# Calculating the partition function of a generalised kinetic energy

I need to calculate the partition function (of the canonical ensemble) given the hamiltonian: $$H=\sum_{i=0}^N A\vert P_i \vert^s$$ where $$s>0$$. and $$p_i$$ are independent of one another. The partition function is given by: $$z=\frac{1}{N!} \idotsint(\prod_i \frac{d^3 q_i d^3 p_i}{(2\pi\hbar)^3})exp(-\beta\sum_{i=0}^N A\vert P_i \vert^s)$$

from here I managed to get:

$$z=\frac{V}{(2\pi\hbar)^3}\idotsint d^3p_i exp(-\beta A\vert P_i \vert^s)$$ but from here I'm stumped. Pretty sure the solution should involve a gamma function, but that's about it. I'd appreciate any help.

You can separate the integrals in $$p$$ as follows:
$$Z\sim\prod_i\left(\int d^3p_i\exp(-\beta A|p_i|^s)\right)$$ You can find the integral as follows
$$\int d^3p\exp(-\beta A|p|^s)=\int 4\pi p^2dp\exp(-\beta A|p|^s)=\int_0^\infty 8\pi p^2dp\exp(-\beta Ap^s)$$ Let $$p^s=x\Rightarrow sp^{s-1}dp=dx$$ $$\int8\pi p^2\frac{dx}{sp^{s-1}} \exp(-\beta Ax)\Rightarrow \int 8\pi x^{3/s-1}dx\exp(-\beta Ax)=8\pi\frac{\Gamma[3/s]}{(\beta A)^{3/s}}$$ $$Z\sim \left(8\pi\frac{\Gamma[3/s]}{(\beta A)^{3/s}}\right)^N$$