Difference between revisions of "Modified Bessel K"

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File:Besselk,n=0plot.png|Graph of $K_0$.
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File:Multiplebesselkplot.png|Graphs of $K_0$, $K_1$, $K_2$, and $K_3$.
 
File:Domaincoloringbesselksub1.png|[[Domain coloring]] of $K_1$.
 
File:Domaincoloringbesselksub1.png|[[Domain coloring]] of $K_1$.
 
File:Page 374 (Abramowitz&Stegun).jpg|Modified Bessel functions from Abramowitz&Stegun.
 
File:Page 374 (Abramowitz&Stegun).jpg|Modified Bessel functions from Abramowitz&Stegun.
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=Properties=
 
=Properties=
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[[Relationship between Airy Ai and modified Bessel K]]
<strong>Proposition:</strong> The following formula holds:
 
$$K_{\frac{1}{2}}(z)=\sqrt{\dfrac{\pi}{2}}\dfrac{e^{-z}}{\sqrt{z}}; z>0.$$
 
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<strong>Proof:</strong> █
 
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{{:Relationship between Airy Ai and modified Bessel K}}
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=References=
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[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_374.htm]
  
<center>{{:Bessel functions footer}}</center>
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[[Category:SpecialFunction]]
  
=References=
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{{:Bessel functions footer}}
[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_374.htm]
 

Latest revision as of 23:46, 10 June 2016

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

Relationship between Airy Ai and modified Bessel K

References

[1]

Bessel functions