Bessel functions

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

Differential equations

${z}^{2} J''_{\nu}\!\left(z\right) + z J'_{\nu}\!\left(z\right) + \left({z}^{2} - {\nu}^{2}\right) J_{\nu}\!\left(z\right) = 0$
${z}^{2} Y''_{\nu}\!\left(z\right) + z Y'_{\nu}\!\left(z\right) + \left({z}^{2} - {\nu}^{2}\right) Y_{\nu}\!\left(z\right) = 0$
${z}^{2} I''_{\nu}\!\left(z\right) + z I'_{\nu}\!\left(z\right) - \left({z}^{2} + {\nu}^{2}\right) I_{\nu}\!\left(z\right) = 0$
${z}^{2} K''_{\nu}\!\left(z\right) + z K'_{\nu}\!\left(z\right) - \left({z}^{2} + {\nu}^{2}\right) K_{\nu}\!\left(z\right) = 0$

Derivatives and recurrence relations

Related topics: Recurrence relations for Bessel functions

$J_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(J_{\nu - 1}\!\left(z\right) + J_{\nu + 1}\!\left(z\right)\right)$
$Y_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(Y_{\nu - 1}\!\left(z\right) + Y_{\nu + 1}\!\left(z\right)\right)$
$I_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(I_{\nu - 1}\!\left(z\right) - I_{\nu + 1}\!\left(z\right)\right)$
$K_{\nu}\!\left(z\right) = -\frac{z}{2 \nu} \left(K_{\nu - 1}\!\left(z\right) - K_{\nu + 1}\!\left(z\right)\right)$
$J'_{\nu}\!\left(z\right) = \frac{J_{\nu - 1}\!\left(z\right) - J_{\nu + 1}\!\left(z\right)}{2}$
$Y'_{\nu}\!\left(z\right) = \frac{Y_{\nu - 1}\!\left(z\right) - Y_{\nu + 1}\!\left(z\right)}{2}$
$I'_{\nu}\!\left(z\right) = \frac{I_{\nu - 1}\!\left(z\right) + I_{\nu + 1}\!\left(z\right)}{2}$
$K'_{\nu}\!\left(z\right) = -\frac{K_{\nu - 1}\!\left(z\right) + K_{\nu + 1}\!\left(z\right)}{2}$

Hypergeometric representations

Related topics: Hypergeometric representations of Bessel functions

$J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, -\frac{{z}^{2}}{4}\right)$
$I_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, \frac{{z}^{2}}{4}\right)$
$K_{\nu}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{-1 / 2} {e}^{-z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 z\right)$

Connection formulas

$J_{-n}\!\left(z\right) = {\left(-1\right)}^{n} J_{n}\!\left(z\right)$
$I_{-n}\!\left(z\right) = I_{n}\!\left(z\right)$
$I_{\nu}\!\left(z\right) = {z}^{\nu} {\left(i z\right)}^{-\nu} J_{\nu}\!\left(i z\right)$
$I_{n}\!\left(z\right) = {i}^{-n} J_{n}\!\left(i z\right)$
$Y_{\nu}\!\left(z\right) = \frac{\cos\!\left(\pi \nu\right) J_{\nu}\!\left(z\right) - J_{-\nu}\!\left(z\right)}{\sin\!\left(\pi \nu\right)}$
$Y_{n}\!\left(z\right) = -\frac{2}{\pi} \left({i}^{n} K_{n}\!\left(i z\right) + \left(\log\!\left(i z\right) - \log(z)\right) J_{n}\!\left(z\right)\right)$

Specific values

Related topics: Specific values of Bessel functions

$J_{1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sin(z)}{z}$
$K_{1 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} \frac{{e}^{-z}}{z}$

Hankel functions

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

Integral representations

$J_{n}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \cos\!\left(n t - z \sin(t)\right) \, dt$
$J_{\nu}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \cos\!\left(\nu t - z \sin(t)\right) \, dt - \frac{\sin\!\left(\pi \nu\right)}{\pi} \int_{0}^{\infty} \exp\!\left(-z \sinh(t) - \nu t\right) \, dt$
$I_{\nu}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \exp\!\left(z \cos(t)\right) \cos\!\left(\nu t\right) \, dt - \frac{\sin\!\left(\pi \nu\right)}{\pi} \int_{0}^{\infty} \exp\!\left(-z \cosh(t) - \nu t\right) \, dt$
$K_{\nu}\!\left(z\right) = \int_{0}^{\infty} \exp\!\left(-z \cosh(t)\right) \cosh\!\left(\nu t\right) \, dt$

Bounds and inequalities

$\left|J_{\nu}\!\left(x\right)\right| \le 1$
$\left|J_{\nu}\!\left(x\right)\right| \le 0.6749 {\nu}^{-1 / 3}$
$\left|J_{\nu}\!\left(x\right)\right| \le 0.7858 {x}^{-1 / 3}$
$\left|J_{n}\!\left(z\right)\right| \le {e}^{\left|\operatorname{Im}(z)\right|}$
$\left|J_{\nu}\!\left(z\right)\right| \le \frac{1}{\Gamma\!\left(\nu + 1\right)} {\left|\frac{z}{2}\right|}^{\nu} {e}^{\left|\operatorname{Im}(z)\right|}$

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

2020-01-31 18:09:28.494564 UTC