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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.
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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: $$\Gamma\left(a,z\right)=\frac{z^{a}e^{-z}}{\Gamma\left(1-a\right)}\... special-functions complex-integration- 2,499
Line integral on complex numbers
I want to solve this integral integral $$\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^{... integration complex-analysis complex-integration- 31
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{... complex-analysis complex-integration- 61
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 ... complex-integration- 229
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. complex-analysis complex-integration residue-calculus analytic-functions- 3
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 ... integration indefinite-integrals complex-integration quantum-mechanics quantum-information- 11
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, ... integration complex-analysis definite-integrals complex-integration- 426
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 ... integration complex-integration- 11
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 ... complex-analysis complex-integration cauchy-integral-formula- 1
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 ... contour-integration complex-integration residue-calculus- 723
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 $$ ... complex-analysis complex-integration residue-calculus- 315
Issues with calculating integral $\int_{0}^{+\infty} \frac{x\sin x}{(x^{2} + a^{2})^{2}} dx$ using residues.
I have this integral $I = \int_{0}^{+\infty} \frac{x\sin x}{(x^{2} + a^{2})^{2}} dx$. I tried to calculate it myself, but apparently I'm using the wrong residues formula, the answer doesn't come out. $... definite-integrals complex-integration residue-calculus- 1
Solve $\int_{-\pi}^{\pi}{\frac{d\phi}{a+b\cos \phi}}$ using residues.
I do a variable substitution by adding $\pi$ to $\phi$ and the sign of $\cos$ changes. $\int_0^{2\pi}{\dfrac{d\phi}{a-b\cos \phi}} = 2\pi \sum_{k=1}^n {res \frac{1}{z} \big [(a-b \frac{1}{2}(z + \frac{... complex-analysis complex-integration residue-calculus- 523
Closed form solution of an improper integral
The integrals $$ 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 $\... integration complex-integration- 1
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 ... complex-analysis complex-integration cauchy-integral-formula- 7
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