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
1
vote
1 answer

Parametric form of curve $\vert z+i\vert = 1$

I need to integrate a complex function through the curve $\vert z+i\vert = 1$. As far as I know I need the parametric form of this curve. I know that when I have $\vert z\vert = 1$, the parametric form is something like $\cos(t) + i\sin(t)$. But…
Gustavo Amgarten
  • 135
  • 4
1
vote
1 answer

Integral of $e^{\overline{z}}$

So I'm suppose to integrate $e^{\overline{z}}$ along the unit circle where $z(t)=e^{it}$ and from t goes from 0 to $2\pi$. This is the work I've done so far $\int e^{\overline{z}}dz = i\int_0^{2\pi}e^{e^{-it}}e^{it}dt =…
Jim
  • 73
1
vote
2 answers

Line integral of complex expression

How can we integrate expressions like these $\int_C \operatorname{Re}(Z) \, dZ$ where $C$ is the shortest path joining the points $1+i$ and $3+2i$. The $\operatorname{Re}(Z)$ in the expression is what confusing me.
mahes
  • 469
1
vote
1 answer

Integrating $z^n$ as a complex valued function

Let $B_1(0) \subset \mathbb{C}$ be the unit ball in the complex plane and $f:B_1(0) \to \mathbb{C}, z \mapsto z^n$. I want to know how one would calculate the integral $$\int_{B_1(0)}f(z) \text{d}z$$ I expect the integral to be 0 because of the…
Flynn Fehre
  • 484
1
vote
3 answers

Contour Integral of $\frac{z^{*}}{z-1}$, where $z^{*}$ is the complex conjugate of $z$.

So I have to evaluate $\oint \frac{z^{*}}{z-1}$ on a circle of radius 5 centered at the origin of the complex plane, where $z^{*}$ is the complex conjugate of $z$ and the orientation is anticlockwise. I know z* isn't analytic anywhere and I can…
Syed Bukhari
  • 369
1
vote
1 answer

Integral involving deforming the contour

Find the value of below integral by deforming the contour. Question 1. $$\int_{|z| = 2} e^z/z(z-3) \, dz $$ Question 2. $$\int_{|z+1| = 2} z^2/(4-z^2) \, dz $$ I tried to use partial fraction on these problems, but is this a right approach when…
mathrock1453
  • 71
1
vote
1 answer

Complex analysis fundamental theorem of calculus on integration

Suppose f is a continuous function on an open set Ω which has an anti-derivative on Ω. if γ, γ' are two paths in Ω with the same beginning point and the same end point, are below statement true? $$\int_{\gamma} f(z) \, dz =\int_{\gamma'} f(z) \, dz…
mathrock1453
  • 71
1
vote
1 answer

integrating $ \frac{\cos(3\theta)}{5-3\cos(\theta)}d\theta$ using residue theorem

So I've been taking complex analysis for fun recently and come across this problem: calculate $\int_{0}^{2\pi} \frac{\cos(3\theta)}{5-3\cos(\theta)} d\theta$ using the residue theorem, after making this a complex integral: $\int_{\partial…
IronicalCoffee
  • 123
1
vote
1 answer

Integral of root of a complex function

I need to find $$\int_{|z|=2} \sqrt[3]{z^3-1}$$ but I'm getting stuck. What I did: We know that $$\sqrt[3]{z^3-1}=z\sqrt[3]{1-\frac{1}{z^3}}$$ From here, we can write $z=2e^{i \theta}$ and $$\int_{|z|=2} z\sqrt[3]{1-\frac{1}{z^3}}=\int_{-…
Idan Daniel
  • 151
1
vote
3 answers

Show $\int_{0}^{2\pi}e^{int}\mathrm dt=0$ for all $n\neq0$

I am struggling to see how this result holds for non-integer $n$ because $$\int_{0}^{2\pi}e^{int}\mathrm dt=\int_{0}^{2\pi}[\cos(nt)+i\sin(nt)]\mathrm dt$$ and this works out to be $$\frac{\sin(2\pi n)}{n}+i\left(-\frac{\cos(2\pi…
math111
  • 326
1
vote
1 answer

Compute $\int_{-\infty}^\infty \frac {\cos(x)}{x^{4} + 1} dx$

This is an integral that has been giving me trouble in my Complex Analysis course. I have already learned residues in one of my physics courses, however, we are not allowed to use it since we have not yet covered it in this course. I am honestly…
ryan6297
  • 21
1
vote
2 answers

problems to calculate $\int_\gamma \frac{e^{iz}}{z^2}dz$

Some ideas to may begin to calculate $\int_\gamma \frac{e^{iz}}{z^2}dz$?(where $\gamma:[0,2\pi]\to\mathbb{C}$ def by $\gamma(t)=e^{it}$). Trying by definition I obtain an expression like $e^{ie^{it}}$ which is weird.
user258738
1
vote
0 answers

Numerically Verify Complex Integrals

Suppose you only have a real integral numerical calculator, how would you use it to verify complex integrals numerically? Well, of course you can use Residue theorem, Cauchy’s integral formula, etc. to theoretically compute integrals. However, I’m…
Szeto
  • 11,159
1
vote
1 answer

Solving definite integrals using complex integration

Can anyone help with solving these two definite integrals using contour integration: $$\int_{0}^{1}\frac{\sqrt{x(1-x)}}{(2x+1)^{2}(x+2)}dx$$ $$\int_{1}^{2}\frac{dx}{x\sqrt[7]{(x-1)^{3}(2-x)^{4}}}$$. For the first one, I use a dog bone contour around…
Tamara
  • 21
1
vote
1 answer

Prove $\int_{C} f(z) dz=0$ where $f(z)=Ln(z-4)$ on a given C

The question I have to answer is Let $C=\{z:|z|=1\}$. Prove that $\int_{C}f(z)dz=0$ where $f(z)=Ln(z-4)$ My attempt: Initially my first idea was to rewrite the complex logarithm in terms of the real logarithm and its argument, namely…
Overclock
  • 265
1 2
3
…
9 10