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
 
$$I_{\nu}(z)=i^{-\nu}J_{\nu}(iz),$$
 
$$I_{\nu}(z)=i^{-\nu}J_{\nu}(iz),$$
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where $J_{\nu}$ is the [[Bessel J sub nu|Bessel function of the first kind]].
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where $i$ denotes the [[imaginary number]] and $J_{\nu}$ denotes the [[Bessel J|Bessel function of the first kind]].
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
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File:Complex modified besselI sub 1.png|[[Domain coloring]] of [[analytic continuation]] of $I_1(z)$.
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File:Besseli,n=0plot.png|Graph of $I_0$.
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File:Besseli,n=1plot.png|Graph of $I_1$.
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File:Multiplebesseliplot.png|Graph of $I_0$,$I_1$,$I_2$, and $I_3$.
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File:Complexbesseli,n=0plot.png|[[Domain coloring]] of $I_0$.
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File:Complexbesseli,n=1plot.png|[[Domain coloring]] of $I_0$.
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File:Page 374 (Abramowitz&Stegun).jpg|Modified Bessel functions from Abramowitz&Stegun.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
  
 
=Properties=
 
=Properties=
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<div class="toccolours mw-collapsible mw-collapsed">
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[[Relationship between Bessel I sub -1/2 and cosh]]<br />
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<strong>Proposition:</strong> The following formula holds:
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[[Relationship between Bessel I sub 1/2 and sinh]]<br />
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$$I_{-\frac{1}{2}}(z)=\sqrt{\dfrac{2}{\pi z}} \cosh(z).$$
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[[Relationship between Bessel I sub n and Bessel J sub n]]<br />
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<div class="mw-collapsible-content">
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[[Relationship between Airy Bi and modified Bessel I]]<br />
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed">
 
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<strong>Proposition:</strong> The following formula holds:
 
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$$I_{\frac{1}{2}}(z)=\sqrt{\dfrac{2}{\pi z}}\sinh(z).$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed">
 
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<strong>Proposition:</strong> The following formula holds:
 
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$$I_{\nu}(z)=\displaystyle\sum_{k=0}^{\infty} J_{\nu+k}(z) \dfrac{z^k}{k!},$$
 
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where $J_{\nu}$ denotes the [[Bessel J sub nu|Bessel function of the first kind]].
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 
  
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{{:Relationship between Bessel I sub n and Bessel J sub n}}
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=References=
  
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{{:Relationship between Airy Bi and modified Bessel I}}
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[[Category:SpecialFunction]]
  
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<center>{{:Bessel functions footer}}</center>
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{{:Bessel functions footer}}

Latest revision as of 23:53, 10 June 2016

The modified Bessel function of the first kind is defined by $$I_{\nu}(z)=i^{-\nu}J_{\nu}(iz),$$ where $i$ denotes the imaginary number and $J_{\nu}$ denotes the Bessel function of the first kind.

  • Graph of $I_0$.

  • Graph of $I_1$.

  • Graph of $I_0$,$I_1$,$I_2$, and $I_3$.

  • Domain coloring of $I_0$.

  • Domain coloring of $I_0$.

  • Modified Bessel functions from Abramowitz&Stegun.

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

References

Bessel functions

Anger function

Bessel-Clifford

Bessel $J_{\nu}$

Bessel $Y_{\nu}$

Bessel $I_{\nu}$

Modified Bessel $K_{\nu}$

Spherical Bessel $j_{\nu}$

Spherical Bessel $y_{\nu}$
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