# 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.

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### How does one prove that a given function is localy summable? [closed]

For example the function ln(x+iy)
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### Methods on finding Automorphism group on simply connected subset of $\mathbb{C}$.

Suppose $D$ is the unit disc and I want to find Aut$(D-\{0\})$ and Aut$(\mathbb{C}-\{0\})$. My approach: For the first automorphism group, I think immediately about Aut$(D)$ and if some how I can pick ...
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### What is $\int_0^{\infty}\sin(ax)\cos(bx)dx$ ? where $a$ and $b$ are constant. [closed]

I am solving an E&M problem and I can't get the solution of integration above. I guess it is something to do with residue theorem, but I still can't solve it. Please help...
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### How to write $\sqrt{-1}$ in the base $\sqrt{-2}$ [closed]

I I’m trying to make a system that would be able to display all numbers (real and nonreal) with one digit string. In theory, you could do this in base $\sqrt{-2}$, so I tried to write some examples by ...
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### Finding $\int_\Gamma \frac{z f'(z)}{f(z)} \, dz$ over a given contour [duplicate]

Let $f(z)=z^4-2z^3+2z^2-3z+60$ and let $\Gamma$ be the circle $|z|=5$. I want to find $$\int_\Gamma \frac{z f'(z)}{f(z)} \, dz$$ Supposing we had $f'(z)$ in the numerator instead of $z f'(z)$, this ...
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### For an analytic function $f$ in a domain $D$, if $\log|f|$ is harmonic in a neighborhood of $\partial D$ then $f\in C(\bar D)$.

I have stuck in one place while reading a paper. If $\phi$ is an analytic function on $D$, where $D$ is bounded multiple connected domains in $\mathbb{C}$. Now given that $\log\lvert \phi\rvert$ is ...
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### How to show a complex-valued function defined by a series converges on complex plane [closed]

I studied about Weierstrass $P$-function (a.k.a Weierstrass elliptic function). In order to prove the fact that this function converges to a meromorphic function on the whole complex plane, my ...
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