# 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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### Prove $f$ is identically zero in $\Omega = \{ z \in \Bbb C:|{\mathop{\rm Re}\nolimits} (z)| < 1,|{\mathop{\rm Im}\nolimits} (z)| < 1\}$

Let $\Omega = \{ z \in \Bbb C:|{\mathop{\rm Re}\nolimits} z| < 1,|{\mathop{\rm Im}\nolimits} z| < 1\}$ and consider the function $f:\bar\Omega\to\Bbb C$ continuous on $\bar\Omega$, analytic in ...
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### Finding domain where this complex logarithm identity holds

I have not had much exposure to more advanced complex analysis exercises, although this one might be easy, and the solution just completely escapes me. I am determining the set of complex numbers for ...
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### Analytic continuation of $\sum_{n=0}^{\infty} e^{-x E_n}$

Suppose we define a function $f(x)$ by the following sum: $$f(x)= \sum_{n=0}^{\infty} e^{-x E_n}$$ where $E_n$ is a sequence which is at most $O(n)$. It is known $f(x)$ does not form a natural boundry ...
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### Existence of biholomorphic map from unit disk to itself that interpolates one set of points

How do you prove that given two points $z_{1}, w_{1} \in D = \{z: |z|<1\}$, there exists a biholomorphic (bijective and analytic) function $f: D \to D$ such that $f(z_{1}) = w_{1}$? Perhaps using ...
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### How does one prove that two punctured disks are conformally equivalent? [closed]

Let D1 = {z: 0 < |z| < R1} and let D2 = {z: 0 <|z| < R2}. Prove that D1 and D2 are conformally equivalent.
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### Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$

Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$ I think I'm supposed to use the following chain of inequalities $$|e^z -1|\leq e^{|z|}-1 \leq |z|e^{|z|}$$ But ...
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### bounded components of the intersection of two planar domains

It seems to be intuitively clear that if U is a domain in the plane having a bounded complementary component C, then C is also a complementary component of the intersection of U with an open disk D ...
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### Are all the zeros of $1-a_2x^2+a_4x^4-a_6x^6+\cdots$ real for $a_{2n}>a_{2(n+1)}$ with $a_{2n+1}=0$ and $a_{2n}>0$?

This question is related to a previous question of mine. I was not pleased about the conditions I provided there. I had something different in mind but I failed in stating it. So here are the ...
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### Prove or refute that $\{p^{1/p}\}_{p\text{ prime}}$ to be equidistributed in $\mathbb{R}/\mathbb{Z}$
I've tried follow the Example 3 (see minute 30'40" of the reference), where is required the related Theorem (stated at minute 21') combined with Serre's formalism for $\mathbb{R}/\mathbb{Z}$ (also ...