Simplify $\displaystyle\prod_{j=1}^{K}\exp(-2\pi\lambda_j(sp_j)^{2/\alpha}\int_0^{\infty}r\int_0^{\infty}e^{-t(1+r^{\alpha})}dtdr)$.

Question

An equation involving the poisson point process is formulated as: $$\prod_{j=1}^{K}\exp(-2\pi\lambda_j(sp_j)^{2/\alpha}\int_0^{\infty}r\int_0^{\infty}e^{-t(1+r^{\alpha})}dtdr).$$

Some algebraic manipulations are carried out and the equation is rewritten as:

$$\exp(-s^{2/\alpha}C(\alpha)\sum_{i=1}^{K}\lambda_ip_i^{2/\alpha}),$$ where $$C(\alpha)=\frac{2\pi^2 csc(\frac{2\pi}{\alpha})}{\alpha}.$$

I only know the manipulation is related to Gamma function. I want to know the datails about the manipulation.

Thanks a lot! : )

Answer 1

Focus on simplifying $$2\pi\int_0^{\infty}r\int_0^{\infty}e^{-t(1+r^{\alpha})}dtdr =2\pi\int_0^{\infty}\frac r{1+r^\alpha}dr$$

Substitute $u=\frac1{1+r^\alpha}$ to get a Beta function form $$\frac{2\pi}{\alpha}\int_0^1(1-u)^{\frac2\alpha-1}u^\frac2\alpha du=\frac{2\pi}{\alpha}\Gamma\left(1-\frac2\alpha\right)\Gamma\left(\frac2\alpha\right)$$

Use the Gamma function property $$\Gamma\left(1-z\right)\Gamma\left(z\right)=\pi\csc(\pi z), z\not\in\mathbb Z$$