Faxén integral

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

[math]\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) \lt 1,\;\operatorname{Re}(\beta)\gt 0). }[/math]

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 [math]\displaystyle{ n }[/math]-dimensional Faxén integral as

[math]\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, }[/math]

with

[math]\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 }[/math] and [math]\displaystyle{ \quad\lambda_n:=\prod\limits_{j=1}^n\mu_j }[/math]

for [math]\displaystyle{ x \in \C }[/math] and

[math]\displaystyle{ (0\lt \alpha_i \lt \mu_i,\;\operatorname{Re}(\beta_i)\gt 0,\; i=1,\dots,n). }[/math]

The parameter [math]\displaystyle{ \lambda_n }[/math] is only for convenience in calculations.

Properties

Let [math]\displaystyle{ \Gamma }[/math] denote the Gamma function, then

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

For [math]\displaystyle{ \alpha=\beta=\tfrac{1}{3} }[/math] one has the following relationship to the Scorer function

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

Asymptotics

For [math]\displaystyle{ x\to \infty }[/math] we have the following asymptotics

• [math]\displaystyle{ \operatorname{Fi}(\alpha,\beta;-x)\sim \frac{\Gamma(\beta/\alpha)}{\alpha y^{\beta/\alpha}}, }[/math]
• [math]\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). }[/math]