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

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3
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1answer
44 views

$\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{1}{s^2}e^{s(t - \frac{1}{2}x^2)}ds$ - different answers depending on value of $t$?

After taking an inverse Laplace transform I have the following - $$y = \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{1}{s^2}e^{s(t - \frac{1}{2}x^2)}ds$$ In my notes it says if $t ...
3
votes
3answers
70 views

Evaluation of real trigonometric integrals using the Cauchy Residue Theorem

$I = \int^{2\pi}_0 \dfrac{d\theta}{2 - \cos \theta}$ This is straight from a book I'm reading, which suggests to convert $\cos\theta$ into $0.5(z+1/z)$ and then solve the integral on the unit circle. ...
7
votes
2answers
290 views

An intuitive definition of contour integration.

Recently I have been trying to learn the method of contour integration, but the Wikipedia article and others don't really help. Is there some resource which provides a definition which can be followed ...
0
votes
0answers
31 views

Integrate function with 2 branch points

Every example I see in textbooks so far has not shown me cases like this, so please help with the following question. I wish to integrate a function $f(z)$ around the contour shown below. $f(z)$ has ...
3
votes
0answers
60 views

Computing an integral using residues

I am trying to find an integral: $$\int_{-\infty}^{+\infty}\frac{e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2}\,\mathrm dx$$ I went about applying contour integral over a semicircle with diameter along $ x = ...
4
votes
3answers
146 views

Compute $\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$

Given $$\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$$ I couldn't evaluate this integral. My only idea here was evaluating this as integration by parts. \begin{align} \int\frac{x ...
0
votes
1answer
44 views

How to use Cauchy's integral formula with more than one pole?

$\int\limits_{\gamma} \frac{z^2}{z(z-2)}$ $\gamma(\theta) = 3e^{i\theta}$, $0 \leq \theta \leq 2\pi$ Cauchy's integral formula is given by: $$\int\limits_{\gamma} \frac{f(z)}{(z-a)^{n+1}} = ...
1
vote
1answer
42 views

Why is $\int\limits_{\gamma} \frac{1}{z-1} \neq 2\pi i$, $\gamma = \{z : \lvert z \rvert = 1\}$?

$\int\limits_{\gamma} \frac{1}{z-1}$ $\gamma = \{z : \lvert z \rvert = 1\}$ I use Cauchy's integral formula, which says $$\int\limits_{\gamma} \frac{f(z)}{(z-a)^{n+1}} = \frac{2\pi i}{n!} ...
0
votes
3answers
31 views

$\int\limits_{\gamma} \frac{1}{z-1}$, $\gamma(\theta) = 2e^{i\theta}$, $0 \leq \theta \leq \frac{\pi}{2}$

$\gamma(\theta) = 2e^{i\theta}$ is a circle centered at $(0,0)$ with radius $2$, so $z = 1$ is inside this path and thus we have to use Cauchy's integral formula for $\int\limits_{\gamma} ...
0
votes
0answers
49 views

Contour integration with a branch cut. Parameterizing f(z) properly

I have a contour integral of a function of the form $(z^6-P)^\alpha z^\beta$ Here $\alpha\in R$, $\beta\in N$ and $P$ is some constant. I therefore have branch points at the sixth roots of $P$. The ...
0
votes
0answers
18 views

Contour Integration example check

I have this question and have solved that the residue is zero? hence the integral is zero by the residue theorem? could someone confirm this please?? Also would the answer to this integral be ...
0
votes
2answers
42 views

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 ...
2
votes
1answer
73 views

how to calculate $\int_{0}^{\infty} \frac{\cos(x)}{(1+x^2)^2} dx$

$$\int_{0}^{\infty} \frac{\cos(x)}{(1+x^2)^2} dx$$ The main problem here is to choose the smart contour integral, but i don't see how. I think i am supposed to do this: note our integral is: $$0.5 ...
5
votes
1answer
173 views

Can $\oint_{|z|=2}z^3 \bar {z} e^\frac{1}{(z-1)} dz$ be solved?

How we can calculate the result of following Integral? $$\oint_{|z|=2}z^3 \bar {z} e^\frac{1}{z-1} \mathrm{d}z$$
0
votes
0answers
54 views

Fourier transform of a tough composite function (sinc, sqrt, polynomial…)

Is it possible to compute the Fourier transform of $\mathrm{sinc}(\sqrt{1+x^4})$ in closed form? It appears the problem to be suited for contour integration, and I started to tackle the mere ...
8
votes
1answer
96 views

The inverse Laplace transform of $ s^{3/2}-a-bs \over s^{3/2}+a+bs$

How can I solve the inverse Laplace transform as below: $$\mathscr{L}^{-1}\left( s^{3/2}-a-bs \over s^{3/2}+a+bs \right) $$ where a and b are constants. Hint: we can consider $${ s^{3/2}-a-bs ...
2
votes
1answer
55 views

Simple Residue calculation

$$\int_{\gamma(0;2)}\frac{e^{i\pi z/2}}{z^2-1} \, dz$$ Using the residue calculus i got $$-2\pi$$But the answer is $$=i$$ I must be wrong at this, but shouldn't the answer have $\pi$ at least since ...
2
votes
2answers
124 views

Using Complex Analysis to Compute $\int_0 ^\infty \frac{dx}{x^{1/2}(x^2+1)}$

I am aware that there is a theorem which states that for $0<a<2$ we have $$\int_0^\infty\frac{x^{a-1}}{x^2+1}dx=\frac{\pi \cos\big(\frac{a\pi }{2}\big)}{\sin (a\pi) }$$ but I prefer to evaluate ...
0
votes
1answer
71 views

The Poisson Integral is harmonic

We have proved that for $h(e^{\mathcal{i}\theta})$ continuous on the unit circle, the Poisson Integral of $h$ defined by ...
1
vote
2answers
158 views

$\int_0^\infty \frac{x^2}{(x^2-4)(x^2-9)}\,\text dx$

I am trying to compute the following contour integration but am quite stuck I have to evaluate it analytically, by extending it to the complex plane and solving an appropriate integral involving a ...
2
votes
2answers
130 views

Contour Integral of $\log(z)/(1+z^a)$ where $a\gt1$

I am asked to prove that: $$ \int_{0}^{+\infty}\frac{\log z}{1+z^{\alpha}}\,dz = -\frac{\pi^2}{\alpha^2}\cdot\frac{\cos\frac{\pi}{\alpha}}{\sin^2\frac{\pi}{\alpha}},$$ provided that $\alpha > 1$, ...
4
votes
1answer
179 views

A bessel function integral

$$\int_{-\infty}^{\infty} dy \frac{J_1 \left ( \pi\sqrt{x^2+y^2} \right )}{\sqrt{x^2+y^2}} = \frac{2 \sin{\pi x}}{\pi x} $$ How do I show this?
1
vote
1answer
58 views

Change the order of integrals:$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$

$$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$$ From this it is obvious that $x\in[0,1],y\in[0,1-x],z\in[0,x+y]$. For it asks for the order to be in $$\int dz\int dx\int f(x,y,z)dy$$ . My method ...
1
vote
1answer
79 views

Show the length of a contour, given by traversing once round a circle radius r, is 2πr

I have tried this problem using the definition for length of a contour $$ L(\gamma) = \int |\gamma'(t)| dt $$ Along the contour $\gamma =Z +re^{it}$ But I cannot get it to work out at $2\pi r$.
4
votes
2answers
70 views

$\int_0^{2\pi} e^{\cos(\phi)}\cos(\phi - \sin(\phi)) d\phi$ via contour integration

Can anyone help me calculating this integral using contour integration? $\int_0^{2\pi} e^{\cos(\phi)}\cos(\phi - \sin(\phi)) d\phi$ I've used the subctraction formula of the cosine: $$\cos(\phi - ...
0
votes
0answers
24 views

Compute the integrals $\int_{C_1}f(z)dz$ and $\int_{C_2}f(z)\,dz$.

Given two smooth contours, $C_1$ and $C_2$, that respectively lie on the upper and lower half plane compute $\int_{C_1}f(z)dz$ and $\int_{C_2}f(z)\,dz$. Let $a$ be a fixed real positive number. ...
3
votes
1answer
88 views

Regarding branch cuts and contour integration

I am trying to compute the following integral through the use contour integration. $$ \int_0^1 \frac{dx}{\sqrt{x^2-1}} $$ So, I am considering the same integrand but from $-1$ to $1$, then doing the ...
1
vote
2answers
78 views

Compute $\int^{2 \pi} _0 \frac{1}{a + \sin \theta} d\theta$

I want to compute $\displaystyle \int^{2 \pi} _0 \frac{1}{a + \sin \theta} d\theta$, with $a > 0$, where we may use the Cauchy Integral Formula. The following hint is given: Write $sin \theta = ...
0
votes
1answer
48 views

Using Cauchy Integral Formula: $\small\displaystyle \int_c \frac{e^{-z^2}}{z^2}dz$

I am going over the solutions to previous problems in order to prepare for a test. I am having a hard time understanding even basic applications of Cauchy's Integral Formula. For example, I have ...
3
votes
0answers
57 views

Integration over a variety

If $ M $ is a differentiable manifold equipped with an Atlas $ \mathcal{A} = ( U_i , \varphi_i )_{ i \in I} $, we can then calculate the integral of a differential form $ \omega $ over $ M $ with the ...
1
vote
1answer
48 views

Calculate $\cos(z)/(z^2-\pi^2)$ using Cauchy integral formula on region |z|=4

I want to verify if my reasoning and answer is correct here. Since $\pi$ and $-\pi$ are both contained within the circle centered at 0 with radius 4, we can use the Cauchy integral formula to deal ...
2
votes
1answer
24 views

Calculate $\sin(z)/(z+i)$ using Cauchy Integral Formula on region $|z+i|=3$

I just want to know what I'm doing wrong here. So we have a singularity at $z=-i$ but this is inside the region of circle centered at $-i$ with radius 3. Hence by Cauchy Integral Formula we have ...
1
vote
1answer
19 views

Circle difference in contour integral

Let's say I am integrating a function over $|z| = 1$ and $|z-1| = 1$, is there any difference? I think the answer for both cases will be same, as in both cases, $$ z = \exp^{i\Theta} $$ and $$ dz = i ...
1
vote
1answer
32 views

Evaluating $\int_\Gamma \frac{2z^2-z+1}{(z-1)^2(z+1)}dz$ along the contour that is shaped by the figure-8 centered at $z=-1$ and $z=1$.

In my answer key, it says this equals $0$, but I get $4 \pi i$. Here's why: $$ \int_\Gamma \frac{2z^2-z+1}{(z-1)^2(z+1)}dz = \int_\Gamma\biggl[\frac{1}{(z-1)^2}+\frac{1}{z-1}+\frac{1}{z+1}\biggr]dz ...
1
vote
1answer
21 views

Confusion regarding contour integral solution

In Schaum's complex variable book, there is an exercise in contour integration: $$ \int \overline{z}^{2} dz $$ over $|z|=1$. The answer seems to be $0$, but when I integrate like this using contour ...
0
votes
1answer
46 views

Find the line integral of $1/(z^2+4)^2$ over region $\gamma$

I have to find: $$I=\oint_{\gamma}\frac{dz}{(z^2+4)^2}.$$ $\gamma$ in this case is a circular curve defined by $|z-i|=2$, which is a circle centered at $i$ with radius $2$. It is clear that the ...
0
votes
0answers
30 views

Find Line integral of $e^{-z} /{z-\pi/2}$ on a region $\gamma$

Let $\gamma$ be the diamond connecting points $x=2, -2$ and $y=2, -2$. and its oriented positively (counter-clockwise, I believe?). I'm not so sure if we can use the Cauchy integral formula here and ...
6
votes
0answers
158 views

Complex contour integral: How does the stationary point method used in this case?

I was reading a paper which has the following integral in order to do the inverse Laplace transformation: $$ I=\frac{1}{2\pi i}\int_{-i\infty+\gamma}^{i\infty+\gamma} ...
0
votes
0answers
104 views

Using the “appropriate” formula

I am asked to solve $$\int_{C}\frac{1}{z+i}dz$$ where $C$ is parametrized $z(t) = 2+e^{it}$ for $t \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ by finding the antiderivative $F(z)$ of $f(z)$ and then ...
0
votes
2answers
43 views

Paremetrising the Contour

I'm trying to paremetrise the Contour of a unit circle descibed anti clockwise. This is so I can integrate $$ \int_{|z| = 1} \frac{e^z}{4z^4} dz $$ Now normally $z(t)=e^{it}$ for $t\in [0,2\pi]$ is ...
0
votes
1answer
40 views

Explaining information on Contour

I wish to compute the following line integral $$\int_{C}(x-iy)dz$$ where $z(t)=(e^t- 1,t)$, $t \in [0,2]$ $dz = dx + idy = (e^t - 1 + i)dt$ We then have $\int_{C}(e^{t} - 1 - it)(e^t - 1 + i)dt$ I'm ...
1
vote
1answer
37 views

Complex integration confusion

I wish to compute $\int_{C}(x^2 - iy^2)dz$, where $C := \{z\mid |z|=1\}$ is positively oriented. I am a bit confused on what $dz$ actually is. I know I have $\int_{C}x^2dz - i\int_C y^2dz$, but I ...
0
votes
0answers
32 views

Integrating along a contour

I wish to compute $\int_{C} \frac{dx}{x^2 + y^2} -2xydy+ i\int_{C}(xdx - ydy)$, where $C$ is the contour is parametrized by $z(t) = (\cos(t),\sin(t))$ for $t \in [0, 2\pi]$. To compute this, I should ...
0
votes
0answers
88 views

Fundamental Theorem of Calculus for Complex Numbers

Lets say we have the integral:$$\int_\gamma\frac{1}{z}+z^2dz$$ and we would like to apply the Fundamental Theorem of Calculus here for complex numbers. Now, we let $\gamma$ be any curve connecting ...
4
votes
1answer
44 views

Simple Question About Contour Integration

If you are integrating $$\int_\gamma y^2\,dz$$ Where $\gamma$ is the line segment from $1$ to $i$. You parameterize the line $$x(t)=1-t$$ $$y(t)=t$$ $$\implies z(t)=1-t+it$$ Now, if you want to use ...
1
vote
1answer
38 views

Complex integration parametric form

Evaluate$\int_{\gamma(0;1)} \frac{\cos z}{z}dz$. Write in parametric form and deduce that$$\int^{2\pi}_0 cos(\cos\theta)\cosh(\sin\theta)d\theta=2\pi$$ By Cauchy's integral formula, ...
0
votes
1answer
50 views

how to calculate $\frac{1}{2\pi i} \int_{\gamma} \frac{2z}{(z-1)^ 4(z-3)}$

How to calculate $\frac{1}{2\pi i} \int_{\gamma} \frac{2z}{(z-1)^ 4(z-3)}dz$ When $\gamma = C_+(0,4)$ and where $\gamma = C_-(0,2)$. I need to use the residuformula which states that is f is ...
0
votes
1answer
39 views

Computing a contour integral of a function that is not analytic inside the contour

I'm wondering if there is another way to calculate the contour integral of $\int(\tan(z/2)/(z-1))$ in the square w/ sides $Re(z)=+/-2$, $Im(z)=+/- 2$ other than using the residue theorem. The cauchy ...
2
votes
0answers
41 views

compute the complex-valued integral for the branch cut

Let $C$ be the circle of radius $2$ centered at origin. Let $f(z)$ be the branch cut of the function $z^{2−i}$ on the domain $−π < θ < π$. Compute the integral $$ \int_C f(z) dz$$ My attempt: ...
1
vote
1answer
48 views

Finite integral with removable singularity

I wanted to integrate $\frac{(exp(-x) -1)^2}{x}$ from $x=0$ to $x=a$ where $a$ is finite. Since the integrand, viz., $\frac{(exp(-x) -1)^2}{x}$ has a removable singularity at $x=0$ , I can take the ...