# Specific values of Bessel functions

Related topics: Bessel functions

## Trigonometric cases

$J_{-1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\cos(z)}{z}$
$J_{1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sin(z)}{z}$
$J_{3 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \left(\frac{\sin(z)}{{z}^{2}} - \frac{\cos(z)}{z}\right)$
$Y_{-1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sin(z)}{z}$
$Y_{1 / 2}\!\left(z\right) = -{\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\cos(z)}{z}$
$Y_{3 / 2}\!\left(z\right) = -{\left(\frac{2 z}{\pi}\right)}^{1 / 2} \left(\frac{\cos(z)}{{z}^{2}} + \frac{\sin(z)}{z}\right)$

## Hyperbolic cases

$I_{-1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\cosh(z)}{z}$
$I_{1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sinh(z)}{z}$
$I_{3 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \left(\frac{\cosh(z)}{z} - \frac{\sinh(z)}{{z}^{2}}\right)$
$K_{-1 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} \frac{{e}^{-z}}{z}$
$K_{1 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} \frac{{e}^{-z}}{z}$
$K_{3 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} {e}^{-z} \left(\frac{1}{z} + \frac{1}{{z}^{2}}\right)$

## Airy function cases

Related topics: Airy functions

$J_{-1 / 3}\!\left(z\right) = \frac{1}{2 \omega} \left(3 \operatorname{Ai}\!\left(-{\omega}^{2}\right) + \sqrt{3} \operatorname{Bi}\!\left(-{\omega}^{2}\right)\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}$
$J_{1 / 3}\!\left(z\right) = \frac{1}{2 \omega} \left(3 \operatorname{Ai}\!\left(-{\omega}^{2}\right) - \sqrt{3} \operatorname{Bi}\!\left(-{\omega}^{2}\right)\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}$
$J_{2 / 3}\!\left(z\right) = \frac{1}{2 {\omega}^{2}} \left(3 \operatorname{Ai}'\!\left(-{\omega}^{2}\right) + \sqrt{3} \operatorname{Bi}'\!\left(-{\omega}^{2}\right)\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}$
$K_{-1 / 3}\!\left(z\right) = \frac{\sqrt{3} \pi}{\omega} \operatorname{Ai}\!\left({\omega}^{2}\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}$
$K_{1 / 3}\!\left(z\right) = \frac{\sqrt{3} \pi}{\omega} \operatorname{Ai}\!\left({\omega}^{2}\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}$
$K_{2 / 3}\!\left(z\right) = -\frac{\sqrt{3} \pi}{{\omega}^{2}} \operatorname{Ai}'\!\left({\omega}^{2}\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}$

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