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.
5,636
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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?
perplexed
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 ...
user9413
184
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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: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?
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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: $$f(z+\omega_1)=f(z)=...
- 78.4k
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votes
18
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Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?
I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute $\sum_{...
- 55.8k
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votes
4
answers
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Frullani 's theorem in a complex context.
It is possible to prove that $$\int_{0}^{\infty}\frac{e^{-ix}-e^{-x}}{x}dx=-i\frac{\pi}{2}$$ and in this case the Frullani's theorem does not hold since, if we consider the function $f(x)=e^{-x}$, we ...
50
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5
answers
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Entire function bounded by a polynomial is a polynomial
Suppose that an entire function $f(z)$ satisfies $\left|f(z)\right|\leq k\left|z\right|^n$ for sufficiently large $\left|z\right|$, where $n\in\mathbb{Z^+}$ and $k>0$ is constant. Show that $f$ is ...
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4
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Show that $\left|\frac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$ when $|\alpha|,|\beta| < 1$
This is the question I'm stumbling with:
When $|\alpha| < 1$ and $|\beta| < 1$, show that:
$$\left|\cfrac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$$
The chapter that contains this ...
- 536
204
votes
28
answers
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What is a good complex analysis textbook, barring Ahlfors's?
I'm out of college, and trying to self-learn complex analysis. I'm finding Ahlfors' text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus ...
49
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6
answers
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A sine integral $\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$
The following question comes from Some integral with sine post
$$\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$$
but now I'd be curious to know how to deal with it by methods of ...
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34
votes
3
answers
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An entire function whose real part is bounded above must be constant.
Greets
This is exercise 15.d chapter 3 of Stein & Shakarchi's "Complex Analysis", they hint: "Use the maximum modulus principle", but I didn't see how to do the exercise with this hint rightaway, ...
- 9,706
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votes
7
answers
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Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$
How would I go about evaluating this integral?
$$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$
What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
user55225
18
votes
5
answers
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Definite integral calculation with poles at $0$ and $\pm i\sqrt{3}$
$$\int_0^\infty \frac{\sin(2\pi x)}{x(x^2+3)} \, dx$$
I looked at $\frac{e^{2\pi i z}}{z^{3}+3z}$, also calculated the residues, but they don't get me the right answer. I used that $\int_{-\infty}^\...
- 2,705
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2
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What is wrong with this fake proof $e^i = 1$?
$$e^{i} = e^{i2\pi/2\pi} = (e^{2\pi i})^{1/(2\pi)} = 1^{1/(2\pi )} = 1$$
Obviously, one of my algebraic manipulations is not valid.
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2
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For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?
Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output.
I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and $\...
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votes
1
answer
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Property of Entire Functions
Suppose $f$ and $g$ are entire functions with $|f(z)|\leq|g(z)|$ for all $z$.
How can we show that $f=cg$ for some complex constant $c$?
Thanks for any help :)
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21
votes
7
answers
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Finite Sum $\sum\limits_{k=0}^{n}\cos(kx)$
I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$
I have some progress made, but I am stuck and could use some help.
What I did:
It ...
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67
votes
8
answers
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Entire one-to-one functions are linear
Can we prove that every entire one-to-one function is linear?
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votes
3
answers
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Showing that the roots of a polynomial with descending positive coefficients lie in the unit disc.
Let $P(z)=a_0+a_1z+\cdots+a_nz^n$ be a polynomial whose coefficients satisfy $$0<a_0<a_1<\cdots<a_n.$$
I want to show that the roots of $P$ live in unit disc. The obvious idea is to use ...
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votes
2
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Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer.
Using residues, try the contour below with $R \rightarrow \infty$ and $$\lim_{R \rightarrow \infty } \int_0^R \frac{1}{1+r^n} dr \rightarrow \int_0^\infty \frac{1}{1+x^n} dx$$
I've attempted ...
- 435
27
votes
7
answers
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Can we characterize the Möbius transformations that maps the unit disk into itself?
The Möbius transformations are the maps of the form $$ f(z)= \frac{az+b}{cz+d}.$$
Can we characterize the Möbius transformations that map the unit disk
$$\{z\in \mathbb C: |z| <1\}$$
into itself?...
- 1,747
14
votes
2
answers
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Intuition behind euler's formula [duplicate]
Possible Duplicate:
How to prove Euler's formula: $\\exp(i t)=\\cos(t)+i\\sin(t)$ ?
Hi, I've been curious for quite a long time whether it is actually possible to have an intuitive ...
- 3,227
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votes
4
answers
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Infinite Series $\sum\limits_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}$
How can we prove that?
$$\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).$$
I think if we write the taylor ...
- 5,626
112
votes
3
answers
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$\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis
Inspired by this recently closed question, I'm curious whether there's a way to do the Gaussian integral using techniques in complex analysis such as contour integrals.
I am aware of the calculation ...
- 4,014
95
votes
10
answers
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"Where" exactly are complex numbers used "in the real world"?
I've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
- 2,771
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5
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How do I rigorously show $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is?
I'm doing a bit of self study, but I'm uncomfortable with a certain idea. I want to show that $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is analytic, and by analytic I mean ...
- 3,081
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The Fourier transform of $1/p^3$
The Fourier transforms we use are
\begin{align}
\tilde{f}(\mathbf{p})&=\int d^3x\,f(\mathbf{x})
e^{-i\mathbf{p}\cdot\mathbf{x}}\\[5pt]
f(\mathbf{x})&=\int \frac{d^3p}{(2\pi)^3}\,\tilde{f}(\...
- 505
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6
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Infinite Series $\sum\limits_{n=2}^{\infty}\frac{\zeta(n)}{k^n}$
If $f\left(z \right)=\sum_{n=2}^{\infty}a_{n}z^n$ and $\sum_{n=2}^{\infty}|a_n|$ converges then,
$$\sum_{n=1}^{\infty}f\left(\frac{1}{n}\right)=\sum_{n=2}^{\infty}a_n\zeta\left(n\right).$$
Since ...
- 5,626
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votes
6
answers
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If $z_n \to z$ then $(1+z_n/n)^n \to e^z$
We are dealing with $z \in \mathbb{C}$.
I know that
$$
\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)...
- 28.1k
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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 ...
- 1,051
13
votes
3
answers
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Complex zeros of the polynomials $\sum_{k=0}^{n} z^k/k!$, inside balls
this is a question from a Temple prelim exam, and i'm trapped in it! We have $p_n(z)=\sum_{k=0}^n\frac{z^k}{k!}$ and we have to prove that $\forall r>0 \quad \exists N\in\mathbb{N}$ s.t. $p_n(z)$ ...
- 2,111
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3
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Proving complex series $1 + \cos\theta + \cos2\theta +... + \cos n\theta $
So I have this result
$1 + z + z^2 + ... + z^n = \frac{z^{n+1}-1}{z-1}$
which I proved already. Now I am supposed to use that result and De Moivre's formula to establish this identity
$1 + \cos\...
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6
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Why: A holomorphic function with constant magnitude must be constant.
How can I prove the following assertion?
Let $f$ be a holomorphic function such that $|f|$ is a constant. Then $f$ is constant.
Edit: The more elementary the proof, the better. I'm working my ...
- 5,812
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votes
2
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Determine and classify all singular points
Determine and find residues for all singular points $z\in \mathbb{C}$ for
(i) $\frac{1}{z\sin(2z)}$
(ii) $\frac{1}{1-e^{-z}}$
Note: I have worked out (i), but (ii) seems still not easy.
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2
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Is L'Hopitals rule applicable to complex functions?
I have a question about something I'm wondering about. I've read somewhere that
L'Hopitals rule can also be applied to complex functions, when they are analytic.
So if have for instance:
$$
\lim_{z \...
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votes
1
answer
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Every closed subset $E\subseteq \mathbb{R}^n$ is the zero point set of a smooth function
In Walter Rudin's Principles of mathematical analysis Exercise 5.21, it is proved that for any closed subset $E\subseteq \mathbb{R}$, there exists a smooth function $f$ on $\mathbb{R}$ such that $E=\{...
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Showing that $|f(z)| \leq \prod \limits_{k=1}^n \left|\frac{z-z_k}{1-\overline{z_k}z} \right|$
I need some help with this problem:
Let $f\colon D \to D$ analytic and $f(z_1)=0, f(z_2)=0, \ldots, f(z_n)=0$ where $z_1, z_2, \ldots, z_n \in D= \{z:|z|<1\}$. I want to show that $$|f(z)|
\...
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14
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Infinite Series $\sum\limits_{n=1}^\infty\left(\frac{H_n}n\right)^2$
How can I find a closed form for the following sum?
$$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$
($H_n=\sum_{k=1}^n\frac{1}{k}$).
user97601
43
votes
5
answers
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Computing $\int_{-\infty}^{\infty} \frac{\cos x}{x^{2} + a^{2}}dx$ using residue calculus
I need to find $\displaystyle\int_{-\infty}^{\infty} \frac{\cos x}{x^{2} + a^{2}}\ dx$ where $a > 0$. To do this, I set $f(z) = \displaystyle\frac{\cos z}{z^{2} + a^{2}}$ and integrate along the ...
- 6,628
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5
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Evaluate: $\int_0^{\pi} \ln \left( \sin \theta \right) d\theta$
Evaluate: $ \displaystyle \int_0^{\pi} \ln \left( \sin \theta \right) d\theta$ using Gauss Mean Value theorem.
Given hint: consider $f(z) = \ln ( 1 +z)$.
EDIT:: I know how to evaluate it, but I am ...
- 2,197
21
votes
2
answers
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Computing $\int_{-\infty}^\infty \frac{\sin x}{x} \mathrm{d}x$ with residue calculus
This refers back to $\int_{-\infty}^\infty \frac{\sin x}{x} \mathrm{d}x = \frac\pi2$ already posted.
How do I arrive at $\frac\pi2$ using the residue theorem?
I'm at the following point:
$$\int \frac{...
- 261
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votes
4
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Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant.
While studying for an exam in complex analysis, I came across this problem. Unfortunately I was not able to solve it. Any help would be greatly appreciated.
Let $U ⊂ \mathbb{C}$ be a domain and $f_1, ...
- 261
16
votes
1
answer
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Zeta function zeros and analytic continuation
I'm learning about the zeta function and already discovered the intuitive proof of the Euler product and the Basel problem proof. I want to learn how to calculate the first zero of the Riemann Zeta ...
- 2,041
46
votes
5
answers
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Characterizing non-constant entire functions with modulus $1$ on the unit circle
Is there a characterization of the nonconstant entire functions $f$ that satisfy $|f(z)|=1$ for all $|z|=1$?
Clearly, $f(z)=z^n$ works for all $n$. Also, it's not difficult to show that if $f$ is ...
- 461
37
votes
1
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Which sets are removable for holomorphic functions?
Let $\Omega$ be a domain in $\mathbb C$, and let $\mathscr X$ be some class of functions from $\Omega$ to $\mathbb C$. A set $E\subset \Omega$ is called removable for holomorphic functions of class $\...
user31373
37
votes
4
answers
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Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.
Find the closed form of $$\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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votes
14
answers
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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 ...
- 2,777
56
votes
2
answers
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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 ...
- 1,289
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votes
2
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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) ...
10
votes
2
answers
3k
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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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