Mathematics Stack Exchange
Mathematics Stack Exchange

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
18
votes
1 answer

Show $|\int f(z)\, dz|\leq4$

[ Remark: The following question was asked yesterday, and obtained 3 votes. Unfortunately it has been deleted by the OP overnight without receiving any answers.] Let $f:\mathbb C \to \mathbb R$ be a continuous real-valued function. Suppose for all…
Christian Blatter
  • 226,825
10
votes
1 answer

Integrate complex conjugate

I'm doing some exercises on complex integration, and stumbled over this: $$\int_\gamma \bar{z} dz,$$ Where $\gamma$ is any closed $C^1$ curve which is the boundary of a bounded, connected, open $U \subset \mathbb{C}$, and $\bar{z}$ denotes the…
Pascal Engeler
  • 711
8
votes
4 answers

What is the value of $\int_{\gamma} \bar{z} dz$?

I could use some help in calculating $$\int_{\gamma} \bar{z} \; dz,$$ where $\gamma$ may or may not be a closed curve. Of course, if $\gamma$ is known then this process can be done quite directly (eg. Evaluate $\int \bar z dz$), though that is not…
Richard Gateson
  • 81
  • 1
  • 2
6
votes
2 answers

The integral $\int_0^\infty \dfrac{x \sin(x)}{x^2+1} dx$

How to calculate: $$\int_0^\infty \frac{x \sin(x)}{x^2+1} dx$$ I thought I should find the integral on the path $[-R,R] \cup \{Re^{i \phi} : 0 \leq \phi \leq \pi\}$. I can easily take the residue in $i$ $$ Res_{z=i} \frac{x \sin(x)}{x^2+1} \quad…
Koenraad van Duin
  • 3,156
4
votes
1 answer

Complex Integral with Pole ON its Contour

Been struggling a while to solve the following integral: $$\int_C\frac{ze^z}{2z-3}$$ where C is $$|z| = 1.5$$. I found a very similar case here but it's outside of what we learned in class. Covered Cauchy's Integral Theorem but not residue theorem…
underdisplayname
  • 1,240
4
votes
2 answers

Complex integration over sphere

Consider complex plane and unit sphere $\mathbb{S}^1 = \lbrace z \in \mathbb{C} : |z|^2 = 1 \rbrace$. It is quite simple that for example by a parameterization, we have $$\int_{\mathbb{S}^1} z\, d z = \int_{0}^{2\pi} e^{i \theta} \, d\theta =…
Barabara
  • 690
4
votes
3 answers

$\oint_C \frac{z^2}{\exp z + 1} dz$ using residue theorem

Evaluate this integral using residues, $$\oint_C \frac{z^2}{\exp z + 1} dz,$$ where $C$ is the contour bounded by $|z| = 4$. My attempt: pole of $\exp z + 1 = +\pi i, -\pi i$. Now what is the order of the pole?
sbp
  • 456
4
votes
3 answers

Complex integration of $\int_{-\infty}^{\infty}\frac{dx}{(x^2+2)^3}$

$$\int_{-\infty}^{\infty}\frac{dx}{(x^2+2)^3}$$ I know that I can use a complex function $f(z)$ and I must deal with: $$\int_{-\infty}^{\infty}\frac{dz}{(z^2+2)^3}$$ So I need the roots of $z$ $$z_0 = i\sqrt2, z_1 = - i \sqrt 2$$ And only work with…
paranoidhominid
  • 643
3
votes
2 answers

How to integrate $\int_{0}^{\infty} \frac{dx}{\sqrt{x}(x^{2}+1)}$ using the residue theorem.

He was doing this integral using the formula $$\int_{0}^{\infty} \frac{dx}{\sqrt{x}(x^{2}+1)}= \frac{2\pi i}{1-e^{-2\pi i\alpha}}(\sum(Res(\frac{F(z)}{z^{\alpha}};z_{k})))$$ where $F(z)=\frac{1}{(x^{2}+1)}$, $\alpha=\frac{1}{2}$ and $z_{k}$ is a…
Jhon Jairo
  • 1,031
3
votes
0 answers

What is the integral of the Zeta function from $-1$ to $i$?

I want to calculate $$\int_{-1}^i \zeta(z) \, dz $$ Using Wolframalpha gives me approx. $0.36$ while using Mathematica gives me approx. $0.07 - 0.28i$. Which of these is correct and why is there a difference?
Tom Bombadil
  • 83
3
votes
3 answers

Use Cauchy's theorem to prove that $\int_{C}{\frac{z^2+2z-5}{(z^2+4)(z^{2}+2z+2)}dz}=0$

Let $C$ the circle $|z|=r$, show that $$\lim_{r\to\infty}\int_{C}{\frac{z^2+2z-5}{(z^2+4)(z^{2}+2z+2)}dz}=0$$ My approach: Note that $|z^2+2z-5|\leq r^2+2r+5$ and $|(z^2+4)(z^{2}+2z+2)|=|(z^2+4)||(z^{2}+2z+2)|\leq (r^2+4)(r^2+2r+2)$. So,…
julios
  • 121
3
votes
1 answer

Riemann Lebesgue Lemma for polynomial?

I was asked to prove that $$\lim_{n\to\infty} \int_{0}^{1} \exp(i\cdot n\cdot p(x))\;dx =0 $$ for nonconstant real polynomial $p(x)$. if $p(x)$ is of degree $1$... It reduces to Riemann-Lebesgue lemma. I think similar motivation will work... but…
Detectives
  • 959
3
votes
1 answer

complex integral over $dz\,dz^*$

the aim is to calculate $$\int dz\,d\bar{z} e^{-z\bar{z}}$$ when interpreting $dz\,d\bar{z}$ term, I got confused: $$dz\,d\bar{z}=(dx+i\,dy)(dx-i\,dy)=(dx)^2-(dy)^2 $$ $$dz\,d\bar{z}=\det\begin{bmatrix} 1 &i \\ 1 &-i…
Jian
  • 285
3
votes
1 answer

complex integral with cauchys integral formula

Integrate $$ \int_{|z-i|=1} \frac{1}{4z^2+1}dz $$ I have used the Cauchy's integral formula and got the answer $\pi/2$. However, my solution manual tells me its $i\pi/2$, who is right?
user269620
2
votes
1 answer

can't follow the steps of a specific complex integration

Hi: I already asked this question on the complex analysis tag but nobody answered it so then I found this complex-integration tag and was hoping that someone might be able to answer it here. It is below and thanks. ========================== I'm…
mark leeds
  • 1,514
1
2 3
…
9 10