# Questions tagged [complex-integration]

This is for questions about integration methods that use results from complex analysis and their applications. The theory of complex integration is elegant, powerful, and a useful tool for physicists and engineers. It also connects widely with other branches of mathematics.

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### Finding unknown from the given complex integral.

Find real number a such that $\oint_c \frac{dz}{z^2-z+a}=π$ where c is the closed contour |z-i|=1 taken in the counter clockwise direction. This is a question that has been asked in the 2021 NBHM ...
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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 ...
1 vote
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### Confusion about the asymptotic behavior of a Fourier integral

I would like to consider the following (inverse) Fourier transform: $$I(t) = \int_{-\infty}^{\infty} d\omega \, e^{i\omega t} (\omega + i \varepsilon)^{-1-i\alpha} \, ,$$ where the pole at $\omega = 0$...
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### Louville theorem and identity principle application on concrete examples

I have a two part question about Louville theorem, and identitiy principle: Loville theorem states that if function $f$ is holomorphic on $\mathbb{C}$ and $|f|$ is bounded $\Rightarrow$ $f$ is ...
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### Calculating the integral $I(w)=\int_\mathbb{R}\exp(-\pi(x+w)^2)dx$ for $w\in\mathbb{C}$

Suppose $I(w)=\int_\mathbb{R}\exp(-\pi(x+w))^2dx$ with $w\in\mathbb{C}$. There are two parts to this problem : i) I have to show that $I(w)$ converges uniformly on compact sets $K\subset\mathbb{C}$ ii)...
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### How to calculate $\int_{\partial D(0;a)} \frac{dz}{(z-a)(z-b)}$

How to calculate $\displaystyle\int_{\partial D(0;r)} \frac{dz}{(z-a)(z-b)}$ when $|a|<r<|b|$?. My first idea: I tried to separate by partial fractions, so for some $A,B$ \begin{equation} \...
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### A question about convergence to infinity of a sequence of integrals of functions with complex values

I have a fixed real number $t\neq 0$ and a function with complex values but defined in $\mathbb{R}$: $$\ell: \mathbb{R} \to \mathbb{C}, \quad \ell(x) = e^{itx} - 1$$ For each $n \in \mathbb{N}$, I ...
1 vote
### compute $\int_0^{\infty}\frac{\cos x}{x^2+a^2}dx$ in complex plane without using the residue
compute $\int_0^{\infty}\frac{\cos x}{x^2+a^2}dx$ in complex plane without using the residue I know that one way is to calculate this integral $\int_{c(r)}\dfrac{e^{iz}}{z^2+a^2}$ over half a circle, (...
I am trying to solve the following integral: \begin{equation}\label{1} \int \sin(\alpha t)\exp(-i\alpha t)\, \text{dt}. \end{equation} Using the exponential form of $\sin(\alpha t)$, the integral ...