# Questions tagged [complex-analysis]

For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.

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### A question related to the complex derivative of a function

I am working on the following problem. Consider a continuous function $\phi:[-1,1]\to \mathbb{C}$ on $[-1,1]$. Let $$g(z):=\int_{-1}^{1}\frac{\phi(t)}{t-z}\:dt.$$ I am interested in finding the ...
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### A question related to an inequality involving two entire functions

I am currently trying to solve the following problem from a previous qualifying exam. Let $\alpha,\beta: \mathbb{C}\to \mathbb{C}$ be two non-constant entire functions with exactly the same zeroes of ...
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### Degree 2 Holomorphic Map Between Annuli

Consider the annulus $A(D) = \{D < |z| < 1\}$ where $0 \leq D < 1$. I need to find all $D$ such that there is a holomorphic map $f: A(D) \rightarrow A(D)$ of degree 2. By degree 2, I mean ...
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### Calculating ζ(2) [closed]

I'm looking for a way to calculate $\zeta(2)$ via the integral $$\zeta(2) = \int_{0}^{\infty} \dfrac{x}{e^x - 1}\, dx$$ without using a contour integral or appealing to the sum of inverse squared ...
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### Can we decompose complex differential equation into two real?

there is a simple system: $$-y''(x) = f(x) \\ y(0) = 0 \\ y(1) = 0$$ where $y$ and $f$ is complex function of real variable $x$. Is it legal to decompose this functions into two real (four real for ...
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1 vote
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### How to apply the second mean value theorem on $\int_{Y^2}^{{Y'}^2} z^{-1/2} e^{2 \pi i N z} dz$?

$Y' > Y > 0$ is given. $N$ is a positive integer. How we can apply the second mean value theorem on $\int_{Y^2}^{{Y'}^2} z^{-1/2} e^{2 \pi i N z} dz$ to conclude that this integral has absolute ...
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### A complex function which is complex differentiable only at 0 but not even continuous elsewhere

Can you give an example of a function $f:\mathbb{C} \to \mathbb{C}$ which is complex differentiable at $0$ but not even continuous at all $z\neq 0$?
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1 vote
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### Contour integration with $\int_{-\infty}^{\infty}{\frac{1-\cos(2z)}{z^2(z^2+1)}}\, dz$

$$\int_{-\infty}^{\infty}{\frac{1-\cos(2z)}{z^2(z^2+1)}}\, dz$$ The contour is shaped like this (image from this answer) With $\epsilon$ being the radius of the smaller semi-circle. The integral over ...
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This is a question from NBHM $2022$ exam. It asks to find the area of the region $\{z+\frac{z^2}{2} \mid z\in \mathbb{C},|z| \leq 1\}$ Now $z+\frac{z^2}{2}$ = $\frac{(z+1)^2}{2}-\frac{1}{2}$. The $-\... 1 vote 1 answer 46 views ### f is holomorphic on the unit disk. Show that there exist an sequence${z_n}$such that$|z_n|$converges to 1 and${f(z_n)}$is bounded [duplicate] I'm trying to prove this by contradiction, which comes out that every limit of$|f(z)|$on boundary tends to infinity. But I have no further idea, any help? • 33 2 votes 2 answers 60 views ### Laurent Series given some annuli The question is to represent$g(z) := \frac{1}{(z-3)(z+1)}$by a Laurent series, with regards to the annuli$A_1:= \{z \in \mathbb{C} : 1 < |z| < 3\}$and$A_2:= \{z \in \mathbb{C} : 3 < |z|\}...
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I am working on a problem in which we take some entire function $F$ which has zeros at every integer square root, i.e. $F(\sqrt{n}) = 0$ for every $n \in \mathbb{Z}^+$. I need to show that, for any ...