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Sommerfeld integral

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An integral representation of the cylinder functions by a contour integral: The Hankel functions of the first kind are given by

where is a curve from to , ; the Hankel functions of the second kind are given by

where is a curve from to , ; the Bessel functions of the first kind are given by

where is a curve from to , . The representation is valid in the domain , and is named after A. Sommerfeld [1].

References

[1] A. Sommerfeld, "Mathematische Theorie der Diffraction" Math. Ann. , 47 (1896) pp. 317–374
[2] E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German)
[3] G.N. Watson, "A treatise on the theory of Bessel functions" , 1–2 , Cambridge Univ. Press (1952)


Comments

The Hankel functions are also called Bessel functions of the first kind.

How to Cite This Entry:
Sommerfeld integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sommerfeld_integral&oldid=48749
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article