# 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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### Examples of Weil's explicit formula

In Bombieri, PROBLEMS OF THE MILLENNIUM: THE RIEMANN HYPOTHESIS, Clay Mathematics Institute (2000), from page 8, V. Further evidence: the explicit formula the author tell us that there is a ...
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### Existence of such a meromorphic function?

Is there a function $f$ that is holomorphic on $\mathbb{C}-\mathbb{Z}$ and maps into or onto $\mathbb{C}-\mathbb{R}$ ? Into or onto $\mathbb{C}-\mathbb{R}^{+}\cup\{ {0} \}$? All I have been able to ...
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### How does the author derive this (Difference of analytic functions evaluated at two points)

The conditions are $f:U\to V$ is holomorphic and injective. I basically have 2 questions: Q1) How did the author get $f(z)-f(z_0)=a(z-z_0)^k+G(z)$? Q2) What does "vanishing to order $k+1$" mean? ...
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### On real part of the complex number $(1+i)z^2$

Find the set of points belonging to the coordinate plane $xy$, for which the real part of the complex number $(1+i)z^2$ is positive. My solution:- Lets start with letting $z=r\cdot e^{i\theta}$. ...
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### find the principal part of the function at its isolated singular point

find the principal part of the function at its isolated singular point, and determine whether that point is a pole, a removable singular point, or an essential singular point. (a) f(z) = z exp(1/z) (...
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### Why is holomorphic function with non-zero derivative a conformal map?

I am new to complex analysis, interested to know why non-zero derivative implies a conformal map. Intuitively, I would think that non-zero derivative means the function is non-constant. Why would ...
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### Are derivatives actually bounded?

Suppose you a function $f$ which is differentiable, with the property that $$f^{(n)} (0) = (n!)^2$$ And in general $$f^{(n)} (a) = O((n!)^2)$$ For any $a \in \mathbb{R}$. This function ...
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### Prove that $u\leq v$ everywhere.

Let $u$ be a subharmonic function on an open set $U$ in $\mathbb{C}$, and let $v$ be an upper semicontinuous function on $U$ such that $u\leq v$ almost everywhere. Prove that $u\leq v$ everywhere. ...
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### Power series expansion of a complex function problem

I don't know what is the function of the "sup" in $\lim \sup_{n\to \infty} |\beta_n|^{1/n}$ and how to Compute the first three terms of the Laurent expansion of $1/f (z)$ about $z = 0$
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### homeomorphisms of the real line

Given a homeomorphism $h$ of the extended real line. Is it true that there exists an extension $\hat h$ of $h$, which is a Mobius transformation of a hyperbolic space $\mathbb{H}$? Any hints are ...
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### Prove that $\overline{f(z)}$ is differentiable at $a \in D(0;1)$ if and only if $f'(a)=0$

Let $f$ be holomorphic in $D(0;1)$ and define $k$ by $k(z)=\overline{f(z)}$. Prove that $k$ is differentiable at $a\in D(0;1)$ if and only if $f'(a)=0$. What I tried was first, assuming $k$ is ...
### If $f$ is a smooth real valued function on real line such that $f'(0)=1$ and $|f^{(n)} (x)|$ is uniformly bounded by $1$ , then $f(x)=\sin x$?
Let $f : \mathbb R \to \mathbb R$ be a smooth ( infinitely differentiable everywhere ) function such that $f '(0)=1$ and $|f^{(n)} (x)| \le 1 , \forall x \in \mathbb R , \forall n \ge 0$ ( as usual ...