Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [contour-integration]

Questions on the evaluation of integrals along a locus in the complex plane.

This is the procedure of calculating the contour integral around a given path/contour. It allows us to evaluate integrals on the real line $\mathbb{R}$ that are not able to be evaluated using real-variable methods.

Links:

Contour Integration at Wolfram MathWorld

3880 questions
0
votes
1 answer

A question on a contour integral in a paper by Hitchin, Karlhede, Lindstrom and Rocek

The paper is entitled "Hyperkahler metrics and Supersymmetry", published in Communications in Mathematical Physics, in 1987. I am a little stuck on a contour integral. My question is related to eq. (2.13). Let $\eta = z - \zeta x -\zeta^2…
Malkoun
  • 5,276
0
votes
2 answers

How to prove contour integral of product is equal to product of contour integrals

Let $f,g$ be analytic on some domain $\Omega \subset \mathbb{C}$. By Cauchy's formula, we have $$ \frac{1}{2\pi i} \oint_{\partial\Omega} \frac{f(z) \, g(z)}{z - z_0} \, dz = f(z_0) \, g(z_0) = -\frac{1}{4\pi^2}…
gTcV
  • 671
0
votes
1 answer

Closed form expression for a contour integral

Let $0\le a \le 1$ and $0\le a_1 \le 1$ and $L \in {\mathbb Z}$. Consider a following contour integral: \begin{equation} S_L(a,a_1) := \oint z^{L+1} \cdot \frac{1}{1-(1-a) z}\cdot \frac{1}{1-(1-a_1) z}\cdot \frac{1}{z-(1-a)}\cdot \frac{1}{z-(1-a_1)}…
Przemo
  • 11,331
0
votes
0 answers

Solution of the integral $\int_{0}^{\omega} \int_{2M}^{2(M-\omega')}{{\frac{{\rm d}r}{1-\sqrt{\frac{2(M-\omega')}{r}}}}(-{\rm d}\omega')}$.

While reading a paper published at Physical Review Letters, I came across an integral, Eqns. (6) and (7) $$\text{Im} \int_{0}^{\omega} \int_{2M}^{2(M-\omega')}{{\frac{{\rm d}r}{1-\sqrt{\frac{2(M-\omega')}{r}}}}(-{\rm d}\omega')}=4 \pi…
Ajit
  • 13
0
votes
0 answers

Evaluating integral e^z over a line

How do i evaluate $$ \int_C e^z dz $$ along the line y=x from (-1, -1) to (1, 1) here $$ z = x+iy $$ since its a straight line, I have z = t where t goes from -1 to 1 ? How do I proceed after this ?
vmark99
  • 29
0
votes
1 answer

Path integral of $1/z$ along $e^{it}$

I tried to calculate the (complex) path integral of $1/z$ along $y: [0, \pi] \rightarrow \mathbb{C}, y(t) = e^{it}$, and got the result $i\pi$. Is this correct?
CHwC
  • 684
0
votes
1 answer

Contour Integration along a line segment

This may sound like a silly question, but was just wondering if someone can clear this up for me Consider the line segment joining the points a,b. Therefore we have f(t)=a+t(b-a) where t is contained between 0 and 1 Now if we were to integrate…
Shalid
  • 1
0
votes
1 answer

Algebraic confusion

How does $$\frac{1}{b^2i}\oint\frac{1}{z}\frac{dz}{\lbrack k + \frac{1}{2i}\left( z - \frac{1}{z}\right) \rbrack^2}$$ become $$\frac{1}{b^2i}\oint\frac{1}{z}\frac{dz}{\lbrack \frac{1}{2iz} \left( z^2 + 2kiz - 1\right) \rbrack^2}$$ Is the book wrong…
paranoidhominid
  • 643
0
votes
1 answer

Let $f$ be an arithmetic function of class $C^2$ with $f_{xx}+f_{yy}=0$. Prove that $\oint\vec{F}•\vec{dr}=0$ where $\vec{F}=(f_y,-f_x)$

I'm studying in preparation for a Mathematical Analysis II examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 4 of 4, part $b$ and graded for 10%. Let $f$ be an arithmetic function of…
Dimitris S.
  • 419
0
votes
2 answers

Contour integration example question

I'm currently trying to solve this however I get to the point where I have, $$\int_{0}^{2\pi} \frac{ie^{\exp(it)}}{\exp(it)+3}.dt$$ am I on the right tracks? if yes could you help with the integration
jimmy
  • 79
-1
votes
1 answer

How can i calculate this contour integral?

Here is my function $f(z)=\frac{1}{(z+1)(z+2)(z+3)...(z+100)}$ I need to find this $\int f(z)dz$ where $C:|z|=150$ counterclockwise I tried to use second residue theorem here. So i get $\int f(z)dz=2\pi i Res_{z=0}[1/z^2 f(1/z)$ so my function…
Aegean
  • 99
-1
votes
1 answer

How to solve integration elegantly using contour integration

Solve this integral analytically and prove the answer: $$ \int_0^{\pi/2}\frac{d\psi}{1-\cos\theta \cdot \cos\psi} = \frac{\pi - \theta}{sin \theta}, \quad \theta \in (0,2\pi) $$ To see the pic I have already proved it numerically using Mathematica.…
Shuofeng Zhang
  • 11
-1
votes
3 answers

Let C be the circle $|z|=1$. Evaluate $\int_{c}\frac{e^{2\pi z}}{(2z+1)^3}dz.$

Let C be the circle $|z|=1$. Evaluate $$\int_{c}\frac{e^{2\pi z}}{(2z+1)^3}dz.$$ Any idea, suggestion, advice or solution.
Runa
  • 1
-3
votes
1 answer

The value of $\int_C \bar{z}dz$?

What is the value of $\int_C \bar{z}dz$ from $z=0$ to $z=4+2\pi i$ along the curve $C$ given by $z=t^2+it$?
F. A. Mala
  • 1
1 2 3 4 5 6 7
8