Faxén integral

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

${\displaystyle \operatorname {Fi} (\alpha ,\beta ;x)=\int _{0}^{\infty }\exp(-t+xt^{\alpha })t^{\beta -1}\mathrm {d} t,\qquad (0\leq \operatorname {Re} (\alpha )<1,\;\operatorname {Re} (\beta )>0).}$

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

More generally one defines the ${\displaystyle n}$-dimensional Faxén integral as^[3]

${\displaystyle I_{n}(x)=\lambda _{n}\int _{0}^{\infty }\cdots \int _{0}^{\infty }t_{1}^{\beta _{1}-1}\cdots t_{n}^{\beta _{n}-1}e^{-f(t_{1},\dots ,t_{n};x)}\mathrm {d} t_{1}\cdots \mathrm {d} t_{n},}$

with

${\displaystyle f(t_{1},\dots ,t_{n};x):=\sum \limits _{j=1}^{n}t_{j}^{\mu _{j}}-xt_{1}^{\alpha _{1}}\cdots t_{n}^{\alpha _{n}}\quad }$ and ${\displaystyle \quad \lambda _{n}:=\prod \limits _{j=1}^{n}\mu _{j}}$

for ${\displaystyle x\in \mathbb {C} }$ and

${\displaystyle (0<\alpha _{i}<\mu _{i},\;\operatorname {Re} (\beta _{i})>0,\;i=1,\dots ,n).}$

The parameter ${\displaystyle \lambda _{n}}$ is only for convenience in calculations.

Let ${\displaystyle \Gamma }$ denote the Gamma function, then

• ${\displaystyle \operatorname {Fi} (\alpha ,\beta ;0)=\Gamma (\beta ),}$
• ${\displaystyle \operatorname {Fi} (0,\beta ;x)=e^{x}\Gamma (\beta ).}$

For ${\displaystyle \alpha =\beta ={\tfrac {1}{3}}}$ one has the following relationship to the Scorer function

${\displaystyle \operatorname {Fi} ({\tfrac {1}{3}},{\tfrac {1}{3}};x)=3^{2/3}\pi \operatorname {Hi} (3^{-1/3}x).}$

For ${\displaystyle x\to \infty }$ we have the following asymptotics^[4]

• ${\displaystyle \operatorname {Fi} (\alpha ,\beta ;-x)\sim {\frac {\Gamma (\beta /\alpha )}{\alpha y^{\beta /\alpha }}},}$
• ${\displaystyle \operatorname {Fi} (\alpha ,\beta ;x)\sim \left({\frac {2\pi }{1-\alpha }}\right)^{1/2}(\alpha x)^{(2\beta -1)/(2-2\alpha )}\exp \left((1-\alpha )(\alpha ^{\alpha }y)^{1/(1-\alpha )}\right).}$