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.
5,646
questions
26
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$\sin(x)$ infinite product formula: how did Euler prove it?
I know that $\sin(x)$ can be expressed as an infinite product, and I've seen proofs of it (e.g. Infinite product of sine function). I found How was Euler able to create an infinite product for sinc by ...
- 2,041
25
votes
4
answers
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Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$.
Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$.
My try :
I consider $h(z)=\frac{f(z)}{g(z)}$. If I prove that $h(z)$ is ...
- 4,141
12
votes
6
answers
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Sum of nth roots of unity
Question: If $c\neq 1$ is an $n^{th}$ root of unity then, $1+c+...+c^{n-1} = 0$
Attempt: So I have established that I need to show that $$\sum^{n-1}_{k=0} e^{\frac{i2k\pi}{n}}=\frac{1-e^{\frac{ik2\pi}...
- 1,191
12
votes
5
answers
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$|e^a-e^b| \leq |a-b|$ for complex numbers with non-positive real parts
Came across this problem on an old qualifying exam: Let $a$ and $b$ be complex numbers whose real parts are negative or 0. Prove the inequality $|e^a-e^b| \leq |a-b|$.
If $f(z)=e^z$ and $z=x+iy$, ...
- 4,666
11
votes
3
answers
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Integral evaluation $\int_{-\infty}^{\infty}\frac{\cos (ax)}{\pi (1+x^2)}dx$ [duplicate]
Possible Duplicate:
Calculating the integral $\int_{0}^{\infty} \frac{\cos x}{1+x^2}\mathrm{d}x$ without using complex analysis
I need help to evaluate the integral of
$$\int_{-\infty}^{\infty}\...
- 3,613
1
vote
3
answers
3k
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How to find $\lim_{n \to \infty}|\left( 1+\frac{z}{n}\right)^{n}|$? [duplicate]
This is a step on the way to proving
$$
\lim_{n \to \infty}\left(1 + \frac{z}{n}\right)^{n} = e^{z}.$$
I'm looking for an answer without
1) a summation or
2) a logarithmic function.
So far, I ...
user100463
29
votes
3
answers
57k
views
Contour integral for $x^3/(e^x-1)$?
What contour and integrand do we use to evaluate
$$ \int_0^\infty \frac{x^3}{e^x-1} dx $$
Or is this going to need some other technique?
- 26.7k
20
votes
1
answer
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Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function
If you map the nth roots of unity $z$ with the function $-W(-z/e)$ you get decent starting points for some root finding algorithm to the roots of the scaled truncated taylor series of $\exp$. Here W ...
user11260
18
votes
1
answer
7k
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Show that if $|f(z)| \leq M |z|^n$ then $f$ is a polynomial max degree n
I can't prove this statement, can anybody show me how to prove it?
$$f:\mathbb{C}\rightarrow \mathbb{C} \in \mathcal{O}(\mathbb{C}), \exists n\in \mathbb{N}, R >0 , M>0 : |f(z)| \le M|z|^{n} \...
- 233
14
votes
4
answers
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Why is $2\pi i \neq 0?$ [duplicate]
We know that $e^{\pi i} = -1$ because of de Moivre's formula. ($e^{\pi i} = \cos \pi + i\sin \pi = -1).$
Suppose we square both sides and get $e^{2\pi i} = 1$(which you also get from de Moivre's ...
- 3,462
13
votes
2
answers
3k
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Complex polynomial and the unit circle
Given a polynomial $ P(z) = z^n + a_{n-1}z^{n-1} + \cdots + a_0 $, such that
$\max_{|z|=1} |P(z)| = 1 $
Prove: $ P(z) = z^n $
Hint: Use cauchy derivative estimation
$$ |f^{(n)} (z_0)| \...
- 582
11
votes
5
answers
8k
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Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis
How do I compute
$$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$
What I am doing is take
$$f(z)=\frac{(\log z)^2}{1+z^2}$$
and calculating
$\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log z)^2}{1+z^2}$
...
- 245
51
votes
3
answers
18k
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Proof that 1-1 analytic functions have nonzero derivative
I recently saw a lecturer prove the following theorem (assuming the result that every analytic function is locally 1-1 whenever its derivative is nonzero): Let $\Omega \subset \mathbb{C}$ be open, and ...
- 2,432
45
votes
16
answers
37k
views
Complex Analysis Book [duplicate]
I want a really good book on Complex Analysis, for a good understanding of theory. There are many complex variable books that are only a list of identities and integrals and I hate it. For example, I ...
34
votes
4
answers
12k
views
How can it be shown that $\mathrm{Aut}(\mathbb{C})=\{f\,|\,f(z)=az+b,a\neq 0\},$ is defined as bijective ...
How can it be shown that
$$\mathrm{Aut}(\mathbb{C})=\{f\,|\,f(z)=az+b,a\neq 0\},$$
where an automorphism of $\mathbb{C}$ is defined as a bijective entire function with entire inverse?
If $f$ is of the ...
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