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Hankel Function

There are two types of functions known as Hankel functions. The more common one is a complex function (also called a Bessel function of the third kind, or Weber Function) which is a linear combination of Bessel functions of the first and second kinds.

Another type of Hankel function is defined by the contour integral

H_epsilon(z)=int_(C_epsilon)((-w)^(z-1)e^(-w))/(1-e^(-w))dw

for I[w]<0, |arg(-w)|<pi, epsilon!=2pik>0, where C_epsilon is a Hankel contour. The Riemann zeta function can be expressed in terms of H_epsilon(z) as

zeta(z)=-(H_epsilon(z))/(2isin(piz)Gamma(z))

for 0<epsilon<2pi and R[z]>1, where Gamma(z) is the gamma function (Krantz 1999, p. 160).

See also

Hankel Contour, Hankel Function of the First Kind, Hankel Function of the Second Kind, Spherical Hankel Function of the First Kind, Spherical Hankel Function of the Second Kind

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References

Arfken, G. "Hankel Functions." §11.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 604-610, 1985.Hankel, H. "Die Cylinderfunctionen erster und zweiter Art." Math. Ann. 1, 467-501, 1869.Hankel, H. "Bestimmte Integrale mit Cylinderfunctionen." Math. Ann. 8, 453-470, 1875.Krantz, S. G. "The Hankel Contour and Hankel Functions." §13.2.4 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 159, 1999.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 623-624, 1953.

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Hankel Function

Cite this as:

Weisstein, Eric W. "Hankel Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HankelFunction.html

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