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Bessel function - Revision history
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 21:58, 30 November 2018</td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''Bessel function''' referes to a solution of the [[Bessel equation]]</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''Bessel function''' referes to a solution<ins class="diffchange diffchange-inline"> $f$</ins> of the [[Bessel equation]]</div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">: </del>$ \!\!\!\!\!\!\!\!\! (1) ~ ~ ~ f''(z)+f'(z)/z+(1-\nu/z^2)f(x) =0$</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>$ \!\!\!\!\!\!\!\!\! (1) ~ ~ ~ f''(z)+f'(z)/z+(1-\nu/z^2)f(x) =0$</div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>with the following asymptotic behaviour:</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>$\displaystyle</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>f(x) \approx x^\nu \left( \frac{2^{-\nu}}{\mathrm{Factorial}(\nu)}+ O(x^2) \right)$</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In particular, </div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>For $\nu=0$, the initial condition is assumed $~f(0)=1~$, $~f'(0)=0$ , and</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>For $\nu=1$, the initial condition is assumed $~f(0)=0~$, $~f'(0)=1$;</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Such a solution $f$ denoted with $\mathrm{BesselJ}[\nu,x]$ or with $J_\nu(x)$.</div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==BesselJ==</div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>+ \frac{\mu\!-\!1}{2(2z)} </div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>+ \frac{\mu\!-\!1}{2(2z)} </div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>+ \frac{ (\mu\!-\!1) (\mu-25)}{6(4z)^3}</div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>+ \frac{ (\mu\!-\!1) (\mu-25)}{6(4z)^3}</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>+ \frac{ (\mu\!-\!1) (\mu^2-114\mu+1073)}{5(4z)^5} </div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>+ \frac{ (\mu\!-\!1) (\mu^2-114\mu+1073)}{5(4z)^5} <ins class="diffchange diffchange-inline">$ $+$ $~\, \displaystyle \frac{ (\mu\!-\!1) (5\mu^3-1535\mu^2+54703\mu-375733) }{14(4z)^7} + ..$</ins></div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>+ \frac{ (\mu\!-\!1) (5\mu^3-1535\mu^2+54703\mu-375733) }{14(4z)^7} + ..$</div></td>
<td colspan="2" class="diff-empty"> </td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>==Integrals==</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Some examples with Mathematica are shown below:</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><nomathjax></div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Integrate[BesselJ[0, x y] y, {y, 0, 1}]</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></nomathjax> $~$</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>does formula</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
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<td colspan="2" class="diff-empty"> </td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>$\displaystyle \int_0^1 J_0(x y) \, y\, \mathrm d y = \frac{ J_1(x)}{x}$</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><nomathjax></div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Simplify[Integrate[1/Sqrt[1-x^2] BesselJ[0, k x], {x, 0, 1}], k > 0]</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></nomathjax> $~$</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>does formula</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>$\displaystyle \int_0^1 \frac{1}{\sqrt{1-x^2}}\, J_0(kx) \, \mathrm d x = \frac{\pi}{2} J_0(k/2)^2$</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><nomathjax></div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Simplify[Integrate[1/Sqrt[1-x^2] BesselJ[0, k x] x, {x, 0, 1}], k > 0] </div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></nomathjax> $~$</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>does formula</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>$\displaystyle \int_0^1 \frac{1}{\sqrt{1-x^2}} \, J_0(kx) \, x \, \mathrm d x = \frac{\sin(k)}{k}$</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>These expressions can be used for testing of the numerical implementations of the [[Bessel transform]].</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Wolfram suggests more functions such that their Bessel transforms are expressed in terms of special functions at </div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>http://mathworld.wolfram.com/HankelTransform.html</div></td>
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<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>==Derivatives of the Bessel==</div></td>
</tr>
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<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>D[BesselJ[v, x], x] $~ ~$ gives</div></td>
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<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>1/2 (BesselJ[-1 + v, x] - BesselJ[1 + v, x])</div></td>
</tr>
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<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In "short" notations, this I can be written with</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>${J_v}^{\prime}(x)=\frac{1}{2}\big( </div></td>
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<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>J_{v-1}(x)-J_{v+1}(x)</div></td>
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<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\big)$</div></td>
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<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>==Zeros of the Bessel==</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Zeros of the Bessel function are demote with identifier BesselJZero; first argument indicates the order of the Bessel function; second one base sense of the radial coordinate.</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>For the numerical evaluation, the asymptotic expression can be used,</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>zo[k_] = Series[BesselJZero[v, k], {k, Infinity, 1}]</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>$\pi \left(k+\frac{v}{2}-\frac{1}{4}\right)-\frac{(2</div></td>
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<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> v-1) (2 v+1)}{8 \pi </div></td>
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<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> \left(k+\frac{v}{2}-\frac{1}{4}\right)}$</div></td>
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<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>For the precise evaluation at integer $v$, three steps of the Newton iteration to adjust the value seem to be sufficient, at least for the "double" precision.</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The [[C++]] implementation can be denoted [[jnz]] or, as in Mathematica, [[BesselJZero]]. The example and the test:</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><poem><nomathjax><nowiki></div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#include<math.h></div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#include<stdio.h></div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#define DB double</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#define DO(x,y) for(x=0;x<y;x++)</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>DB jnp(int n,DB x){ return .5*( jn(n-1,x)-jn(n+1,x) ) ; } // Derivative of n th Bessel</div></td>
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<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>DB jnz(int v, int k){ DB x,t; t=M_PI*(k+.5*v-.25); x= t - (v*v-.25)*.5/t;</div></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> x-= jn(v,x)/jnp(v,x); // Newton adjustment of the root</div></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> x-= jn(v,x)/jnp(v,x);</div></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> x-= jn(v,x)/jnp(v,x);</div></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> return x; } // the k th zero of v th Bessel</div></td>
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<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>int main(){ int n,v,k; DB x,y; // evaluation of zeros of Bessel of order 0,1,2,3 and the testing</div></td>
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<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>for(n=1;n<101;n++){ printf("%3d",n);</div></td>
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<tr>
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<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> DO(v,4){x=jnz(v,n); y=jn(v,x);</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> printf("%21.16lf %19.16lf",x,y);</div></td>
</tr>
<tr>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> }</div></td>
</tr>
<tr>
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<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> printf("\n");</div></td>
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<tr>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> }</div></td>
</tr>
<tr>
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<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>return 0;</div></td>
</tr>
<tr>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>}</div></td>
</tr>
<tr>
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<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></nowiki></nomathjax></poem></div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The first 10 lines of the output are copupasted below:</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> 1 2.4048255576957729 -0.0000000000000001 3.8317059702075125 -0.0000000000000000 5.1356223018406828 -0.0000000000000001 6.3801618959239841 -0.0000000000000002</div></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> 2 5.5200781102863106 -0.0000000000000000 7.0155866698156188 0.0000000000000000 8.4172441403998643 -0.0000000000000001 9.7610231299816697 -0.0000000000000000</div></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> 3 8.6537279129110125 -0.0000000000000001 10.1734681350627216 0.0000000000000001 11.6198411721490587 0.0000000000000002 13.0152007216984344 -0.0000000000000000</div></td>
</tr>
<tr>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> 4 11.7915344390142813 -0.0000000000000001 13.3236919363142228 -0.0000000000000001 14.7959517823512599 -0.0000000000000002 16.2234661603187682 0.0000000000000000</div></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> 5 14.9309177084877867 -0.0000000000000001 16.4706300508776344 -0.0000000000000003 17.9598194949878263 0.0000000000000000 19.4094152264350122 -0.0000000000000001</div></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> 6 18.0710639679109235 0.0000000000000002 19.6158585104682430 0.0000000000000002 21.1169970530218443 -0.0000000000000002 22.5827295931044425 0.0000000000000001</div></td>
</tr>
<tr>
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<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> 7 21.2116366298792585 0.0000000000000001 22.7600843805927724 -0.0000000000000001 24.2701123135731009 0.0000000000000003 25.7481666992949769 0.0000000000000001</div></td>
</tr>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> 8 24.3524715307493018 -0.0000000000000001 25.9036720876183821 -0.0000000000000001 27.4205735499845566 -0.0000000000000001 28.9083507809217579 0.0000000000000000</div></td>
</tr>
<tr>
<td colspan="2" class="diff-empty"> </td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> 9 27.4934791320402532 0.0000000000000002 29.0468285349168553 -0.0000000000000000 30.5692044955163986 -0.0000000000000002 32.0648524070977103 -0.0000000000000001</div></td>
</tr>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td>
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https://mizugadro.mydns.jp/t/index.php?title=Bessel_function&diff=775&oldid=prev
Maintenance script at 05:59, 20 June 2013
2013-06-20T05:59:27Z
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<p><b>New page</b></p><div>'''Bessel function''' referes to a solution of the [[Bessel equation]]<br />
<br />
: $ \!\!\!\!\!\!\!\!\! (1) ~ ~ ~ f''(z)+f'(z)/z+(1-\nu/z^2)f(x) =0$<br />
<br />
<br />
==BesselJ==<br />
<br />
Due to singularity of the equation at $z=0$, the regular solution should have specific behavior. This solution is called <br />
$J_\nu$. For $\nu=0$ and $\nu=1$, there are specific implementations [[BesselJ0]] and [[BesselJ1]].<br />
Many formulas about the Bessel functions below are borrowed from the handbook by [[Abramowirtz,Stegun]]<br />
<ref name="a"><br />
http://people.math.sfu.ca/~cbm/aands/page_365.htm<br />
Abramovitz, Stegun. Handbook on mathematical functions.<br />
</ref>; the numeration of formulas from there is used below.<br />
<br />
==Integral representations==<br />
<br />
: $ \!\!\!\!\!\!\!\!\!\! (9.1.20) ~ ~ ~ \displaystyle<br />
J_\nu(z) = \frac{(z/2)^{\nu}}{\pi^{1/2} ~(\nu-1/2)!}<br />
~<br />
\int_0^\pi<br />
~<br />
\cos(z \cos(t)) \sin(t)^{2 \nu} ~t~ \mathrm d t<br />
$<br />
<br />
===Sonin representation===<br />
The [[Mehler,Sonin]] formulas <br />
<ref><br />
http://dlmf.nist.gov/10.9 Digital library of mathematical functions<br />
</ref> suggest that<br />
<br />
: $\displaystyle J_\nu(z)=\frac{2}{\pi} \int_0 ^\infty \sin(x \cos(t) - \pi \nu/2) \cos(\nu t) \mathrm d t$<br />
<br />
: $\displaystyle Y_\nu(z)=\frac{-2}{\pi} \int_0 ^\infty \cos(x \cos(t) - \pi \nu/2) \cos(\nu t) \mathrm d t$<br />
<br />
and, in particular,<br />
<br />
: $\displaystyle J_0(x)=\frac{2}{\pi} \int_0^\infty \sin(x \cosh(t)) \mathrm d t$<br />
<br />
: $\displaystyle Y_0(x)=\frac{-2}{\pi} \int_0^\infty \cos(x \cosh(t)) \mathrm d t$<br />
<br />
Also,<br />
: $\displaystyle J_\nu(x)=\frac{2 (x/2)^{-\nu}}{\pi^{1/2} \Gamma(1/2-\nu)}<br />
\int_1^\infty \frac{\sin(xt)~ \mathrm d t} { (t^2-1)^{\nu+1/2}}$<br />
<br />
: $\displaystyle Y_\nu(x)=-\frac{2 (x/2)^{-\nu}}{\pi^{1/2} \Gamma(1/2-\nu)}<br />
\int_1^\infty \frac{\cos(xt)~ \mathrm d t} { (t^2-1)^{\nu+1/2}}$<br />
<br />
<br />
However the reason of the suggested restriction $x>0$ is not clear.<br />
Peerhaps, these expressions can be used to deduce the expansion suitable for the numerical implementation.<br />
<br />
==Expansion of $J_\nu$ at zero==<br />
: $\!\!\!\!\!\!\!\!\!\!\!\!\! (9.1.10) ~ ~ ~ \displaystyle<br />
J_\nu(z)=\left(\frac{z}{2}\right)^{\!\nu}~<br />
\sum_{k=0}^{\infty} ~<br />
\frac{(-z^2/4)^k}<br />
{k!~ (\nu\!+\!k)!}$<br />
<br />
==Expansion of $Y_n$ at zero==<br />
<br />
The similar expansion for $Y_n$ at natural $n\!+\!1$ looks ugly:<br />
<br />
: $\!\!\!\!\!\!\!\!\!\!\!\!\! (\mathrm{GR} 8.403) ~ ~ ~ \displaystyle<br />
\pi Y_n(z)= 2 J_n(z) ( \ln(z/2) + C ) - \sum_{k=0}^{n-1} \frac{(n\!-\!k\!-\!1)!}{k!} (z/2)^{2k-n} -$<br />
: $ \displaystyle<br />
- (z/2)^n \frac{1}{n!} \sum_{k=1}^n \frac{1}{k} - \sum_{k=0}^{\infty} \frac{(-1)^k (z/2)^{n+2k}}{k! ~(n\!+\!k)!}<br />
\left(<br />
\sum_{m=1}^{n+k} \frac{1}{m} + \sum_{m=1}^k \frac{1}{m} \right)<br />
$<br />
where $C$ is [[Euler constant]], called also [[EulerGamma]]<br />
: $\displaystyle<br />
C=- \int_0^\infty \exp(-t)~ \ln(t) ~\mathrm d t \approx 0.57721566490$<br />
<br />
Up to year 2012, no beautiful representation for the expansion coefficients is available.<br />
<br />
==Expansion at infinity by [[Gradshtein,Ryzhik]]==<br />
<br />
[[Gradshtein,Ryzhik]] <ref name="gr"><br />
http://books.google.co.jp/books?id=aBgFYxKHUjsC&pg=PA859&hl=ja&source=gbs_toc_r&cad=4#v=onepage&q&f=false<br />
[[Izrail Solomonovich Gradshtein]], [[Iosif Moiseevich Ryzhik]], Alan Jeffrey, Daniel Zwillinger.<br />
Table of Integrals, Series, And Products. <br />
</ref> suggest the following expansions (See 8.4.5.5)<br />
<!--<br />
; the coefficient of order of 2 seems to be lost:<br />
!--><br />
<br />
: $ \displaystyle J_{\pm \nu}(z)=$<br />
: $ \displaystyle = \sqrt{\frac{2}{\pi z}} \cos\!\Big(z-(\pm 2 \nu\!+\!1)\pi/4 \Big) ~ \left( ~ \sum_{k=0}^{n-1} ~<br />
\left(\frac{-1}{4z^2}\right)^{\!\! k}<br />
\frac{ \Gamma(\nu+2k+1/2)}{(2k)!~ \Gamma(\nu-2k+1/2)} + R_1 \right) -$<br />
:$ \displaystyle - \sqrt{\frac{2}{\pi z}} \sin\!\Big(z-(\pm 2 \nu\!+\!1)\pi/4 \Big)~ \left( \frac{1}{2z} ~ \sum_{k=0}^{n-1} ~ <br />
\left(\frac{-1}{4z^2}\right)^{\!\! k}<br />
\frac{ \Gamma(\nu+2k+3/2)}{(2k\!+\!1)! ~\Gamma(\nu-2k-1/2)} + R_2 \right)$<br />
<br />
: $ \displaystyle Y_{\pm \nu}(z)=$<br />
: $ \displaystyle = \sqrt{\frac{2}{\pi z}} \sin\!\Big(z-(\pm 2 \nu\!+\!1)\pi/4 \Big) ~ \left( ~ \sum_{k=0}^{n-1} ~<br />
\left(\frac{-1}{4z^2}\right)^{\!\! k}<br />
\frac{ \Gamma(\nu+2k+1/2)}{(2k)!~ \Gamma(\nu-2k+1/2)} + R_1 \right) +$<br />
:$ \displaystyle + \sqrt{\frac{2}{\pi z}} \cos\!\Big(z-(\pm 2 \nu\!+\!1)\pi/4 \Big)~ \left( \frac{1}{2z} ~ \sum_{k=0}^{n-1} ~ <br />
\left(\frac{-1}{4z^2}\right)^{\!\! k}<br />
\frac{ \Gamma(\nu+2k+3/2)}{(2k\!+\!1)! ~\Gamma(\nu-2k-1/2)} + R_2 \right)$<br />
<br />
: $ \displaystyle H_{\nu}(z)=$<br />
: $ \displaystyle = \sqrt{\frac{2}{\pi z}} \exp\!\Big(z-(\pm 2 \nu\!+\!1)\pi/4 \Big) ~ \left( ~ \sum_{k=0}^{n-1} ~<br />
\left(\frac{\mathrm i}{2z}\right)^{\!\! k}<br />
\frac{ \Gamma(\nu+n+1/2)}{(2n)!~ \Gamma(\nu-n+1/2)} + <br />
\theta_1 \left(\frac{\mathrm i}{2z}\right)^{\!\! k}<br />
\frac{ \Gamma(\nu+n+1/2)}{(2n)!~ \Gamma(\nu-n+1/2)} \right) $<br />
<br />
: $ \displaystyle<br />
|R_1|< \left|<br />
\frac{\Gamma(\nu+2n+1/2)}{(2z)^{2n} ~ (2n)! ~ \Gamma(\nu-2n+1/2)}<br />
\right|$<br />
<br />
: $ \displaystyle<br />
|R_2|< \left|<br />
\frac{\Gamma(\nu+2n+3/2)}{(2z)^{2n+1} ~ (2n\!+\!1)! ~ \Gamma(\nu-2n-1/2)}<br />
\right|$<br />
<br />
: while $\Im(z)\le 0$, the esitmate $|\theta_1| < 1 $<br />
<br />
For half–natural $\nu$, the singularity of $\Gamma$ terminates the series and they become finite sums.<br />
<br />
==Expansion at infinity from [[Abramowitz,Stegun]]==<br />
Let $\mu=4 \nu^2$. Define two series $P_\nu(z)$ and $Q_\nu(z)$ with<br />
<br />
: $\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9.2.09) ~ ~ ~ \displaystyle<br />
P_\nu(z)=1 <br />
-\frac{(\mu\!-\!1)(\mu\!-\!9)}{2! ~ (8z)^2}<br />
+\frac{(\mu\!-\!1)(\mu\!-\!9)(\mu\!-\!25)(\mu\!-\!49)}{4! ~ (8z)^4}<br />
- ..$<br />
: $\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9.2.10) ~ ~ ~ \displaystyle<br />
Q_\nu(z)=<br />
\frac{(\mu\!-\!1)}{1! ~ (8z)}<br />
+\frac{(\mu\!-\!1)(\mu\!-\!9)(\mu\!-\!25)}{3! ~ (8z)^3}<br />
- \frac{(\mu\!-\!1)(\mu\!-\!9)(\mu\!-\!25)(\mu\!-\!49)(\mu\!-\!81)}{5! ~ (8z)^5}<br />
+..$<br />
Let $x=z-\Big(\nu/2+\pi/4\Big)\pi~$;<br />
then<br />
: $\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9.2.05) ~ ~ ~ \displaystyle<br />
J_\nu(z_)=\sqrt{\frac{2}{\pi z}}\Big(P_\nu(z) \cos(z)-Q_\nu(z) \sin(z) \Big)$<br />
<br />
: $\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9.2.06) ~ ~ ~ \displaystyle<br />
Y_\nu(z_)=\sqrt{\frac{2}{\pi z}}\Big(P_\nu(z) \sin(z)+Q_\nu(z) \cos(z) \Big)$<br />
<br />
: $\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9.2.07) ~ ~ ~ \displaystyle<br />
H_\nu(z_)=\sqrt{\frac{2}{\pi z}}\Big(P_\nu(z) + \mathrm i ~ Q_\nu(z) \Big) \mathrm e^{\mathrm i z}$<br />
<br />
while $|\mathrm{Arg}(z)|<\pi$.<br />
<br />
==Asymptotic expansion for modulus and phase==<br />
<br />
: $ \!\!\!\!\!\!\!\!\! (9.2.19) ~ ~ ~ J_\nu(z)= M \cos(\theta) ~~, ~~$ $~ Y_\nu(z)= M \sin(\theta) ~~, ~~$<br />
<br />
: $ \!\!\!\!\!\!\!\!\! (9.2.17) ~ ~ ~ M = \sqrt{J_\nu(z)^2 + Y_\nu(z)^2}$ <br />
<br />
In certain range, while $ |\Re(\theta)|<\pi$, also<br />
: $ \!\!\!\!\!\!\!\!\! (9.2.0) ~ ~ ~ \theta = \mathrm{atan2}( Y_\nu(z)/ J_\nu(z))$<br />
<br />
$M$ is called "modulus" and $\theta$ is called "argument" <br />
<ref name="a"><br />
http://people.math.sfu.ca/~cbm/aands/page_365.htm<br />
Abramovitz, Stegun. Handbook on mathematical functions.<br />
</ref>. Let $\mu=4\nu^2$.<br />
<br />
At large values of the argument, it worth to expand $M$ and $\theta$ instead of $J$:<br />
<br />
: $ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9.2.28) \displaystyle ~ ~ ~ M^2 = \frac{2}{\pi z} \left( 1 + <br />
\frac{1}{2} \frac{\mu \!-\!1}{(2z)^2} +<br />
\frac{1\cdot 3}{2 \cdot 4} \frac{(\mu \!-\!1)(\mu\!-\!9)}{(2z)^4} +<br />
\frac{1\cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{(\mu\!-\!1)(\mu\!-\!9)(\mu\!-\!25)}{(2z)^6} +..<br />
\right)<br />
$<br />
<br />
: $ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9.2.29) \displaystyle ~ ~ ~ \theta = z - \left(\frac{\nu}{2}+\frac{1}{4}\right) \pi <br />
+ \frac{\mu\!-\!1}{2(2z)} <br />
+ \frac{ (\mu\!-\!1) (\mu-25)}{6(4z)^3}<br />
+ \frac{ (\mu\!-\!1) (\mu^2-114\mu+1073)}{5(4z)^5} <br />
+ \frac{ (\mu\!-\!1) (5\mu^3-1535\mu^2+54703\mu-375733) }{14(4z)^7} + ..$<br />
<br />
==References==<br />
<references/><br />
<br />
http://en.wikipedia.org/wiki/Bessel_function<br />
<br />
http://en.citizendium.org/wiki/Bessel_functions<br />
<br />
==Keywords==<br />
[[BesselJ0]],<br />
[[BesselJ1]],<br />
[[BesselY0]],<br />
[[BesselY1]],<br />
<br />
[[Category:Bessel function]]<br />
[[Category:Cylindric function]]<br />
[[Category:BesselJ0]]<br />
[[Category:BesselJ1]]<br />
[[Category:Entire function]]<br />
[[Category:Articles in English]]</div>
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