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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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Integrate $\int_{-\pi}^{\pi} \frac{\sin \theta}{a - \sin\theta}$ where $a > 1$

This is a question from the textbook Mathematical Methods for Physics and Engineering, Ex 24.13. I am having some issues to solve the definite integral by complex contour in the title. I tried to use a complex unit circle contour and convert the…
Benjamin_Gal
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The process of complex integration is the same as that of real integration

If I compute the complex integral (along $\gamma(t)=e^{it}\,;\;t\in[0,\pi]$) \begin{align*} \int_\gamma z^2dz &=\left[\frac{z^3}{3}\right]_\gamma \\ &=\left[\frac{e^{3it}}{3}\right]_0^\pi=-2/3 \end{align*} But if I expand the…
johnbear02
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Complex analysis contour integral: $\int_0^{\infty}\frac{\log(x)}{x^2-1}$

I am working on the integral $\displaystyle\int_0^{\infty}\frac{\log(x)}{x^2-1}$. I see it done here $\int_0^\infty\frac{\log x dx}{x^2-1}$ with a hint. but I am wondering if it is possible to integrate on a different contour, the keyhole. I know…
LightbulbBob
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Complex Integral of a meromorphic function

Please help with the following prelim problem. Thanks !. Express the integral as a complex integral of a meromorphic function, where $\rho > 0$ and $a$ is complex valued $$ \int_{\left\vert z\right\vert\ =\ \rho}\ \left\vert z -…
Eager Student
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I want to prove: $|f'(a)| \leq \frac{M}{1-|a|^2}$ so, what is the technic for use this Hint in down:

We have $f$ is a function holomorphic on $D(0,1)$, bounded by $M>0$, how to use this hint : $$\frac{1}{1-|a|^2}= \frac{1}{2i\pi} \int_{|z|=1} \frac{1}{z|z-a|^2} \,dz$$
مصطفى أحمد
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What is the meaning of $c+i\times\infty$ when integrating a complex function

My friend gave me a complex function to plot in MATLAB. The problem is that the integration interval or path (I do not know much about complex analysis so I don't know which is correct here path or interval) is from $c-i\times\infty$ to…
Dante
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can we use indefinite integration on $1/z$ (complex)

I'm just learning in my class at college about the cauchy integral theorem. We learned is that $\oint_c \frac{dz}{z} = 2\pi i$ where c is the unit circle oriented counter clockwise. I'm wondering what happens if it is not a closed loop. As far as I…
user395375
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Solving integral through contour integration

I'm self studying contour integration and I found a problem I can't solve. Prove for a real number $k$ that $$\int_{0}^{\pi} e^{k \cos(\theta)} \cos(k \sin(\theta))d \theta = \pi$$ I tried doing the change of variable $x = \cos \theta$ which gets…
Apo
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Proof integral:$\lim_{x \rightarrow +\infty} \int_0^{x} dx'e^{-i(k-k_0)x'} = \pi\delta(k-k_0) - P(\frac{i}{k-k_0})$,where $P$ stands for...

Where can I find the full derivation or the proof of the following integral? $$\lim_{x \rightarrow +\infty} \int_0^{x} dx'e^{-i(k-k_0)x'} = \pi\delta(k-k_0) - P(\frac{i}{k-k_0})$$, where $P$ stands for "Cauchy principal value". I saw this formula in…
WishBeLeibniz
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$|\int\limits_a^b f(t)dt| \leq \int\limits_a^b |f(t)|dt$, where $f(t) \in \mathbb C$

Let $f: [a,b] \to \mathbb C, t \to f(t) = \text {Re } f(t) + i\text{ Im } f(t)$. Suppose $f$ is continuous. Let $\int\limits_a^b f(t)dt = \rho e^{i\theta}$, where $\rho \geq 0$ is the module. Then: $$|\int\limits_a^b f(t)dt| = \rho =…
user2016
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Prove that: $\oint_c{\dfrac{1}{z^{2n+1}}(z^2+1)^{2n}}=\binom {2n} {n} 2\pi i$

In a mathematical methods problem, where the $c$ is a the unit circle around the origin and in counterclockwise, I need to use a step that I'm not so sure about (Because I don't know how to develop it) How can I affirm…
Victor F Salazar G
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contour integral complex conjugate

I'm having trouble trying to find this integral, where $C$ is the semicircle, centre $z = 1$, of radius $1$, lying in the upper half-plane $$ \int_C \bar{z}\ {dz} $$ Currently I have that, $c(t)=1+e^{it}$, where $t$ is in $(0,\pi)$, and where $c$…
sean
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Complex Line Integral $\int_{i}^{i+1}{z\>dz}$ along a straight line parallel to the $x$ axis.

PROBLEM Integrate $\int_{i}^{i+1}{zdz}$ along a straight line parallel to the $x$ axis. The definition of a complex line integral states let $f(z)$ be a continuous complex-valued function of a complex variable, and let $\mathbb{C}$ be a smooth…
Joshua Burrow
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How can I solve $\int \left| \sum_{i=0}^{N} f_i(x) \right|^2 d x $ for complex functions $f_i$?

The $f_i(x)$ are complex functions and I know that $\int_{-\infty}^{\infty} f_i (x ) \, d x=c_i$. How can solve this integral? $$\int_{-\infty}^{\infty} \left| \sum_{i=0}^{N} f_i(x) \right|^2 d x $$
Abolfazl Abdollahi
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Complex integral with different contours

If I have a complex integral to solve using the Cauchy Integral formula with the same point but with different contours, in which the point used is inside both contours, is the result the same? Say for example that I have $$ \int_C…
Gustavo Amgarten
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