Difference between revisions of "Modified Bessel I"
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The modified Bessel function of the first kind is defined by | The modified Bessel function of the first kind is defined by | ||
$$I_{\nu}(z)=i^{-\nu}J_{\nu}(iz),$$ | $$I_{\nu}(z)=i^{-\nu}J_{\nu}(iz),$$ | ||
| − | where $J_{\nu}$ is the [[Bessel J | + | where $J_{\nu}$ is the [[Bessel J|Bessel function of the first kind]]. |
<div align="center"> | <div align="center"> | ||
Revision as of 20:10, 9 June 2016
The modified Bessel function of the first kind is defined by $$I_{\nu}(z)=i^{-\nu}J_{\nu}(iz),$$ where $J_{\nu}$ is the Bessel function of the first kind.
Domain coloring of analytic continuation of $I_1(z)$.
Modified Bessel functions from Abramowitz&Stegun.
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Properties
Relationship between Bessel I sub -1/2 and cosh
Relationship between Bessel I sub 1/2 and sinh
Relationship between Bessel I sub n and Bessel J sub n
Relationship between Airy Bi and modified Bessel I
Proposition: The following formula holds: $$I_{\nu}(z)=\displaystyle\sum_{k=0}^{\infty} J_{\nu+k}(z) \dfrac{z^k}{k!},$$ where $J_{\nu}$ denotes the Bessel function of the first kind.
Proof: █
Bessel $I_{\nu}$
