The discussion focuses on evaluating the integral $$\int_0^{\infty} \frac{p^2 dp}{\mathrm{exp}(a\sqrt{p^2+b^2}) \pm 1}$$ using Wolfram Alpha. The transformation to $$x(p) = a\sqrt{p^2 + b^2}$$ simplifies the integral to $$\frac{1}{a^3} \int_0^{\infty} \frac{\sqrt{x^2-(b/a)^2}}{e^x \pm 1} dx$$. Reference to Schaum's Outline of Mathematical Handbook of Formulas and Tables indicates that the solution involves the Gamma function, specifically formulas 18.81 and 18.82. Users are advised to verify their algebra when changing variables to ensure the bounds of the integral are correct.
PREREQUISITESMathematicians, physicists, and students engaged in advanced calculus or statistical mechanics who are looking to solve complex integrals and utilize computational tools effectively.
You might want to double check your algebra for changing variables (e.g. do the bounds of the integral over ##x## make sense?)ergospherical said:Looking to evaluate an integral of the form $$\int_0^{\infty} \frac{p^2 dp}{\mathrm{exp}(a\sqrt{p^2+b^2}) \pm 1} $$Changing to ##x(p) = a\sqrt{p^2 + b^2}## gives $$\frac{1}{a^3} \int_0^{\infty} \frac{\sqrt{x^2-(b/a)^2}}{e^x \pm 1} dx$$Wolfram alpha doesn't tell me anything useful, sadly.