Faxén Integral

In mathematics, the Faxén integral (also named Faxén function) is the following integral
:$\operatorname(\alpha,\beta;x)=\int_0^ \exp(-t+xt^)t^\mathrmt,\qquad (0\leq \operatorname(\alpha) <1,\;\operatorname(\beta)>0).$

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
:$I_n(x)=\lambda_n\int_0^\cdots \int_0^t_1^\cdots t_n^e^\mathrmt_1\cdots \mathrmt_n,$
with
:$f(t_1,\dots,t_n;x):=\sum\limits_^n t_j^-xt_1^\cdots t_n^\quad$ and $\quad\lambda_n:=\prod\limits_^n\mu_j$
for $x \in \C$ and
:$(0<\alpha_i <\mu_i,\;\operatorname(\beta_i)>0,\; i=1,\dots,n).$
The parameter $\lambda_n$ is only for convenience in calculations.

Properties

Let $\Gamma$ denote the Gamma function, then
*$\operatorname(\alpha,\beta;0)=\Gamma(\beta),$
*$\operatorname(0,\beta;x)=e^\Gamma(\beta).$
For $\alpha=\beta=\tfrac$ one has the following relationship to the Scorer function
:$\operatorname(\tfrac,\tfrac;x)=3^\pi \operatorname(3^x).$

Asymptotics

For $x\to \infty$ we have the following asymptotics
*$\operatorname(\alpha,\beta;-x)\sim \frac,$
*$\operatorname(\alpha,\beta;x)\sim \left(\frac\right)^(\alpha x)^\exp\left((1-\alpha)(\alpha^y)^{1/(1-\alpha)}\right).$

References

Mathematical analysis
Functions and mappings
Definitions of mathematical integration