I'm trying to do the Gamma function of 3/2, so $$\int_0^{\infty } e^{-x} \sqrt{x} \, dx$$ So far I have this u substitution $$u=\sqrt{x}$$ $$du=\frac{dx}{2 \sqrt{x}}$$ $$\int_0^{\infty } e^{-u^2} u2u \, du$$ $$2\int_0^{\infty } u^2e^{-u^2}\, du$$ and then integration by parts (I'm going to abuse some notation because I don't understand LaTeX, please forgive me) $$2\int_0^{\infty } u^2e^{-u^2}\, du$$ $$2((\frac{-\infty e^{-\infty ^2}}{2}-\frac{-0 e^{-0^2}}{2})-\int_0^{\infty } \frac{-e^{-u ^2}}{2} \, du)$$ $$\int_0^{\infty } e^{-u^2}\, du$$ After all that we are left with the familiar Gaussian integral. From what I've read on the internet, first you $$\sqrt{(\int_0^{\infty } e^{-u^2}\, du)^2}$$ $$\sqrt{(\int_0^{\infty } e^{-u^2}\, du)(\int_0^{\infty } e^{-u^2}\, du)}$$ $$\sqrt{(\int_0^{\infty } e^{-u^2}\, du)(\int_0^{\infty } e^{-v^2}\, dv)}$$ $$\sqrt{\int_0^{\infty }\int_0^{\infty } e^{-u^2}e^{-v^2}\, du\, dv}$$ $$\sqrt{\int_0^{\infty }\int_0^{\infty } e^{-(u^2+v^2)}\, du\, dv}$$ So far so good. However, I'm having trouble switching to polar coordinates here. The internet just kind of says $$r^2=u^2+v^2$$ $$dxdy=rdrdt$$ as a given, after which point it becomes literally free. I'm trying to figure out how to come up with that but I'm struggling. I've tried doing the following substitutions $$r=u\sec(t),dr=du\sec(t)$$ $$t=\arcsin(v/r),dt=\frac{dv}{r\sqrt{1-\frac{v}{r}^2}}$$ which transforms the integrand to $$e^{-r^2} r \cos ^2(t)$$ which can be integrated, but now we need to find the limits.
The integral with respect to r needs to be from 0 to infinity in order for both u and v to range from 0 to infinity. However, I can't figure out how to come up with the right limit for t. I would presume that it would be from 0 to Pi/2, since u and v both range from 0 to infinity. I presume that should mean that we are integrating just the 1st quadrant. But unfortunately, $$\int_0^{\frac{\pi }{2}} \left(\int_0^{\infty } e^{-r^2} r \cos ^2(t) \, dr\right) \, dt=\frac{\pi }{8}\neq\frac{\pi }{4}$$ Setting the upper limit to Pi gets us the right answer of Pi/4, but the only reason I know that is because I already know what the answer is supposed to be. I'm still in the dark as to why that's the limit. Would anyone know what I'm doing wrong with the substitutions or the limits?
