All Questions
6
questions
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vote
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Proving a function is holomorphic in $\mathbb{C}\setminus[0,1]$
Let $h(t):[0,1]\to\mathbb C$ be a continuous function. Prove that $f:\mathbb C\setminus[0,1]\to\mathbb C $ defined by $f(z)=\int_0^1\frac{h(t)}{z-t}$ is holomorphic.
My attempt:
$$\lim_{z\to0}\frac{...
- 4,653
0
votes
1
answer
81
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Unable to think about argument involving integratibility of a function in a research paper
I am self studying a research paper in analytic number theory and I am unable to think about how G(x,z) is integrable with z belonging to Complex Number and |z|$\geq$1.
Adding relevant image
My ...
user775699
-1
votes
2
answers
684
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Why is $\int_{[0,1]} \frac{dw}{1-wz}$ is holomorphic in unit disc?
Expanding on the question and answer in: Prove $f(z)=\int_{[0,1]}\frac{1}{1-wz}dw$ is holomorphic in the open unit disk.
A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis ...
- 13.7k
3
votes
2
answers
1k
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Prove $f(z)=\int_{[0,1]}\frac{1}{1-wz}dw$ is holomorphic in the open unit disk.
Define $f:D[0,1] \rightarrow \mathbb C$ through
$$f(z)=\int_{[0,1]}\frac{1}{1-wz}dw$$
The integration path is from 0 to 1 along the real line.
Prove that $f$ is holomorphic in the open unit disk $D[0,...
- 2,098
0
votes
1
answer
81
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Show that $\lim_{r \to 0}\int_{0}^{2\pi}f(r e^{i \varphi})d\varphi = 2 \pi f(0) $
I need to show that $\lim_{r \to 0} \int_{0}^{2 \pi}f(r e^{i \varphi})d \varphi = 2 \pi f(0)$ if $f$ is continuous on a neighborhood of $z = 0$.
I was given the following hint: Write $\int_{0}^{2 \...
user100463
2
votes
1
answer
149
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Question about Continuity of Path Integrals
I have this continuous function $f:\mathbb{C}\rightarrow\mathbb{C}$ defined on an open set $\Omega$.
I also have a family of identical smooth curves up to translation $z_{t}:[a,b]\rightarrow\mathbb{C}$...
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