# 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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### Ordering complex numbers - When is it right and when is it not?

I know there are a lot of questions related to this topic. But, I have one specific doubt. If we order complex numbers, does that mean that we are wrong all the time? If we say $4 + 3i < 5 + 7i$. I ...
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### Modern complex analysis book (any suggestions ?)

Can someone suggest any modern books for complex analysis (in pdf if possible)? With nice looking text and formulas, examples, chapters, chapter reviews and so on. It is easier and more interesting ...
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### A Proof of Residue Theorem on a Compact Riemann Surface

Usually a proof of the Residue Theorem on a Compact Riemann Surface uses the crucial fact that Holomorphic forms are closed. I tried to write a proof and somehow I didn't use that fact anywhere. Can ...
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### Explicit holomorphic differentials on $y^3 = P(z)$ riemann surface

Let X be a compact Riemann surface of genus $g$ corresponding to the equation $w^3 = P(z)$, P is a polynomial without multiple roots and $degP = 8$ It's known that there are $g$ holomorphic ...
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### What are some resources for learning mathematical writing?

I'm looking for recommendations for some resources which will teach me how to convey ideas through consistent mathematical writing? For example, the book mathematical thinking and writing is one ...
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### An application of the Paley-Wiener theorem

Extract of an article: "the Laplace transform of $T(t)f$ is an entire function and since the resolvent is meromorphic of finite exponential type it must be of finite exponential type, say $\nu$, too. ...
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### Describe the Riemann Surface for $z^2 = w + \frac{1}{w}$

Describe the Riemann Surface for $z^2 = w + \frac{1}{w}$. Not sure if my process is correct. We can see $z = \frac{\sqrt{w^2+1}}{\sqrt{w}}$. So there are branch points at 0, i, -i. We have two layers, ...
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### identities other than $(a-b)^2=(a+b)^2-4ab$? [on hold]

We know that for any integer (or real in general) the following equation $$(a-b)^2=(a+b)^2-4ab$$ holds for all $a,b$. I am looking for some other identities which involves $a+b$ and $ab$ and ...
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### How to find the branch cuts of $\sqrt{g(z)}$ and the contour integral $\int_{z_1}^{z_2}d z\sqrt{g(z)}$

I need to evaluate the following integral: \begin{equation} \int_{z_1}^{z_2} d z \sqrt{g(z)}, \end{equation} where the function $g(z)$ is given by \begin{equation} g(z)=-\left(\alpha-\frac{\beta}{...