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Let $\Sigma$ be a compact Riemann surface of genus $g$, and $f: \Sigma \to \mathbb{C}$ a meromorphic function. Take $U \subset \Sigma$ an open disk in $\Sigma$ biholomorphic to a disk in $\mathbb{C}$, …

There are many elliptic integrals, so to show my point let me just pick one of them (complete elliptic integral of the first kind [1]): $$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})} …

Let $F$ be a generating function $F(x) = \sum_{i=0}^\infty f_i x^i$, and suppose that we can do operations formally without worrying about convergence issues. Define the coefficients \begin{gather*} a …

Consider a polynomial in one variable with complex coefficient $$f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$ we are interested in its roots. Babylonian solved for $n = 2$, and Cardano did it …

I was recently introduced to analytic combinatorics, and found the method of removing poles astonishing. More precisely, I was reading the last chapter of the popular "generatingfunctionology", in whi …

I am interested in the number of complex structures on a surface. More precisely, given a genus $g$ surface (topological manifold of real dimension 2) with $n$ punctures $X_{(g,n)}$, how many complex …