Faxén integral

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

$\mathrm{Fi}(\alpha ,\beta ;x)=\underset{0}{\overset{\mathrm{\infty }}{\int }}\exp (-t+x{t}^{\alpha })t^{\beta -1}\mathrm{d}t,(0\le \mathrm{Re}(\alpha )<1,\mathrm{Re}(\beta )>0).$

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

$I_n(x)={\lambda }_n\underset{0}{\overset{\mathrm{\infty }}{\int }}\cdots \underset{0}{\overset{\mathrm{\infty }}{\int }}{t}_1^{{\beta }_1-1}\cdots t_n^{{\beta }_n-1}{e}^{-f(t_1,\dots ,t_n;x)}\mathrm{d}{t}_1\cdots \mathrm{d}{t}_n,$

with

$f(t_1,\dots ,t_n;x):=\sum _{j=1}^n{t}_j^{{\mu }_j}-x{t}_1^{{\alpha }_1}\cdots t_n^{{\alpha }_n}$ and ${\lambda }_n:=\prod _{j=1}^n{\mu }_j$

for $x\in ℂ$ and

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

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

Properties

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

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

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

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

Asymptotics

For $x\to \mathrm{\infty }$ we have the following asymptotics

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