# 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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### Double periodic entire function

Suppose f is entire and $f(z)=f(z+1)=f(z+\pi)$. Does this imply $f$ is constant? I want to prove that it is constant.I see that it is enough to consider the value of $f(z)$ in between the lines $z=1$ ...
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### Difference between branches $[-\pi, \pi)$ and $[\pi, 3\pi)$ of the complex logarithm

I think that both branches just exclude the negative part of the real line. So what's the difference between them then?
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### A problem about the property of limit of holomorphic function

Suppose $G\subset\mathbb{C}$ is open and connected,let $\left\{ f_{n}:n=1,2\ldots \right\}$ be a uniformly bounded sequence of holomorphic functions on $G$ that convergences uniformly on compact ...
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### Why are $L$-functions a big deal?

I've been studying modular forms this semester and we did a lot of calculations of $L$-functions, e.g. $L$-functions of Dirichlet-characters and $L$-functions of cusp-forms. But I somehow don't see, ...
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### Rewriting a complex matrix as a real matrix and changing basis to $(z,\bar z)$

If we have a complex $n\times n$ matrix $A$ and we rewrite it as a real $2n\times 2n$ matrix $B$ by using the identity $z=x+iy$. I don't get how if we change basis from $(x,y)$ to $(z,\bar z)$ that is ...
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### Calculating the convergence radius of a Taylor expansion

Find the radius of convergence for the Taylor series $$\left(\cot\dfrac{\pi}{100}z\right)=\sum^{\infty}_{n=0}a_{n}\left(z-20\pi\right)^{n}$$ The singularities of this function are the $100n$ where $n$...
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### For $f$ analytic on $|z|<1$, $|f|\le M$ with $a_1,\ldots,a_n\in \Bbb{D}$ zeros of $f$ show that $|f(0)|\le M \prod |a_j|$

For $f$ analytic in unit disk $\Bbb{D}$ where $|f|\le M$ with $a_1,\ldots,a_n\in \Bbb{D}$ such that $f(a_1)=\cdots=f(a_n)=0$ show that $|f(0)|\le M \prod |a_j|$. I have tried many approaches ...
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