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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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\...
- 1,542
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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
answers
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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.
- 3,967
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votes
2
answers
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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:
$$
\lim_{z \...
- 1,822
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votes
1
answer
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Every closed subset $E\subseteq \mathbb{R}^n$ is the zero point set of a smooth function
In Walter Rudin's Principles of mathematical analysis Exercise 5.21, it is proved that for any closed subset $E\subseteq \mathbb{R}$, there exists a smooth function $f$ on $\mathbb{R}$ such that $E=\{...
- 8,549
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votes
1
answer
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Showing that $|f(z)| \leq \prod \limits_{k=1}^n \left|\frac{z-z_k}{1-\overline{z_k}z} \right|$
I need some help with this problem:
Let $f\colon D \to D$ analytic and $f(z_1)=0, f(z_2)=0, \ldots, f(z_n)=0$ where $z_1, z_2, \ldots, z_n \in D= \{z:|z|<1\}$. I want to show that $$|f(z)|
\...
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votes
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
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5
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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
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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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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, ...
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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 ...
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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