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.
5,649
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Property of Entire Functions
Suppose $f$ and $g$ are entire functions with $|f(z)|\leq|g(z)|$ for all $z$.
How can we show that $f=cg$ for some complex constant $c$?
Thanks for any help :)
- 2,583
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7
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Finite Sum $\sum\limits_{k=0}^{n}\cos(kx)$
I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$
I have some progress made, but I am stuck and could use some help.
What I did:
It ...
- 23.2k
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8
answers
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Entire one-to-one functions are linear
Can we prove that every entire one-to-one function is linear?
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3
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Showing that the roots of a polynomial with descending positive coefficients lie in the unit disc.
Let $P(z)=a_0+a_1z+\cdots+a_nz^n$ be a polynomial whose coefficients satisfy $$0<a_0<a_1<\cdots<a_n.$$
I want to show that the roots of $P$ live in unit disc. The obvious idea is to use ...
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2
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Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer.
Using residues, try the contour below with $R \rightarrow \infty$ and $$\lim_{R \rightarrow \infty } \int_0^R \frac{1}{1+r^n} dr \rightarrow \int_0^\infty \frac{1}{1+x^n} dx$$
I've attempted ...
- 435
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7
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Can we characterize the Möbius transformations that maps the unit disk into itself?
The Möbius transformations are the maps of the form $$ f(z)= \frac{az+b}{cz+d}.$$
Can we characterize the Möbius transformations that map the unit disk
$$\{z\in \mathbb C: |z| <1\}$$
into itself?...
- 1,747
14
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2
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Intuition behind euler's formula [duplicate]
Possible Duplicate:
How to prove Euler's formula: $\\exp(i t)=\\cos(t)+i\\sin(t)$ ?
Hi, I've been curious for quite a long time whether it is actually possible to have an intuitive ...
- 3,227
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4
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Infinite Series $\sum\limits_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}$
How can we prove that?
$$\sum_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}=\frac{1}{3}e^{\frac{-x}{2}}x\left(e^{\frac{3x}{2}}-2\sin\left(\frac{\pi+3\sqrt{3}x}{6}\right)\right).$$
I think if we write the taylor ...
- 5,626
112
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3
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$\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis
Inspired by this recently closed question, I'm curious whether there's a way to do the Gaussian integral using techniques in complex analysis such as contour integrals.
I am aware of the calculation ...
- 4,014
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10
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"Where" exactly are complex numbers used "in the real world"?
I've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
- 2,771
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5
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How do I rigorously show $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is?
I'm doing a bit of self study, but I'm uncomfortable with a certain idea. I want to show that $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is analytic, and by analytic I mean ...
- 3,081
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The Fourier transform of $1/p^3$
The Fourier transforms we use are
\begin{align}
\tilde{f}(\mathbf{p})&=\int d^3x\,f(\mathbf{x})
e^{-i\mathbf{p}\cdot\mathbf{x}}\\[5pt]
f(\mathbf{x})&=\int \frac{d^3p}{(2\pi)^3}\,\tilde{f}(\...
- 505
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6
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Infinite Series $\sum\limits_{n=2}^{\infty}\frac{\zeta(n)}{k^n}$
If $f\left(z \right)=\sum_{n=2}^{\infty}a_{n}z^n$ and $\sum_{n=2}^{\infty}|a_n|$ converges then,
$$\sum_{n=1}^{\infty}f\left(\frac{1}{n}\right)=\sum_{n=2}^{\infty}a_n\zeta\left(n\right).$$
Since ...
- 5,626
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6
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If $z_n \to z$ then $(1+z_n/n)^n \to e^z$
We are dealing with $z \in \mathbb{C}$.
I know that
$$
\left(1+ \frac{z}{n} \right)^n \to e^{z}
$$
as $n \to \infty$. So intuitively if $z_n \to z$ then we should have
$$
\left(1+ \frac{z_n}{n} \right)...
- 28.1k
14
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3
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Entire function with vanishing derivatives?
Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be an entire function.
And assume that at each point, one of it's derivatives vanishes.
What can you say about $f$?
A hint suggests that $f$ must be a ...
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