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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

3109 questions
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Find the fallacy in the argument leading to the result $2\pi i=0$

If $C$ is a positively oriented circle$ |z|=1$then by cauchy integral formula $\int_C \frac{dz}{z} =2\pi i$ Decompose$ C$ in the form $C=C_1+C_2$, Where $C_1$ is the upper half of C from $1$ to $ -1$ and $C_2$ is lower half of$ C$ from$ - 1$ to…
Abdullah O. Alfaqir
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Evaluate this complex line integral

I think I'm close but there's a part I'm not sure of. I have to find the value of $$\int|z|^2(\bar z + i)dz$$ and $\gamma$ is the circle given by $|z - i| = 1$ with counterclockwise orientation. I already figured out that $z(t) = cos(t) + i(1 +…
HokieFan7
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Complex Line Integral Limits

Let $\gamma$ be the circle in the complex plane given by $|z-i| = 1$, with counterclockwise orientation. Evaluate $$\int \frac{|z|^2}{(z-i)^2} dz$$ The function $|z|^2$ is not analytic. How would I set up the limits for this integral, is it just 0…
HokieFan7
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Stuck on Complex Contour Integration

I don't know where to begin with this question. Can anyone help me? Let be a semicircular path with centre at and with radius of 1 in the left half plane, that is, it is the straight line from 2 to 0 followed by left half a circle from 0 back to…
Kat L
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Complex integral on a circumference

I have to integrate the function $$f(z) = \frac{1}{1+z^2} e^{-2\pi i z \xi}$$ on the upper part of the circumference with center $0$ and radius $R$ in the complex plane. $\xi$ is a real number. I tried to use polar coordinates but it doesn't help.…
Nicola
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How to calculate the definite integral of the complex function $\frac{1}{(a+ib + (a-ib)x^2)}$ for $x\in(-\infty,\infty)$

How do I calculate the following integral: $\int_{-\infty}^{\infty} \frac{1}{a+bi+(a-bi)x^2}dx$, where $a,b\in\mathbb{R}$.
Tamir Moshe
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Integration with removable singularity

I have to evaluate the complex integral $$\oint_c\dfrac{z^2dz}{\sin^2(4z)}$$ where $c$ is a circle in the complex plane centered at $z=1/4$ and having radius $\pi/4$. $z=0$ is a removable singularity while $z=+\pi/4$ is within $c$ and is a second…
Purushothaman
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On the influence of $\|dz\|$ on the result of a complex integral

Take the following complex integral: $$ \int_\gamma (z^2+2z)dz = -\frac{20}{3}+\frac{40}{3}i $$ where $\gamma$ is parameterised as $z(t)=1+t(1+2i)$ (so $dz=(1+2i)dt$) with $0\le t\le 1$. For that particular integral, I solved it by doing the…
condosz
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Complex Integration with Euler's formula

I'm now practicing integration with Euler's Formula $$\int{xe^{x}sin(x)dx}$$ $$ = Im\biggr[\int{xe^{x}e^{ix}}dx\biggr] = Im\biggr[\int{xe^{(1+i)x}dx\biggr]}$$ $$= Im\biggr[\frac{1}{1+i}\int{xd(e^{(1+i)x})}\biggr]$$ $$ =…
Verdehile
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How do I evaluate this complex integral?

I've been given a integral to solve: For $r>2 \ $ let $ \ C_r = \{r e^{it}: 0 \le t \le \pi\ \} \ $ show that $$ \lim_{r \to \infty} \int_{C_r} \frac{1}{(z^2 + 4)^2}dz = 0 $$ Could someone give my a hint on where to start?
user486957
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Contour integral $\frac{1}{z^2 -1}$ questions

just in a bit of a situation and hoping someone can help i've been asked to find the following integral $$\int_{S_{1}^{+}(-1)} \frac{dz}{z^2-1}$$ where $$S_{1}^{+}(-1)$$ is the positively orientated circle of radius 1 centred at -1 So far i've…
Vaas
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Find primitives ( = anti-derivatives) of the functions $sin(z)$ and $cos(z)$.

Find primitives ( = anti-derivatives) of the functions $sin(z)$ and $cos(z)$. Since both functions are holomorphic, we can certainly use their power series expression to get their primitives, also, in $\Bbb R$, we know that $sin(x)$ has…
WaterBro
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value of complex integral $\int_C {\frac{e^{-z}}{z+1}}dz$ over the circle $|Z| = 1/2$

As per my understanding, $f(z)= {\frac{e^{-z}}{z+1}} $ is not analytic at $z=0$, so we cannot apply Cauchy's Fundamental Theorem. Which states " if $f(z)$ is analytic at all points within and on the closed contour C, then $\int_C f(z) dz = 0$" But…
Sarkar
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Complex integration of a function around a triangle.

Please help me how to solve this problem. I have no idea how to initiate the solution.
Mathforjob
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How to integrate e$^{int}$ with respect to $t$?

I am wanting to show that $\int_{0}^{2\pi}e^{nit}dt=0$ for $n\neq0$, but I am unsure if it is correct to write $$\int_{0}^{2\pi}e^{int}\mathrm dt=\left.\frac{e^{int}}{ni}\right|_{0}^{2\pi}$$ or $$\int_{0}^{2\pi}e^{int}\mathrm…
math111
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