Faxén integral

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

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

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

n-dimensional Faxén integral

More generally one defines the $n$-dimensional Faxén integral as

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

with

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

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

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

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

Properties

Let $\Gamma$ denote the Gamma function, then

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

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

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

Asymptotics

For $x\to \infty$ we have the following asymptotics

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