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

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

 

with

  and  

for   and

 

The parameter   is only for convenience in calculations.

Properties

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Let   denote the Gamma function, then

  •  
  •  

For   one has the following relationship to the Scorer function

 

Asymptotics

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For   we have the following asymptotics[4]

  •  
  •  

References

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  1. ^ Olver, Frank W. J. (1997). Asymptotics and Special Functions. A K Peters/CRC Press. p. 332. doi:10.1201/9781439864548. ISBN 978-0-429-06461-6.
  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. 3 (6). A K Peters/CRC Press: 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 (3): 311–325. doi:10.4310/MAA.1997.v4.n3.a5.