# Questions tagged [complex-analysis]

For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.

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### Witt's proof of Gelfand-Mazur / Ostrowski's theorem

Now asked on MathOverflow. Background: It seems that, after his groundbreaking work on quadratic forms and inventing Witt vectors, Ernst Witt developed the hobby of giving extremely short proofs to ...
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### A conformal mapping onto a region bounded by convex contours (Ahlfors)

I want to solve the following exercise (from Ahlfors' text, page 261) *3. Using Ex. 2, show that $p + q$ maps $\Omega$ in a one-to-one manner onto a region bounded by convex contours. Comments: (i)...
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### What can be said about the level set of the real part of an analytic function?

I am working with a function $F(z;a)$, for $z\in \mathbb{C}$ and $a$ being a set of parameters, from which I need to analyze the level set $\text{Re}(F(z))=0$ (for a fixed set of parameters $a$, which ...
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### How to find area of a polygon built on the roots of a given polynomial?

How to find the area of a (maximum area convex) polygon, built on the roots of a given polynomial in the complex plane? For example, consider the equation: $$2x^5+3x^3-x+1=0$$ It has one real and four ...
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### When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ for $\Gamma$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least ...
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### How large must $f(f(z))-z^2$ be in the unit disk?

This is a follow-up question to $|f(f(z))-z^2|$ must be large somewhere in the disc $\mathbb{D}$?, where the following was proven: Let $f$ be holomorphic in the unit disk $\Bbb D$ with $f(0) = 0$ and ...
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### How to recognize when a function is secretely holomorphic

Let $f : M \rightarrow N$ be a holomorphic map between complex manifolds (I'd be interested even in the case $M=N=\mathbb{C}$ which should not be much different). Now take $K$ a compact subset of $M$,...
Recall that the argument principle states that given a meromorphic function $f$ and a compact region $K \subseteq \mathbb{C}$ whose boundary determines a simple contour and on which $f$ has no ...