Questions tagged [complex-analysis]
For questions mainly about the 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 variable.
54,267 questions
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Complex logarithm base 1
Is a logarithm with base 1 defined in the field of complex numbers? I have not found any information about this. In real numbers, this is uncertain because $ \ln(1) = 0 $ and
$ \log_a(b)= \frac {\ln(...
- 10.6k
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Generalization of Schwarz's Lemma
I am reading Lectures on Riemann Surfaces by Otto Forster. He says: (p.110)
The following lemma may be viewed as a generalization of Schwarz's lemma. Let $D,D'$ be a pair of open subsets of $\mathbb{...
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Showing $\int_0^{2\pi} \log|1-ae^{i\theta}|d\theta=0$
This is a homework problem for a second course in complex analysis. I've done a good bit of head-bashing and I'm still not sure how to solve it-- so I might just be missing something here. The task is ...
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Finding zeros of the product of two analytic functions
I know when finding the order of a zero, $z_0$, of an analytic function, you can just differentiate the function until substituting $z_0$ in does not give zero. Instead, to save time on ...
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Showing a function map to itself
Let $ D = \{ z \in \mathbb{C} : |z| < 1\}$. Fix $ w \in D$ and define $f: \bar{D} \to \mathbb{C}$ by $$f(z) = \frac{w-z}{1-\bar{w}z}$$
Show the following:
$f$ maps $D$ to $D$ and $\partial D$ to $\...
CommunityBot
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$T$ is torus. $H_1(T,\mathbb Z)\to \operatorname{Hom}(\Omega^1(T),\mathbb C)$ is injection.
Let $T=\frac{V}{\Lambda}$ be a torus where $V$ is a complex vector space of dimension $n$ and $\Lambda$ is a rank $2n$ lattice in $V$. $\Omega^1(T)$ is the space of holomorphic 1-forms of $T$. $\dim_C(...
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problems on Analytic function [closed]
I have a problem with this
Let S be the disk $|z|<3$ in the complex plane and let $f:S \rightarrow C$ be an analytic function such that $f(1+\frac{\sqrt 2}{n}i)=-\frac{2}{n^2}$ for each natural ...
CommunityBot
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Fourier transform of a function with exponential and powers [closed]
How can I calculate this Fourier transform $F(y)$ ?
$$F(y)= \int_0^{\infty}(1+x)^{\frac{1}{2}} x^{-\frac{1}{2}-a} e^{-a x} \cos(2 \pi xy) dx$$
with $a$ complex ($0<Re(a)<\frac{1}{2}$)
This is ...
CommunityBot
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Problem evaluating a contour integral using parametrization
I tried to solve the following contour integral:
$$
\oint_\gamma {\frac{{dz}}{{z - c}}}
$$
Where $\gamma$ is a disk centered at the origin. In order to do so, I used the following parametrization:
$$
...
- 443k
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Is there a dense set of Lipschitz functions on the unit ball which all peak at the same point on the boundary?
I posted this same question over 2 years ago on MO but didn't get any comments or answers. The question is admittedly quite narrow, but I am still very interested in its resolution.
Let $U$ be the ...
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Asymptotic evaluation of integral method of steepest descent
The question asks to show that the leading term of the integral
$$
\int_{-\infty}^\infty (1+t^2)^{-1}\exp\left(ik(t^5/5+t)\right) dt
$$
for large $k$ using the method of steepest descent is equal to
...
- 13.4k
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(2,3,7) tiling of Hurwitz surface
I'm reading through the paper On the Geometry of Hurwitz Surfaces for an undergraduate project. Apologies for my basic questions; I've not met these ideas before.
In the abstract, the source says:
By ...
- 21.1k
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Is the set of finite Blaschke products a graded ring?
I have heard tell that there are many analogies between Blaschke products and polynomials.
A finite Blaschke product is a rational function on the complex unit disk $B(z)=\mu\prod_{i=1}^{\deg B}\frac{...
- 3,745
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Does the closest root of a n-th partial sum of a power series to a root of the power series converge?
Let
$$
f(z)=\sum_{k\ge \text{0}} b_{k}\, z^{k}
$$
be a power series with complex coefficients, and suppose $a\in\mathbb {C}$ satisfies $f(a)=0$.
For each index $n$, consider
$$
f_{n}(z)=\sum_{k=\text{...
- 30.9k
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Uniqueness of holomorphic function satisfying differential equation
I am asked to show that there exists at most one holomorphic function $f: \mathbb{C} \to \mathbb{C}$, such that both
$$ f'(z) = \sin(z)f(z) + e^{z^2}$$
and $f(0) = 3+2i$ are satisfied. I have, to be ...
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