Difference between revisions of "Modified Bessel I"
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$$I_{\nu}(z)=i^{-\nu}J_{\nu}(iz),$$ | $$I_{\nu}(z)=i^{-\nu}J_{\nu}(iz),$$ | ||
where $J_{\nu}$ is the [[Bessel J sub nu|Bessel function of the first kind]]. | where $J_{\nu}$ is the [[Bessel J sub nu|Bessel function of the first kind]]. | ||
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| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Complex modified besselI sub 1.png|[[Domain coloring]] of [[analytic continuation]]. | ||
| + | </gallery> | ||
| + | </div> | ||
=Properties= | =Properties= | ||
Revision as of 06:23, 18 May 2015
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.
Contents
Properties
Proposition: The following formula holds: $$I_{-\frac{1}{2}}(z)=\sqrt{\dfrac{2}{\pi z}} \cosh(z).$$
Proof: █
Proposition: The following formula holds: $$I_{\frac{1}{2}}(z)=\sqrt{\dfrac{2}{\pi z}}\sinh(z).$$
Proof: █
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: █
Theorem
The following formula holds: $$\mathrm{Bi}(z)=\sqrt{\dfrac{z}{3}} \left( I_{\frac{1}{3}}\left(\frac{2}{3}x^{\frac{3}{2}} \right) + I_{-\frac{1}{3}} \left( \frac{2}{3} x^{\frac{3}{2}} \right) \right),$$ where $\mathrm{Bi}$ denotes the Airy Bi function and $I_{\nu}$ denotes the modified Bessel $I$.
