In mathematics, the Faxén integral (also named Faxén function) is the following integral[1]

The integral is named after the Swedish physicist Olov Hilding Faxén, who published it in 1921 in his PhD thesis.[2]

n-dimensional Faxén integral

More generally one defines the -dimensional Faxén integral as[3]

with

and

for and

The parameter is only for convenience in calculations.

Properties

Let denote the Gamma function, then

For one has the following relationship to the Scorer function

Asymptotics

For we have the following asymptotics[4]

References

  1. Olver, Frank W. J. (1997). Asymptotics and Special Functions. A K Peters/CRC Press. p. 332. doi:10.1201/9781439864548.
  2. Faxén, Hilding (1921). Einwirkung der Gefässwände auf den Widerstand gegen die Bewegung einer kleinen Kugel in einer zähen Flüssigkeit (PhD). Uppsala University.
  3. Paris, Richard Bruce (2010). "Asymptotic expansion of n-dimensional Faxén-type integrals". European Journal of Pure and Applied Mathematics. A K Peters/CRC Press. 3 (6): 1006–1031.
  4. Kaminski, David; Paris, Richard B. (1997). "Asymptotics via iterated Mellin–Barnes integrals: Application to the generalised Faxén integral". Methods and applications of analysis. 4: 311–325.
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