Does anyone know of any interesting proofs using complex analysis not covered in an intro sequence? For my final I have to give a 40 minute presentation on some topic (preferably one we haven't covered). I was hoping to prove some interesting theorem, maybe one that wasn't so fundamental to complex analysis but that used complex analysis to approach an interesting topic from another direction. I'd also like the result to be intuitively "cool" (like the PNT). Does anyone know of such topics, and where I could find the proofs?
As a reference, we have done prime number theorem, riemann mapping theorem, little picard, dirichlet's primes, and elliptic functions (p functions and on)
Thanks so much!
 A: I've always liked covering the theory of complex infinite products. There are a lot of really cool applications to this, like Dedekind's $\eta$ function
$$
\eta(\tau) = e^{i\pi\tau/12}\prod_{n=1}^{\infty}(1-e^{2\pi in\tau})
$$
for $\tau \in \mathcal{H} = \lbrace z \in \mathbb{C} \; | \; \Im(z) > 0 \rbrace$. This function is an modular form of half integral weight, and proving its functional equation
$$
\sqrt{\frac{\tau}{i}}\eta(\tau) = \eta\left(\frac{-1}{\tau}\right)
$$
is a nice illustration of analytic continuation and logarithmic derivatives, as well as manipulating normal convergence. This function is also extremely useful because
$\eta(\tau)^{24}$ is a cusp form of weight 12 of full level, which is the first nontrivial example of a full weight cusp form one can find. These have applications in the theory of modular forms and in automorphic representation theory, and hence are extremely useful in number theory and the Langlands' Programme.
Perhaps a more natural complex analytic answer would be to prove things about the Weierstrass $\Delta$ function, for $\gamma$ the Euler-Mascheroni constant,
$$
\Delta(z) = ze^{\gamma z}\prod_{n=1}^{\infty}\left(1-\frac{z}{n}\right)e^{-z/n},
$$
like functional equations and whathaveyou, and then show that there is an equality
$$
\Delta(z) = \frac{1}{\Gamma(z)},
$$
where $\Gamma$ is the Gamma function! This makes finding the residues and stuff about $\Gamma$ easier and is a nice illustration of using infinite products in complex analysis.
