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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.

2,772 questions

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Can this integral be related to the incomplete Gamma function or another known special function?

In the link one can find many integral represntations of the incomplete Gamma function. In particular: $$\Gamma\left(a,z\right)=\frac{z^{a}e^{-z}}{\Gamma\left(1-a\right)}\... user avatar Medo

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Line integral on complex numbers

I want to solve this integral integral $$\int_{\gamma} \frac{e^{iz}}{z^2},\quad \gamma(t)=e^{it}\text{ and } 0 \leq t\leq 2\pi$$ My attempt: $$f(\gamma(t))=\frac{e^{ie^{it}}}{{(e^{it})}^2}={(e^i)^{e^{... user avatar Fermatto

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How to prove that the following complex integral vanishes to 0?

Consider the path $\mu:[0,2\pi]\rightarrow\mathbb{C}, t \mapsto \left\{\begin{matrix} Re^{it} & \text{if } 0 \leq t\leq \pi\\ \frac{R}{\pi}(2t-3\pi) & \text{if } \pi \leq t\leq 2\pi\\ \end{... user avatar cnymfais

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Find $\int_{C} \sec^2{z} \cdot dz$ Where C be any path from $\frac{\pi i}{4}$ to $\frac{\pi}{4}$ in the unit disk.

We have to find $$ \int_{C} \sec^2{z} \cdot dz$$ Where C be any path from $\frac{\pi i}{4}$ to $\frac{\pi}{4}$ in the unit disk. We know that, $$Z = x + iy$$ so, $$dz = dx + idy$$ so the integral ... user avatar newdeep ji

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Finding Residues by Calculus of Residues [closed]

Find the residues of the function ${\dfrac{e^z}{z^4}}$. Have no clue on how to solve this, help. user avatar sarthakjoleya

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integrating multiplication of complex functions

In my calculations on a problem of quantum optics, I faced an integral of the following form \begin{align} \frac{1}{\pi}\int e^{(\beta^* \alpha)} f(\alpha^*) d^2 \alpha \end{align} with $\alpha$ and ... user avatar Eliii

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How to integrate the following $\int\limits_{-\infty}^{\infty} \dfrac{1}{t - i \alpha} \dfrac{1}{t + i \beta/2} \dfrac{1}{(\tau - t) - i\alpha} dt$

Recently encountered an integral of the following kind: $$ \int\limits_{-\infty}^{\infty} \dfrac{1}{t - i \alpha} \dfrac{1}{t + i \beta/2} \dfrac{1}{(\tau - t) - i\alpha} dt ,$$ where $\alpha, \beta, ... user avatar Sl0wp0k3

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Integral of multiplication of complex functions [closed]

In my calculations on a problem of quantum optics, I faced an integral of the following form \begin{align} \frac{1}{\pi}\int e^{(\beta^* \alpha)} f(\alpha^*) d^2 \alpha \end{align} with $\alpha$ and ... user avatar Eliii

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Contour integration in complex analysis [duplicate]

How to carry out the following integration?$$\int_{0}^{\infty}\frac{x^6}{(x^4+a^4)^2}dx$$I tried doing it using contour integration but it turned out to be extremely complicated. A help would be ... user avatar user231188

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How to deal with numerator in contour integration

Consider the integral $$ \oint dx\, \frac{g(x)}{x - x_{0}} $$ where $x_{0} \in \mathbb{C}$ is the position of a pole for the integrand that we assume to be integrated along a closed path encircling ... user avatar lux

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Interesting result in a complex integration

Studying for my complex-analysis exam I found an interesting integral. The activity consisted of calculating a complex integral around a curve using the residue theorem. The integral had the form $$ ... user avatar AlexSp3

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Issues with calculating integral $\int_{0}^{+\infty} \frac{x\sin x}{(x^{2} + a^{2})^{2}} dx$ using residues.

I have this integral $I = \int_{0}^{+\infty} \frac{x\sin x}{(x^{2} + a^{2})^{2}} dx$. I tried to calculate it myself, but apparently I'm using the wrong residues formula, the answer doesn't come out. $... user avatar Varga

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Solve $\int_{-\pi}^{\pi}{\frac{d\phi}{a+b\cos \phi}}$ using residues.

I do a variable substitution by adding $\pi$ to $\phi$ and the sign of $\cos$ changes. $\int_0^{2\pi}{\dfrac{d\phi}{a-b\cos \phi}} = 2\pi \sum_{k=1}^n {res \frac{1}{z} \big [(a-b \frac{1}{2}(z + \frac{... user avatar H-a-y-K

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Closed form solution of an improper integral

The integrals $$ I_1=\int_0^{+\infty} \frac{\exp(-\sqrt{x^2 + c})}{\sqrt{x^2 + c}} \, dx,\qquad I_2 = \int_0^{+\infty} \sqrt{x^2 + c} \, \exp(-\sqrt{x^2 + c}) \, dx $$ where $c\in \mathbb{C}$ and $\... user avatar GentleMin

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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 ... user avatar Aiswarya T

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