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Bessel functions

Table of contents: Differential equations - Derivatives and recurrence relations - Hypergeometric representations - Connection formulas - Specific values - Hankel functions - Integral representations - Bounds and inequalities

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Symbol: BesselJ Jν ⁣(z)J_{\nu}\!\left(z\right) Bessel function of the first kind
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Symbol: BesselY Yν ⁣(z)Y_{\nu}\!\left(z\right) Bessel function of the second kind
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Symbol: BesselI Iν ⁣(z)I_{\nu}\!\left(z\right) Modified Bessel function of the first kind
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Symbol: BesselK Kν ⁣(z)K_{\nu}\!\left(z\right) Modified Bessel function of the second kind

Differential equations

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z2Jν ⁣(z)+zJν ⁣(z)+(z2ν2)Jν ⁣(z)=0{z}^{2} J''_{\nu}\!\left(z\right) + z J'_{\nu}\!\left(z\right) + \left({z}^{2} - {\nu}^{2}\right) J_{\nu}\!\left(z\right) = 0
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z2Yν ⁣(z)+zYν ⁣(z)+(z2ν2)Yν ⁣(z)=0{z}^{2} Y''_{\nu}\!\left(z\right) + z Y'_{\nu}\!\left(z\right) + \left({z}^{2} - {\nu}^{2}\right) Y_{\nu}\!\left(z\right) = 0
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z2Iν ⁣(z)+zIν ⁣(z)(z2+ν2)Iν ⁣(z)=0{z}^{2} I''_{\nu}\!\left(z\right) + z I'_{\nu}\!\left(z\right) - \left({z}^{2} + {\nu}^{2}\right) I_{\nu}\!\left(z\right) = 0
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z2Kν ⁣(z)+zKν ⁣(z)(z2+ν2)Kν ⁣(z)=0{z}^{2} K''_{\nu}\!\left(z\right) + z K'_{\nu}\!\left(z\right) - \left({z}^{2} + {\nu}^{2}\right) K_{\nu}\!\left(z\right) = 0

Derivatives and recurrence relations

Related topics: Recurrence relations for Bessel functions

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Jν ⁣(z)=z2ν(Jν1 ⁣(z)+Jν+1 ⁣(z))J_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(J_{\nu - 1}\!\left(z\right) + J_{\nu + 1}\!\left(z\right)\right)
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Yν ⁣(z)=z2ν(Yν1 ⁣(z)+Yν+1 ⁣(z))Y_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(Y_{\nu - 1}\!\left(z\right) + Y_{\nu + 1}\!\left(z\right)\right)
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Iν ⁣(z)=z2ν(Iν1 ⁣(z)Iν+1 ⁣(z))I_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(I_{\nu - 1}\!\left(z\right) - I_{\nu + 1}\!\left(z\right)\right)
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Kν ⁣(z)=z2ν(Kν1 ⁣(z)Kν+1 ⁣(z))K_{\nu}\!\left(z\right) = -\frac{z}{2 \nu} \left(K_{\nu - 1}\!\left(z\right) - K_{\nu + 1}\!\left(z\right)\right)
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Jν ⁣(z)=Jν1 ⁣(z)Jν+1 ⁣(z)2J'_{\nu}\!\left(z\right) = \frac{J_{\nu - 1}\!\left(z\right) - J_{\nu + 1}\!\left(z\right)}{2}
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Yν ⁣(z)=Yν1 ⁣(z)Yν+1 ⁣(z)2Y'_{\nu}\!\left(z\right) = \frac{Y_{\nu - 1}\!\left(z\right) - Y_{\nu + 1}\!\left(z\right)}{2}
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Iν ⁣(z)=Iν1 ⁣(z)+Iν+1 ⁣(z)2I'_{\nu}\!\left(z\right) = \frac{I_{\nu - 1}\!\left(z\right) + I_{\nu + 1}\!\left(z\right)}{2}
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Kν ⁣(z)=Kν1 ⁣(z)+Kν+1 ⁣(z)2K'_{\nu}\!\left(z\right) = -\frac{K_{\nu - 1}\!\left(z\right) + K_{\nu + 1}\!\left(z\right)}{2}

Hypergeometric representations

Related topics: Hypergeometric representations of Bessel functions

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Jν ⁣(z)=(z2)ν0F1 ⁣(ν+1,z24)J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, -\frac{{z}^{2}}{4}\right)
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Iν ⁣(z)=(z2)ν0F1 ⁣(ν+1,z24)I_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, \frac{{z}^{2}}{4}\right)
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Kν ⁣(z)=(2zπ)1/2ezU ⁣(ν+12,2ν+1,2z)K_{\nu}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{-1 / 2} {e}^{-z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 z\right)

Connection formulas

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Jn ⁣(z)=(1)nJn ⁣(z)J_{-n}\!\left(z\right) = {\left(-1\right)}^{n} J_{n}\!\left(z\right)
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In ⁣(z)=In ⁣(z)I_{-n}\!\left(z\right) = I_{n}\!\left(z\right)
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Iν ⁣(z)=zν(iz)νJν ⁣(iz)I_{\nu}\!\left(z\right) = {z}^{\nu} {\left(i z\right)}^{-\nu} J_{\nu}\!\left(i z\right)
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In ⁣(z)=inJn ⁣(iz)I_{n}\!\left(z\right) = {i}^{-n} J_{n}\!\left(i z\right)
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Yν ⁣(z)=cos ⁣(πν)Jν ⁣(z)Jν ⁣(z)sin ⁣(πν)Y_{\nu}\!\left(z\right) = \frac{\cos\!\left(\pi \nu\right) J_{\nu}\!\left(z\right) - J_{-\nu}\!\left(z\right)}{\sin\!\left(\pi \nu\right)}
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Yn ⁣(z)=2π(inKn ⁣(iz)+(log ⁣(iz)log(z))Jn ⁣(z))Y_{n}\!\left(z\right) = -\frac{2}{\pi} \left({i}^{n} K_{n}\!\left(i z\right) + \left(\log\!\left(i z\right) - \log(z)\right) J_{n}\!\left(z\right)\right)

Specific values

Related topics: Specific values of Bessel functions

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J1/2 ⁣(z)=(2zπ)1/2sin(z)zJ_{1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sin(z)}{z}
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K1/2 ⁣(z)=(πz2)1/2ezzK_{1 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} \frac{{e}^{-z}}{z}

Hankel functions

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Symbol: HankelH1 Hν(1) ⁣(z)H^{(1)}_{\nu}\!\left(z\right) Hankel function of the first kind
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Symbol: HankelH2 Hν(2) ⁣(z)H^{(2)}_{\nu}\!\left(z\right) Hankel function of the second kind
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Hν(1) ⁣(z)=Jν ⁣(z)+iYν ⁣(z)H^{(1)}_{\nu}\!\left(z\right) = J_{\nu}\!\left(z\right) + i Y_{\nu}\!\left(z\right)
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Hν(2) ⁣(z)=Jν ⁣(z)iYν ⁣(z)H^{(2)}_{\nu}\!\left(z\right) = J_{\nu}\!\left(z\right) - i Y_{\nu}\!\left(z\right)

Integral representations

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Jn ⁣(z)=1π0πcos ⁣(ntzsin(t))dtJ_{n}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \cos\!\left(n t - z \sin(t)\right) \, dt
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Jν ⁣(z)=1π0πcos ⁣(νtzsin(t))dtsin ⁣(πν)π0exp ⁣(zsinh(t)νt)dtJ_{\nu}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \cos\!\left(\nu t - z \sin(t)\right) \, dt - \frac{\sin\!\left(\pi \nu\right)}{\pi} \int_{0}^{\infty} \exp\!\left(-z \sinh(t) - \nu t\right) \, dt
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Iν ⁣(z)=1π0πexp ⁣(zcos(t))cos ⁣(νt)dtsin ⁣(πν)π0exp ⁣(zcosh(t)νt)dtI_{\nu}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \exp\!\left(z \cos(t)\right) \cos\!\left(\nu t\right) \, dt - \frac{\sin\!\left(\pi \nu\right)}{\pi} \int_{0}^{\infty} \exp\!\left(-z \cosh(t) - \nu t\right) \, dt
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Kν ⁣(z)=0exp ⁣(zcosh(t))cosh ⁣(νt)dtK_{\nu}\!\left(z\right) = \int_{0}^{\infty} \exp\!\left(-z \cosh(t)\right) \cosh\!\left(\nu t\right) \, dt

Bounds and inequalities

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Jν ⁣(x)1\left|J_{\nu}\!\left(x\right)\right| \le 1
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Jν ⁣(x)0.6749ν1/3\left|J_{\nu}\!\left(x\right)\right| \le 0.6749 {\nu}^{-1 / 3}
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Jν ⁣(x)0.7858x1/3\left|J_{\nu}\!\left(x\right)\right| \le 0.7858 {x}^{-1 / 3}
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Jn ⁣(z)exp ⁣(Im(z))\left|J_{n}\!\left(z\right)\right| \le \exp\!\left(\left|\operatorname{Im}(z)\right|\right)
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Jν ⁣(z)1Γ ⁣(ν+1)z2νexp ⁣(Im(z))\left|J_{\nu}\!\left(z\right)\right| \le \frac{1}{\Gamma\!\left(\nu + 1\right)} {\left|\frac{z}{2}\right|}^{\nu} \exp\!\left(\left|\operatorname{Im}(z)\right|\right)

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-11-19 15:10:20.037976 UTC