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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Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.
Find the closed form of $$\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}.$$
We can use the Fourier series of $e^{-bx}$ ($|x|<\pi$) to find $$\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}.$$ But here ...
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What's the difference between $\mathbb{R}^2$ and the complex plane?
I haven't taken any complex analysis course yet, but now I have this question that relates to it.
Let's have a look at a very simple example. Suppose $x,y$ and $z$ are the Cartesian coordinates and ...
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What is the intuition behind the Wirtinger derivatives?
The Wirtinger differential operators are introduced in complex analysis to simplify differentiation in complex variables. Most textbooks introduce them as if it were a natural thing to do. However, I ...
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About the limit of the coefficient ratio for a power series over complex numbers
This is my first question in mathSE, hope that it is suitable here!
I'm currently self-studying complex analysis using the book by Stein & Shakarchi, and this is one of the exercises (p.67, Q14) ...
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Maximum of sum of finite modulus of analytic function.
Let $f_1,f_2,\ldots,f_n $ be analytic complex functions in domain $D$. and $f = \sum_{k=1}^n|f_k|$ is not constant.
Can I show the maximum of $f$ only appears on boundary of $D\,$?
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The Integral that Stumped Feynman?
In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only ...
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Determining precisely where $\sum_{n=1}^\infty\frac{z^n}{n}$ converges?
Inspired by the exponential series, I'm curious about where exactly the series $\displaystyle\sum_{n=1}^\infty\frac{z^n}{n}$ for $z\in\mathbb{C}$ converges.
I calculated
$$
\limsup_{n\to\infty}\...
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How to rigorously justify "picking up half a residue"?
Often in contour integrals, we integrate around a singularity by putting a small semicircular indent $\theta \rightarrow z_0 + re^{i\theta}$, $0 \leq \theta \leq \pi$ around the singularity at $z_0$.
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half iterate of $x^2+c$
I'm looking for literature on fractional iterates of $x^2+c$, where c>0. For c=0, generating the half iterate is trivial.
$$h(h(x))=x^2$$
$$h(x)=x^{\sqrt{2}}$$
The question is, for $c>0,$ and $x&...
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Infinite Series $\sum\limits_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}$
I'm looking for a way to prove
$$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$$
I know that
$$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{1}{4^{2m+...
user97601
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Let $q \in \mathbb C$, $\|q\|=1$ and $q^n \neq 1, \forall n \in \mathbb N$. Show that $\{q^n: n \in \mathbb N\}$ is dense in $S^1$
Let $q \in \mathbb C$, $|q|=1$ and $q^n \neq 1, \forall n \in \mathbb N$. Show that $\{q^n: n \in \mathbb N\}$ is dense in $S^1$.
My attempt: As $(q^n)$ is limited, there is subsequence $(q^{n_j})$ ...
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When is a function satisfying the Cauchy-Riemann equations holomorphic?
It is, of course, one of the first results in basic complex analysis that a holomorphic function satisfies the Cauchy-Riemann equations when considered as a differentiable two-variable real function. ...
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Uniform limit of holomorphic functions
Let $\{f_n\}$ be a sequence of holomorphic functions defined in a generic domain $D \subset \mathbb{C}$. Suppose that there exist $f$ such that $f_n \to f$ uniformly.
My question is: is it true that $...
user67133
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Application of Liouville's Theorem
Let $f(z)$ be an entire function such that $$|f(z)|<\frac{1}{|\text{Im}(z)|},\qquad z\in\Bbb C-\Bbb R.$$ The question asked me to prove that $f(z)=0$. At least looking at it, it really seems to ...
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Inequality with complex numbers
Consider the following problem.
Fix $n \in \mathbb N$. Prove that for every set of complex numbers $\{z_i\}_{1\le i \le n}$, there is a subset $J\subset \{1,\dots , n\}$ such that
$$\left|\sum_{j\...
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