In mathematics, the Faxén integral (also named Faxén function) is the following integral[1]
![{\displaystyle \operatorname {Fi} (\alpha ,\beta ;x)=\int _{0}^{\infty }\exp(-t+xt^{\alpha })t^{\beta -1}\mathrm {d} t,\qquad (0\leq \operatorname {Re} (\alpha )<1,\;\operatorname {Re} (\beta )>0).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d165d7ed30fcc9f372a232a27379eaf52a39c1cf)
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]
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with
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for and
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The parameter is only for convenience in calculations.
Let denote the Gamma function, then
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For one has the following relationship to the Scorer function
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For we have the following asymptotics[4]
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