Modified Bessel functions

Builds on Bessel function of the first kind Gamma function

Definition

Let $\nu \in \C$ , let $\Gamma$ be the gamma function, and let $z \in \C \setminus (-\infty, 0]$ with the principal branch of $z^\nu$ .

The modified Bessel function of the first kind of order $\nu$ is $I_\nu(z) = \sum_{m=0}^{\infty} \frac{1}{m!\, \Gamma(m + \nu + 1)} \left( \frac{z}{2} \right)^{2m + \nu},$ where $\frac{1}{\Gamma(m + \nu + 1)}$ is read as $0$ when $m + \nu + 1$ is a non-positive integer.
For $\nu \notin \Z$ , the modified Bessel function of the second kind of order $\nu$ is $K_\nu(z) = \frac{\pi}{2}\, \frac{I_{-\nu}(z) - I_\nu(z)}{\sin(\nu \pi)},$ and for an integer $n$ it is $K_n(z) = \lim_{\nu \to n} K_\nu(z)$ .

Remarks

Both functions solve the modified Bessel equation $z^2 y'' + z y' - (z^2 + \nu^2) y = 0$ . In terms of the Bessel function of the first kind, $I_\nu(z) = i^{-\nu} J_\nu(iz)$ for $-\pi < \arg z \le \frac{\pi}{2}$ . Example: for $x > 0$ , $I_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sinh x$ and $K_{1/2}(x) = \sqrt{\frac{\pi}{2x}}\, e^{-x}$ . For $x > 0$ there is the integral form $K_\nu(x) = \int_0^\infty e^{-x \cosh t} \cosh(\nu t)\, dt$ . A non-example: $I_{-n}$ for an integer $n$ is not a second solution, since $I_{-n} = I_n$ ; this is why $K_n$ is defined as a limit. $K_\nu$ is also called the Macdonald function.

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Bessel function of the first kind concept
Gamma function concept
Modified Bessel functions definition

Tag	`VSD1D6ASVH`
URI	`m2://VSD1D6ASVH`
Kind	definition
Writing	W0 stub
MSC	33C10, 34B30
Defines	Modified Bessel functions
Uses	Bessel function of the first kind, Gamma function

Modified Bessel functions

Definition

Remarks

Reading path

Further reading

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