# 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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### Entire function approaching zero along upper half plane

Suppose $f$ is entire, i.e, $\;f: \Bbb C \to \Bbb C$ is analytic. Let $\Bbb H:= \{ z: Im(z)>0\}$ be the upper half plane. Suppose that $$\lim_{\substack{z \to \infty \\ z \in \Bbb H}} f(z)=0$$ ...
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### Show that $f(z) = \ln r + i \varphi$ is differentiable in a neighborhood of $z_{0}$

I am faced with the following problem: Let $z_{0}\neq 0$ and let $f(z) = \ln r + i \varphi$, where $r = |z|$, $\varphi \in arg z$, and $\varphi$ is chosen so that $f$ is continuous in a neighborhood ...
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### Complex Analysis with differential forms

I'm studying a little of Complex Anlysis and I have seen that I can thing the integrals of complex functions as integrals of differential forms in $\mathbb{R}^n$. For example I know that Cauchy ...
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### Mistake in this definition from Conway's complex analysis book

I'm reading Conway's complex analysis book and on page 64 the author has enunciated the following definition: However, on page 81 the author has stated that: I think Conway was mistaken in the ...
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### Complex Root of Unity Analogue of Forward Difference Operator

In my studies I have come across a couple of operators; in particular; $$\Delta[f(x)]=f(x+1)-f(x)$$ $$\Delta^*[f(x)]=f(x+1)+f(x)$$ $\Delta$ has been called the Forward Difference Operator. I was ...
I found this question in Conway, and really have no idea how to answer it. Can anyone provide any hints? For each integer $n\geq 1$ determine all meromorphic functions on $\mathbb{C}$ $f$ and $g$ ...