I've learned factorial. But today I saw a question which I don't know how to start with:


Can anyone explain how to solve it? Thanks


As suggested by Workaholic, $$I=\left(-\frac{1}{2}\right)!=\Gamma\left(\frac{1}{2}\right) = \int_{0}^{+\infty}x^{-1/2}e^{-x}\,dx = 2\int_{0}^{+\infty}e^{-y^2}\,dy = \int_{\mathbb{R}}e^{-y^2}\,dy $$ and by Fubini's theorem: $$ I^2 = \int_{\mathbb{R}^2}e^{-(y^2+z^2)}\,dy\,dz = \int_{0}^{2\pi}\int_{0}^{+\infty}\rho\, e^{-\rho^2}\,d\rho\,d\theta = \pi \int_{0}^{+\infty}2\rho\, e^{-\rho^2}\,d\rho = \color{red}{\pi} $$ so:

$$ \left(-\frac{1}{2}\right)! = \color{red}{\sqrt{\pi}}. $$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.