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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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Placement of singularities in the residue theorem

Why do the singularities in Cauchy's residue theorem have to be within the contour, and why do they still count if they're not on the path of integration, like I'd suspect for real integrals? Sorry if this is basic but I haven't found it anywhere…
Pineapple Fish
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Calculate the complex integral

I have $$ \int{\frac{dz}{z^2+9}} $$ Also I'm given 2 different conditions. First is $|z|=\pi$, second is $|z-2i|=2$. Okay, so for the integral i have $\int{\frac{dz}{(z+3i)(z-3i)}}$. For the first condition, if I draw a circle, then $\pi$ will be…
user3132457
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Find the complex integral

I have $$\int_{\gamma} {\frac{z}{\overline z}}dz$$ where $\gamma$ is the edge of $\{1 < |z| < 2\ $and $\Im (z) > 0\}$. I think the way to solve this is to calculate the integral for $|z|=1$ and then $|z|=2$, then subtract the first result from the…
user3132457
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Evaluate a complex integral and determine if it contradicts Cauchy's theorem

I'm trying to determine if $\oint_C \frac{dz}{z-3}=0$ where $|z-2|=5$ and if it contradicts Cauchy's theorem. I assume it won't be $0$, and then it won't contradict Cauchy's theorem because $C$ isn't centered at the origin, it's centered at…
AdamK
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Small question on improper complex integral intuition

Recently, I am dealing with an integral $$\int^\infty_{a+bi}f(z)dz$$ where $f(z)$ is a meromorphic function on $\mathbb C$. The $\infty$ shall be understood as the real positive infinity, and conventionally the integral is along a straight line…
Szeto
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Determine $\int_{C}{\frac{e^{z^{2}}}{z-1}dz} $

Determine $\displaystyle\int_{\gamma}{\frac{e^{z^{2}}}{z-1}dz}$, where $\gamma$ is the rectangule given by $x=0$, $x=3$, $y=-1$ e $y=1$. My approach: If we consider the rectangule, then each side is given by…
julios
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Is the answer zero?

My question for my exam on linear algebra was to evaluate the integral $\displaystyle \oint \operatorname{im}(z^2)dz$ , where integration is done under area bounded by vertices $0$, $1$, $i$ in the complex plane. Since…
Arun Sunny
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integrating 1/(cos(z)-1) over a circle with radius 7

I want to evaluate the integral $$\displaystyle\oint_C\left(\frac{1}{\cos(z)-1}\right)dz$$ where $C$ is circle with radius $7$, counterclockwise. I kept having problems with it. If someone can help, it would really be appreciated.
Matan Gelber
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Integrating using real parts

Taking this from https://en.wikipedia.org/wiki/Integration_using_Euler%27s_formula problem 3 Understanding Euler's formula I still don't understand how: "cos x is the real part of e^(i x)" and how the "Real part" affects the…
TheAutomaton
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Compute $\int_{|z|=R} \frac{|dz|}{|z-a|^4}$ when $R>0, |a|\neq R$

I want to compute $\int_{|z|=R} \frac{|dz|}{|z-a|^4}$ when $R>0, |a|\neq R$. I did so by parametrizing the circle, but this way required many pages of tedious calculations. Do you have a more straightforward way to do it? Let me mention that my…
perlman
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Prove that $\int_{\gamma} f(z)\ dz =0$.

Let $f$ be continuous in $B(z_0 ; R)$ and analytic in $B(z_0 ; R) \setminus \{a\}$ for some $a \in B(z_0 ; R)$. Then $\int_{\gamma} f(z)\ dz =0$ for every closed contour $\gamma$ in $B(z_0 ; R)$. I find lots of difficulty to prove the above…
Arnab Chattopadhyay.
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Difficulty in proving Estimation lemma for a contour.

I have already proved Estimation lemma when the complex integration is taken over a smooth curve but I find difficulty to prove it when the complex integration is taken over a piecewise smooth curve i.e. over a contour. Please help me in proving…
Arnab Chattopadhyay.
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Gaussian integral via complex variable

I have a doubt on how to cancel certain poles around a given domain in an integral. Consider $$ \int_{\partial D_R} \frac{e^{\pi i(z-1/2)^2}}{1-e^{2\pi iz}}dz, $$ where $\partial D_R$ is the parallelogram with vertices $\pm1/2\pm(1+i)R$. Use the…
user2820579
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Not sure why this question is taking about half a circle?

I'm trying to solve this question. Let $C_{r}$ be the half-circle $|z|=R$, $0\leq \arg(z) \leq 2\pi$. Show that $$\int_{C_{r}}\frac{1}{x^{4}+x^{2}+1}dx$$ Show that the integral approaches 0 as $R \rightarrow \infty$ What I'm confused about is that…
user262291
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Use ML lemma to show $\lim_{R \to \infty} \int_{C_R} \frac{z^2}{z^4+1} dz =0$

I am meant to use the ML lemma to deduce that $\lim_{R \to \infty} \int_{C_R} \frac{z^2}{z^4+1} dz =0$ where $C_R$ is the semi circle in the upper half of the complex plane oriented from $z=R$ to $z=-R$, centered a t the origin. So I am trying to…
mcmapple
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