Difference between revisions of "Modified Bessel K"
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(Created page with "The modified Bessel function of the second kind is defined by $$K_{\nu}(z)=\dfrac{\pi}{2} \dfrac{I_{-\nu}(z)-I_{\nu}(z)}{\sin(\nu \pi)},$$ where $I_{\nu}$ is the Modified Be...") |
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$$K_{\nu}(z)=\dfrac{\pi}{2} \dfrac{I_{-\nu}(z)-I_{\nu}(z)}{\sin(\nu \pi)},$$ | $$K_{\nu}(z)=\dfrac{\pi}{2} \dfrac{I_{-\nu}(z)-I_{\nu}(z)}{\sin(\nu \pi)},$$ | ||
where $I_{\nu}$ is the [[Modified Bessel I sub nu|modified Bessel function of the first kind]]. | where $I_{\nu}$ is the [[Modified Bessel I sub nu|modified Bessel function of the first kind]]. | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Proposition:</strong> The following formula holds: | ||
| + | $$K_{\frac{1}{2}}(z)=\sqrt{\dfrac{\pi}{2}}\dfrac{e^{-z}}{\sqrt{z}}; z>0.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
Revision as of 05:25, 16 May 2015
The modified Bessel function of the second kind is defined by $$K_{\nu}(z)=\dfrac{\pi}{2} \dfrac{I_{-\nu}(z)-I_{\nu}(z)}{\sin(\nu \pi)},$$ where $I_{\nu}$ is the modified Bessel function of the first kind.
Properties
Proposition: The following formula holds: $$K_{\frac{1}{2}}(z)=\sqrt{\dfrac{\pi}{2}}\dfrac{e^{-z}}{\sqrt{z}}; z>0.$$
Proof: █
