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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To sum $1+2+3+\cdots$ to $-\frac1{12}$
$$\sum_{n=1}^\infty\frac1{n^s}$$
only converges to $\zeta(s)$ if $\text{Re}(s)>1$.
Why should analytically continuing to $\zeta(-1)$ give the right answer?
274
votes
32
answers
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Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$?
A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral:
$$\int\limits_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$$
Well, can ...
184
votes
17
answers
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How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?
Could you provide a proof of Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?
24
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2
answers
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Proving that a doubly-periodic entire function $f$ is constant.
Let $f: \Bbb C \to \Bbb C$ be an entire (analytic on the whole plane) function such that exists $\omega_1,\omega_2 \in \mathbb{S}^1$, linearly independent over $\Bbb R$ such that: $$f(z+\omega_1)=f(z)=...
123
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18
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Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?
I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute $\sum_{...
22
votes
4
answers
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Frullani 's theorem in a complex context.
It is possible to prove that $$\int_{0}^{\infty}\frac{e^{-ix}-e^{-x}}{x}dx=-i\frac{\pi}{2}$$ and in this case the Frullani's theorem does not hold since, if we consider the function $f(x)=e^{-x}$, we ...
50
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5
answers
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Entire function bounded by a polynomial is a polynomial
Suppose that an entire function $f(z)$ satisfies $\left|f(z)\right|\leq k\left|z\right|^n$ for sufficiently large $\left|z\right|$, where $n\in\mathbb{Z^+}$ and $k>0$ is constant. Show that $f$ is ...
17
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4
answers
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Show that $\left|\frac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$ when $|\alpha|,|\beta| < 1$
This is the question I'm stumbling with:
When $|\alpha| < 1$ and $|\beta| < 1$, show that:
$$\left|\cfrac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$$
The chapter that contains this ...
204
votes
28
answers
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What is a good complex analysis textbook, barring Ahlfors's?
I'm out of college, and trying to self-learn complex analysis. I'm finding Ahlfors' text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus ...
49
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6
answers
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A sine integral $\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$
The following question comes from Some integral with sine post
$$\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$$
but now I'd be curious to know how to deal with it by methods of ...
34
votes
3
answers
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An entire function whose real part is bounded above must be constant.
Greets
This is exercise 15.d chapter 3 of Stein & Shakarchi's "Complex Analysis", they hint: "Use the maximum modulus principle", but I didn't see how to do the exercise with this hint rightaway, ...
39
votes
7
answers
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Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$
How would I go about evaluating this integral?
$$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$
What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
18
votes
5
answers
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Definite integral calculation with poles at $0$ and $\pm i\sqrt{3}$
$$\int_0^\infty \frac{\sin(2\pi x)}{x(x^2+3)} \, dx$$
I looked at $\frac{e^{2\pi i z}}{z^{3}+3z}$, also calculated the residues, but they don't get me the right answer. I used that $\int_{-\infty}^\...
50
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2
answers
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What is wrong with this fake proof $e^i = 1$?
$$e^{i} = e^{i2\pi/2\pi} = (e^{2\pi i})^{1/(2\pi)} = 1^{1/(2\pi )} = 1$$
Obviously, one of my algebraic manipulations is not valid.
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2
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For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?
Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output.
I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and $\...
14
votes
1
answer
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Property of Entire Functions
Suppose $f$ and $g$ are entire functions with $|f(z)|\leq|g(z)|$ for all $z$.
How can we show that $f=cg$ for some complex constant $c$?
Thanks for any help :)
21
votes
7
answers
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Finite Sum $\sum\limits_{k=0}^{n}\cos(kx)$
I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$
I have some progress made, but I am stuck and could use some help.
What I did:
It ...
67
votes
8
answers
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Entire one-to-one functions are linear
Can we prove that every entire one-to-one function is linear?
40
votes
3
answers
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Showing that the roots of a polynomial with descending positive coefficients lie in the unit disc.
Let $P(z)=a_0+a_1z+\cdots+a_nz^n$ be a polynomial whose coefficients satisfy $$0<a_0<a_1<\cdots<a_n.$$
I want to show that the roots of $P$ live in unit disc. The obvious idea is to use ...
38
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2
answers
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Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer.
Using residues, try the contour below with $R \rightarrow \infty$ and $$\lim_{R \rightarrow \infty } \int_0^R \frac{1}{1+r^n} dr \rightarrow \int_0^\infty \frac{1}{1+x^n} dx$$
I've attempted ...
27
votes
7
answers
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Can we characterize the Möbius transformations that maps the unit disk into itself?
The Möbius transformations are the maps of the form $$ f(z)= \frac{az+b}{cz+d}.$$
Can we characterize the Möbius transformations that map the unit disk
$$\{z\in \mathbb C: |z| <1\}$$
into itself?...
14
votes
2
answers
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Intuition behind euler's formula [duplicate]
Possible Duplicate:
How to prove Euler's formula: $\\exp(i t)=\\cos(t)+i\\sin(t)$ ?
Hi, I've been curious for quite a long time whether it is actually possible to have an intuitive ...
10
votes
4
answers
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Infinite Series $\sum\limits_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}$
How can we prove that?
$$\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).$$
I think if we write the taylor ...
112
votes
3
answers
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$\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis
Inspired by this recently closed question, I'm curious whether there's a way to do the Gaussian integral using techniques in complex analysis such as contour integrals.
I am aware of the calculation ...
95
votes
10
answers
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"Where" exactly are complex numbers used "in the real world"?
I've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
38
votes
5
answers
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How do I rigorously show $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is?
I'm doing a bit of self study, but I'm uncomfortable with a certain idea. I want to show that $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is analytic, and by analytic I mean ...
9
votes
1
answer
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The Fourier transform of $1/p^3$
The Fourier transforms we use are
\begin{align}
\tilde{f}(\mathbf{p})&=\int d^3x\,f(\mathbf{x})
e^{-i\mathbf{p}\cdot\mathbf{x}}\\[5pt]
f(\mathbf{x})&=\int \frac{d^3p}{(2\pi)^3}\,\tilde{f}(\...
43
votes
6
answers
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Infinite Series $\sum\limits_{n=2}^{\infty}\frac{\zeta(n)}{k^n}$
If $f\left(z \right)=\sum_{n=2}^{\infty}a_{n}z^n$ and $\sum_{n=2}^{\infty}|a_n|$ converges then,
$$\sum_{n=1}^{\infty}f\left(\frac{1}{n}\right)=\sum_{n=2}^{\infty}a_n\zeta\left(n\right).$$
Since ...
17
votes
6
answers
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If $z_n \to z$ then $(1+z_n/n)^n \to e^z$
We are dealing with $z \in \mathbb{C}$.
I know that
$$
\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)...
14
votes
3
answers
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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 ...