-.72-3.403635c-.654545-1.341818-1.592727-2.050909-1.701818-2.050909c-.065455 0-.109091 .043636-.109091 .109091c0 .032727 0 .054545 .207273 .250909c1.069091 1.08 1.690909 2.814545 1.690909 5.094544c0 1.865454-.403636 3.785453-1.756363 5.159999c-.141818 .130909-.141818 .152727-.141818 .185454c0 .065455 .043636 .109091 .109091 .109091c.109091 0 1.090909-.741818 1.734545-2.127272c.556363-1.2 .687273-2.410908 .687273-3.327272z"></path> | |
| <path id="s63g3-50" d="m1.385454-.84l1.156363-1.123636c1.701818-1.505454 2.356363-2.094545 2.356363-3.185454c0-1.243636-.981818-2.116363-2.312727-2.116363c-1.232727 0-2.039999 1.003636-2.039999 1.974545c0 .610909 .545454 .610909 .578182 .610909c.185454 0 .567273-.130909 .567273-.578182c0-.283636-.196364-.567273-.578182-.567273c-.087273 0-.109091 0-.141818 .010909c.250909-.709091 .84-1.112727 1.472727-1.112727c.992727 0 1.461818 .883636 1.461818 1.778181c0 .872727-.545454 1.734545-1.145454 2.410908l-2.094545 2.334545c-.12 .12-.12 .141818-.12 .403636h4.047272l.305454-1.898181h-.272727c-.054545 .327273-.130909 .807273-.24 .970909c-.076364 .087273-.796363 .087273-1.036363 .087273h-1.963636z"></path> | |
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| <script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("8.30655|39|18.75|"+window.location,"*");</script> | |
| </svg> and <svg class="svg-preview" id="s2tex_46" style="vertical-align:-8.30655pt; opacity: 1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="52" height="25" viewBox="1872.019782 1483.074098 31.083437 14.94396"> | |
| <defs> | |
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| <script type="text/ecmascript">if(window.parent.postMessage)window.parent.postMessage("8.30655|39|18.75|"+window.location,"*");</script> | |
| </svg> are already within 3% of each other), resulting in loss of accuracy.</li> | |
| </ol> | |
| <p class="line" data-line="163">Acknowledgements. This note is originally motivated by <a href="https://www.shadertoy.com/view/MXGSRc">this approximation</a>. | |
| The proof is inspired by Ward identities, as outlined in <a href="https://mathoverflow.net/questions/340108/challenge-non-gaussian-quartic-integral-and-path-integral-in-quantum-field-theo">this</a> | |
| topic. See also e.g. <a href="http://dx.doi.org/10.1088/1126-6708/2009/12/002">"Introduction to integral discriminants"</a> for more information. | |
| This document was prepared using <a href="https://upmath.me/">https://upmath.me/</a> editor.</p> |
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