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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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Meaning of Complex Definite Integral

I have come across some complex integrals with limits when doing questions but have not been able to make sense of what they mean due to my lack of understanding. We usually define integrals along a path and I have been told that taking a suitable…
user258521
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Complex integral problem disguised as a real integral

I'm self studying complex analysis, and I've found the following problem which I can't solve. Prove that $$\int_{-\infty}^{\infty} e^{-x^2}Im(e^{-2ix}p(x+i))dx = 0$$ Where p(z) is a polynomial of real coefficients. As a hint the books says to…
Apo
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integrate $\frac{1}{(z-e^{-z})}$ on upper half unit circle

I am struggling to integrate $\frac{1}{(z-e^{-z})}$ on upper half unit circle. I would appreciate any support as I am very new to complex integration.
vyaman
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Finding $\int_{\partial B_{2}(0)} \frac{1}{(z^n-1)^2}dz$.

I'd really like some help with this problem. I'm supposed to find $$ \int_{\partial B_{2}(0)} \frac{1}{(z^n-1)^2}dz,$$ where $B_2(0) = \{ z \in \mathbb{C} \; | \; |z|<2 \}$ (ie. the ball of radius 2 and centered at 0). This, of course, amounts to…
Peter Watts
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Evaluate the complex integral $\int_{|z|=2}\frac{e^z}{z(z^2-1)(z+1)}dz$.

How I can solve this complex integral? $$\int_{|z|=2}\dfrac{e^z}{z(z^2-1)(z+1)}dz$$ I know how to find the singularities but I do not understand in this case how to distribute the fractions of the integral. I mean $A/(x-a)$ etc.... The…
user495372
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Evaluate indefinite integral $\int e^{-3t} \cos(2-\sqrt{3}t)dt $ using complex exponential

$$\int e^{-3t} \cos(2-\sqrt{3}t)dt $$ I got to here $$\int e^{-3t}\left(\frac{1}{2}\left(e^{2i-i\sqrt{3}t} + e^{2i+i\sqrt{3}t}\right)\right)dt$$ But I'm not sure how to go on from here
randb
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Complex Integral Cauchy Integral Formula

Im trying to calculate $$\int_{\gamma}\frac{e^{iz}}{z^2} dz$$ where $\gamma(t)=e^{it}, 0\le t \le 2\pi$ My problem is that $z^2$ in the denominator. If the function was $\frac{e^{iz}}{z}$ as $f(z)=e^{iz}$ is holomorphic…
idk
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Contour Integration on unit circle.

Find The Following : $$\int_{|z|=1}\frac{z^n}{z-1}\,dz$$ N.B Cauchy Residue formula does not work. Since the pole is on the boundary.
mudok
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Why does a different parametrization give a different answer?

I was doing this complex integration question: $$\int_{c}f(z)dz$$ where $f(z) = z^{2}-3$ and the curve is the union of segments $[0,i]$ and $[i,1+2i]$. I tried to parametrize the curve in two different ways and see what I got. First…
user262291
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A complex integral over two lines?

Let C denote the union of segments $[-1-i,2]$ and $[2,3+2i]$. Find $$\int_{c}(z^{2}-z+1)dz$$ Sorry about not showing any work. I know we are supposed to parameterize the lines but I'm have a lot of difficulty with that.
user262291
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Complex Integration Over a Semicircle

I don't really understand how I solve ii) in the problem linked. I'm quite sure that it involves splitting up the integral in i) to parts over $C_R$ and $[-R,R]$, but I don't know how to solve the latter. Since it's just along the real axis, I can…
user3255234
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complex integration $\frac{1-|a|^{2}}{\pi}\int_{L}\frac{|dz|}{{|z+a|}^{2}}$

How to solve complex integration $\frac{1-|a|^{2}}{\pi}\int_{L}\frac{|dz|}{{|z+a|}^{2}}$, where $L$ is the simple closed curve $|z|=1$ and $a\in\mathbb{C}$ with $|a|<1.$ $z=e^{it}$ is the parametrization of the circle $|z|=1.$If we put values it…
neelkanth
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Finding a Taylor series and using it to evaluate a complex integral

I have a problem with a few parts (possibly related to each other) that I'm not sure about: Show that $f(z) = z/(z^2 - 4z +1)^2$ is holomorphic except at two points, $\alpha$ and $\beta$ with $|\alpha| < 1 < |\beta|$ This part I'm OK with, since…
pizzaroll
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Find, using only Cauchy's integral formula, $\int_{\gamma} \frac{\Re(z)}{2z-i} dz$

I have to evaluate: $$I=\int_{\gamma} \frac{\Re(z)}{2z-i} dz$$ where $\gamma$ is the unit circle centered at $0$. Cauchy's integral formula tells us that: $$f^{(n)}(a) = \frac{n!}{2 \pi i} \int_{\gamma} \frac{f(z)}{(z-a)^{n+1}} dz$$ But this only…
pizzaroll
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Complex integration (zill exercise)

Can any one help me with this path integral? $$\int_C\dfrac{dz}{z^2-2i}$$ where $C$ is the right half of the circle $|z| = 6$ from $z = −6i$ to $z = 6i$. Thanks.
wggv
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