# Tagged Questions

The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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### Distance to infinity in the complex plane

I am trying to understand calculating the distance between a point in the complex plane and infinity by using the project of the point, say $z_0$ onto the Riemann sphere. We can define the distance ...
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### $f(z)$ analytic in the panctured disk $\{|z|<1 \}\setminus\{0\}, \text{Im}(f(z))>0$, then $z=0$ is removable singularity

Let $f(z)$ be analytic in the panctured disk $\{|z|<1 \}\setminus\{0\}$ and let $\text{Im}(f(z))>0$. Prove $z=0$ is removable singularity. I try to show that $f(z)$ is bounded near $z=0$ but ...
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### Proving $f$ has at least one zero inside unit disk

Let $f$ be a non-constant and analytic on a neighborhood of closure of the unit disk such that $|f(z)|=\text{constant}$ for $|z|=1$. Prove $f$ has at least one zero inside unit disk. I thought of ...
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### $\wp$ via Jacobi triple product

$$\wp(z;\tau) = -(\log \vartheta_{11}(z;\tau))'' + c$$ $$\vartheta_{11}(z|q) = -2 q^{1/4}\sin(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 - 2 \cos(2 \pi z)q^{2m}+q^{4m}\right)$$ Then ...