# 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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### For which alpha does this system of equations spiral towards the orgin?

I am stuck with a math problem and I hope that somebody could help me in the right direction. Question: For which values of $\alpha$ do the trajectories of the solutions of the systems $x^{\prime}=A x$...
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### How objectively do we evaluate likelihood of two very rare events?

I've been pondering the concept of objectiveness to evaluate likelihood and predictability of extremely rare events. I'd like to demonstrate my questions over trivial hypothethical examples below. ...
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### Properties of the critical strip [closed]

I'm searching for the known properties of the critical strip. I have searched a bit on the web but did not find the awnsers i need. Regards, Justin Edit: I’m searching for facts that are known ...
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### $z_0+r*e^{2\pi imt}$ meaning of $m$

I know that $f:[0,1] \to \mathbb{C}, f(t)=z_0+r*e^{2\pi imt}, r>0, m\in\mathbb{Z},z_0\in\mathbb{C}$ parameterizes the circle around $z_0$ with radius $r$. But how does $m$ affect the ...
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### Evaluate $PV \int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2}dx. a>0$

Evaluate $PV \int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2}dx. a>0$ using residues. So I have a theory how to calculate $PV \int_{-\infty}^{\infty} f(x)e^{iax}dx$ a>0, but I don’t know how to ...
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### When will the quadratic equation $z^2+z_1z+z_0=0$ have a root lie on the unit circle

Very similar to this question, but what if the coefficients are complex? Is there a necessary and sufficient condition to guarentee that there is at least a root on unit circle for $z^2+z_1z+z_0=0$ ...
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### $f:D(0,1)\to \mathbb{C}$ , $e^{f(z)}$ is constant $\Rightarrow f$ is constant

Let $f:D(0,1)\to \mathbb{C}$ continuous and $e^{f(z)}$ is constant $\forall z\in D(0,1)$ show $f$ is constant. $\cdot$ $e^f=c=e^{x_0+iy_0}$ and $c\neq 0$ $\Rightarrow f-(x_0+iy_0)=k(z)2\pi i$, ...
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### how to find the table integral that contains factorial of n?

I read in this page that" There are lots of definite integrals that depend on a parameter n∈N and whose result contains factorials of n " I am really struggle with this integral \begin{align*...
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### Description of complete analytic function for $\sqrt{4z-\sqrt[3]{z}}$.

The problem is to describe all branches and all the curves of analytic continuation of $\sqrt{4z-\sqrt[3]{z}}$. I started with representing function in a way $\sqrt{4w^3-w}\circ\sqrt[3]{z}$ So, there ...
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### Derivation of Area theorem (conformal mapping).

While solving the Area theorem , i'm facing trouble in understanding the equation in these two black boxes, i know how they write $\displaystyle A=\frac{1}{2}\int_{c} R^2 d\phi$, but how they got its ...
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### Contour integration of $\int_{0}^{+\infty}\frac{x^2\cos x}{\cosh x} \,{\rm d}x$

Using contour integration, find $$\int_{0}^{+\infty}\frac{x^2\cos x}{\cosh x} \,{\rm d}x$$ How to calculate it? I never worked with integrals of this type.
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