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]]. | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Proposition:</strong> The following formula holds: | ||
| + | $$I_{-\frac{1}{2}}(z)=\sqrt{\dfrac{2}{\pi z}} \cosh(z).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Proposition:</strong> The following formula holds: | ||
| + | $$I_{\frac{1}{2}}(z)=\sqrt{\dfrac{2}{\pi z}}\sinh(z).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Proposition:</strong> 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 J sub nu|Bessel function of the first kind]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
Revision as of 05:27, 16 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.
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: █
