# Hypergeometric representations of Bessel functions

Related topics: Bessel functions, Confluent hypergeometric functions

$J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, -\frac{{z}^{2}}{4}\right)$
$I_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, \frac{{z}^{2}}{4}\right)$
$Y_{\nu}\!\left(z\right) = \frac{1}{\sin\!\left(\pi \nu\right)} \left(\cos\!\left(\pi \nu\right) {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, -\frac{{z}^{2}}{4}\right) - {\left(\frac{z}{2}\right)}^{-\nu} \,{}_0{\textbf F}_1\!\left(1 - \nu, -\frac{{z}^{2}}{4}\right)\right)$
$K_{\nu}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{-1 / 2} {e}^{-z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 z\right)$
$K_{\nu}\!\left(z\right) = \frac{1}{2} \frac{\pi}{\sin\!\left(\pi \nu\right)} \left({\left(\frac{z}{2}\right)}^{-\nu} \,{}_0{\textbf F}_1\!\left(1 - \nu, \frac{{z}^{2}}{4}\right) - {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(1 + \nu, \frac{{z}^{2}}{4}\right)\right)$
$J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \frac{{e}^{-i z}}{\Gamma\!\left(\nu + 1\right)} \,{}_1F_1\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)$
$J_{\nu}\!\left(z\right) = \frac{{z}^{\nu}}{{\left(2 \pi\right)}^{1 / 2}} \left({\left(i z\right)}^{-1 / 2 - \nu} {e}^{i z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, -2 i z\right) + {\left(-i z\right)}^{-1 / 2 - \nu} {e}^{-i z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)\right)$
$J_{\nu}\!\left(z\right) = {\left(2 \pi z\right)}^{-1 / 2} \left({e}^{-i \theta} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, -2 i z\right) + {e}^{i \theta} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)\right)\; \text{ where } \theta = \frac{\pi \left(2 \nu + 1\right)}{4} - z$

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

2021-03-15 19:12:00.328586 UTC