View source for Hankel functions

[[Cylinder functions|Cylinder functions]] of the third kind. They may be defined in terms of [[Bessel functions|Bessel functions]] as follows:

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046310/h0463103.png" /> is not an integer. This implies the important relations

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Hankel functions are complex for real values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046310/h0463106.png" />; however,

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are real if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046310/h0463108.png" /> is real and positive. Hankel functions have simple asymptotic representations for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046310/h0463109.png" />:

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The Hankel function of a  "half-integral"  argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046310/h04631012.png" /> can be expressed in terms of elementary functions, in particular:

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These functions were introduced by H. Hankel in 1869.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Jahnke,   F. Emde,   F. Lösch,   "Tafeln höheren Funktionen" , Teubner  (1966)</TD></TR></table>



====Comments====
See [[Cylinder functions|Cylinder functions]] for additional references.