integration - $int_{-infty}^{+infty} e^{-x^2} dx$ with complex analysis

Inspired by this recently closed question, I'm curious whether there's a way to do the Gaussian integral using techniques in complex analysis such as contour integrals.

I am aware of the calculation using polar coordinates and have seen other derivations. But I don't think I've ever seen it done with methods from complex analysis. I am ignorant enough about complex analysis to believe it can somehow be done without knowing how it would be done.

Answer

What follows is a list of solutions that I enjoy, and use complex analysis either implicitly or explicitly. I will update the list as I come up with more. (Note: Solution 4 is my favorite, and is completely complex analysis oriented. I also quite like Solution 6.)

First, let $u=x^{2}$, $du=2xdx$. Then our integral becomes $$\int_{-\infty}^\infty e^{-x^2}dx=\int_{0}^{\infty}u^{-\frac{1}{2}}e^{-u}du=\Gamma\left(\frac{1}{2}\right).$$ where $\Gamma(s)$ is the Gamma function.

Solution 1:
Since $$\Gamma(1-s)\Gamma(s)=\frac{\pi}{\sin\pi s}$$ for all complex $s$, we conclude $$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}.$$

Solution 2: Recall the Beta function, $$\text{B}(x,y)=\int_{0}^{1}t^{x-1}(1-t)^{y-1}dt=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}.$$ Setting $x=y=\frac{1}{2}$ we have

$$\left(\Gamma\left(\frac{1}{2}\right)\right)^{2}=\int_{0}^{1}\frac{1}{\sqrt{t(1-t)}}dt.$$

To evaluate this, set $t=\sin^{2}(x)$ to find $$\left(\Gamma\left(\frac{1}{2}\right)\right)^{2}=\int_{0}^{\frac{\pi}{2}}\frac{2\sin x\cos x}{\sin x\cos x}dt=\pi.$$
Alternatively, we could evaluate the last integral by choosing branch's such that the integrand is analytic on $\mathbb{C}-[0,1]$ and then integrating around this cut. (The residue then comes from the residue at infinity)

Solution 3: Setting $s=\frac{1}{2}$ in the duplication formula, $$\Gamma(s)\Gamma\left(s+\frac{1}{2}\right)=\sqrt{\pi}2^{1-2s}\Gamma(2s),$$ yields $$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}.$$

Solution 4: My personal favorite:
Recall the functional equation for the zeta function, namely that $$\pi^{-\frac{z}{2}}\Gamma\left(\frac{z}{2}\right)\zeta(z)=\pi^{-\frac{1-z}{2}}\Gamma\left(\frac{1-z}{2}\right)\zeta(1-z).$$ Taking the limit as $z\rightarrow1$, we know that $\zeta(z)\sim\frac{1}{z-1}$ and $\Gamma\left(\frac{1-z}{2}\right)\sim2\frac{1}{\left(z-1\right)}$ so that we must have the equality $$\pi^{-\frac{1}{2}}\Gamma\left(\frac{1}{2}\right)=2\zeta(0).$$ By taking the limit in the right half plane as $s\rightarrow0$ using the identity $$\zeta(s)=\frac{s}{s-1}-s\int_{1}^{\infty}\{u\}u^{-s}du,$$ which holds for $\sigma>0$, we can find that $\zeta(0)=\frac{1}{2}.$ (notice the pole/zero cancellation). Consequently $$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}.$$

Solution 5: From complex integration, for $a,b>0$ we have the identity $$\int_{-\infty}^{\infty}(1-ix)^{-a}(1+ix)^{-b}dx=\frac{2^{2-a-b}\pi\Gamma(a+b-1)}{\Gamma(a)\Gamma(b)}.$$ Set $a=\frac{1}{2},b=\frac{3}{2}$ to find that
$$\int_{-\infty}^{\infty}\frac{1-ix}{\left(1+x^{2}\right)^{\frac{3}{2}}}dx=\frac{2\pi}{\Gamma\left(\frac{1}{2}\right)^{2}}.$$ Hence $$\int_{0}^{\infty}\frac{1}{\left(1+x^{2}\right)^{\frac{3}{2}}}dx=\frac{\pi}{\Gamma\left(\frac{1}{2}\right)^{2}}.$$ Since the integrand on left hand side has anti derivative $\frac{x}{\sqrt{x^{2}+1}}+C$, it follows that the integral is $1$ and hence $$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}.$$

Solution 6: More with the Beta function. Consider the Mellin Transform $$\mathcal{M}\left(\frac{1}{\left(1+t\right)^{a}}\right)(z):=\int_{0}^{\infty}\frac{t^{z-1}}{(1+t)^{a}}dt=\text{B}(a-z,z).$$ The last equality follows by substituting $v=\frac{1}{1+t}$, and then rewriting the integral as $\int_{0}^{1}v^{a-z-1}(1-v)^{z-1}dv.$ Now, plug in $a=1$ and $z=\frac{1}{2}$ to get $$\int_{0}^{\infty}\frac{1}{\sqrt{t}(1+t)}dt=\Gamma\left(\frac{1}{2}\right)^{2}$$ and then let $t=x^{2}$ to find
$$2\int_{0}^{\infty}\frac{1}{1+x^{2}}dx=\pi=\Gamma\left(\frac{1}{2}\right)^{2}.$$

Solution 7: We can also prove the result by using Stirling's formula. Admittedly, this isn't really using complex analysis, but I find it interesting.

Since $z\Gamma(z)=\Gamma(z+1)$ we see that
$$\Gamma\left(n+\frac{1}{2}\right)=\Gamma\left(\frac{1}{2}\right)\cdot\left(\frac{1}{2}\right)\left(\frac{3}{2}\right)\cdots\left(\frac{2n-1}{2}\right)=\Gamma\left(\frac{1}{2}\right)\left(\frac{(2n)!}{n!4^{n}}\right)=\Gamma\left(\frac{1}{2}\right)\binom{2n}{n}\frac{n!}{4^{n}}.$$
By Stirling's formula,
$$\binom{2n}{n}\frac{1}{4^{n}}\sim\frac{1}{\sqrt{\pi n}}\ \text{as}\ n\rightarrow\infty$$
and
$$\frac{\Gamma\left(n+\frac{1}{2}\right)}{n!}\sim\frac{1}{\sqrt{ne}}\frac{\left(n+\frac{1}{2}\right)^{n}}{n^{n}}.$$
Using the fact that $\lim_{n\rightarrow\infty}\left(1+\frac{a}{n}\right)^{n}=e^{a}$, it then follows that

$$\frac{\Gamma\left(n+\frac{1}{2}\right)}{n!}\sim\frac{1}{\sqrt{n}}.$$
Consequently, taking the limit as $n\rightarrow\infty$ in the formula $$\Gamma\left(\frac{1}{2}\right)=\frac{\Gamma\left(n+\frac{1}{2}\right)}{n!}\frac{4^{n}}{\binom{2n}{n}}$$ yields $$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}.$$

Hope that helps,

Remark: All of the formulas used here can be proven without use the fact that $\Gamma(1/2)=\sqrt{\pi}$, so that none of these are cyclic. This is mainly worth pointing out for $4$.

Edit: I put what were solutions 2 and 3 together since they were not different.