Difference between revisions of "Hankel H (1)"

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The Hankel functions of the first kind are defined by
 
The Hankel functions of the first kind are defined by
 
$$H_{\nu}^{(1)}(z)=J_{\nu}(z)+iY_{\nu}(z),$$
 
$$H_{\nu}^{(1)}(z)=J_{\nu}(z)+iY_{\nu}(z),$$
where $J_{\nu}$ is the [[Bessel J sub nu|Bessel function of the first kind]] and $Y_{\nu}$ is the [[Bessel Y sub nu|Bessel function of the second kind]]. Note the similarity of these functions to the [[Hankel H sub nu (2)|Hankel functions of the second kind]].
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where $J_{\nu}$ is the [[Bessel J|Bessel function of the first kind]] and $Y_{\nu}$ is the [[Bessel Y sub nu|Bessel function of the second kind]]. Note the similarity of these functions to the [[Hankel H sub nu (2)|Hankel functions of the second kind]].
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
File:Complex hankel H1 sub 1.png|[[Domain coloring]] of [[analytic continuation]] of $H_1^{(1)}(z)$.
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File:Complex hankel H1 sub 1.png|[[Domain coloring]] of $H_1^{(1)}(z)$.
 
File:Page 359Abramowitz-Stegun(Bessel functions).jpg|Bessel functions from [http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/ Abramowitz&Stegun]
 
File:Page 359Abramowitz-Stegun(Bessel functions).jpg|Bessel functions from [http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/ Abramowitz&Stegun]
 
</gallery>
 
</gallery>
 
</div>
 
</div>
  
<center>{{:Hankel functions footer}}</center>
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=See Also=
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[[Bessel J|Bessel $J$]]<br />
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[[Bessel Y|Bessel $Y$]]<br />
  
=See Also=
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=References=
[[Bessel J]]<br />
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Bessel Y|next=Hankel H (1) in terms of csc and Bessel J}}: 9.1.3
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{{:Hankel functions footer}}
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 23:59, 22 December 2016

The Hankel functions of the first kind are defined by $$H_{\nu}^{(1)}(z)=J_{\nu}(z)+iY_{\nu}(z),$$ where $J_{\nu}$ is the Bessel function of the first kind and $Y_{\nu}$ is the Bessel function of the second kind. Note the similarity of these functions to the Hankel functions of the second kind.

See Also

Bessel $J$
Bessel $Y$

References

Hankel functions