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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.

The concept of definite integral of real functions does not directly extend to the case of complex functions, since real functions are usually integrated over intervals but complex functions are integrated over curves.

Surprisingly complex integration are not so complex to evaluate, oftenly simpler than the evaluation of real integrations. Some real integrations which are otherwise difficult to evaluate can be evaluated easily by complex integration, and moreover, some basic properties of analytic functions are established by complex integration only.

References:

https://en.wikipedia.org/wiki/Contour_integration

"An Introduction to the Theory of Analytic. Functions of One Complex Variable" by Lars Ahlfors

"Complex Variables and Applications " by James Ward Brown and Ruel V. Churchill

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how to evaluate this complex integral

$I=\int_{L} \frac{z}{\bar{z}}dz$, where L is $1\le|z|<2$, $Im(z)\ge 0$ traversed counterclockwise. thanks for helping. $I=\int_{L} \frac{z}{\bar{z}}dz=\int_{L}{z^2\over |z|}dz$,
Myshkin
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Integration in the complex plane of abs(z)

I have to integrate the function $|z|$ on the circle $|z-1|=1$. I tried to parameterize the circle by $z-1=\exp(it)$, but I don't know how to proceed. Can somebody help me please? thanks.
Rina
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A question about complex integral: $\int_{|z| = 1}\frac{1}{z^{2}\sin z}\,dz.$

I am stuck computing the following complex integral $$\int_{|z| = 1}\frac{1}{z^{2}\sin z}\,dz.$$ Any help will be appreciated.
z3wood
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Can I solve this using Complex integration?

I am trying to solve this question. $$\int_0^\infty \frac{1-\cos(x)}{x^2}dx$$ As this function $f(x) =\frac{1-\cos(x)}{x^2} $ has a removable singularity at $z = 0$.Is it possible to solve this using complex integration.If not why ? Other solutions…
ANUPAM BISHT
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How to take a complex integral?

I wish to understand how to take complex integrals and derivatives. For the simplest case, $\int{x+iy, dz}$ $f(z) dz = (u + iv)(dx + idy) = (u + iv) dx + (iu − v) dy = (udx − vdy) + i(vdx + udy)$ So $\int({x+iy})(dx+idy)$ $=(\int{xdx}-\int…
User3910
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Complex integration with trigonometric functions

I'm not sure how to solve this integral: $$ \int_{0}^{2\pi} [\frac{a+\cos(n\theta)}{a^2+1+2acos(n\theta)}] \ d\theta $$ SUGGGESTION: Use the function $ f(z)=\frac{1}{z^n+a} $ Solution (I developed some steps but I dont know how can I continue) Let…
Renato Collado Tello
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How to find integrals of the form $\int_{-r}^{+r}\frac{\gamma^{2}-1-y^{2}}{y-z\gamma} dy$

I'm trying to solve the above integral which will give a dispersion relation. Note: $\gamma$ and y are real but z is a complex variable.
S.S
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Can we multiply by -1 when flipping limits for integration over the complex plane?

I have this integral that goes from $ci$ to $0$, along the imaginary axis, and I'm finding that I seem to get different answers if I (1) Parametrize and evaluate it. (2) Multiply the whole thing by $-1$, flip the limits to going from $0$ to $ci$,…
Borealis
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$\int_{[1+iR, 1+2iR]}\frac{e^{z^2}}{z^2}dz \to 0$ as $R \to \infty$

Please help me with this problem, $\int_{[1+iR, 1+2iR]}\frac{e^{z^2}}{z^2}dz \to 0$ as $R \to \infty$ My attempt : $r(t)= 1+ itR$, $1\leq t\leq 2$ Then the integral becomes $\int_1^2\frac{e^{(1+ itR)^2}}{(1+ itR)^2}(1+ iR)dt$. Then how can I…
sukanta
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complex integral for Trigonometry integral solution

I have two integral that must solve with complex integrals I know how to solve it in normal way but my university professor told me to solve it in complex integral solution way. I know that it would solve by residue theorem but when I try to solve…
Mahdi Ahmadi
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Integrate $\int{ \frac{\sin(\pi z^2)+\cos(\pi z^2)}{\{(z-1)(z-2)\}^{4}} dz }$

I want to evaluate $$\int{ \frac{\sin(\pi z^2)+\cos(\pi z^2)}{\{(z-1)(z-2)\}^{4}} dz }.$$ This is the contour integration I came across. I know Cauchy's integral formula and Cauchy's integral formula for higher derivatives. First I separate $\cos$…
hood
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complex integration with elementary fuctions help

I needed some help on how to solve this problem. Integrate $$\oint_C \frac{\sin z}{4z^2-8iz}dz $$ where $C$ consists of boundaries of the squares with vertices $\pm3,\pm3i$ counterclockwise and $\pm1,\pm i$ clockwise. My approach was like, $$\oint_C…
therewasaduck
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Finding the limits when integrating a complex number

Evaluate $\int_c f(z) dz$ from $z(0,0)$ to $z=2+4i$ where $f(z)=x^2 -iy^2$ I know how to work this out and I know the answer is $24+\frac{8}{5}i$ However I do not understand why the limits for x are $0 \to 2$ and for y are $0 \to 4$
Al jabra
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Complex analysis integration method

How do you solve the integral $$\int^\infty_{-\infty}\frac{cos z}{z^2+9}dz$$ If I first find the roots, I get $z=-3i$ and $z=3i$ I also know that $$\int^\infty_{-\infty} f(x) dx=2 \pi i \sum^m_{k=1} res(f,c_k)$$ How do I apply this?
Al jabra
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contour integration complex variables

Using Contour integration, evaluate $$ \int_{-\infty}^{\infty} \frac{\cos x}{ (x^2 +1)^2}\, dx $$
vishal Singh
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