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

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2
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3answers
60 views

Computing $\int_{0}^{\infty} \frac{x^2 + a}{x^6 + a^3}dx$

I wish to compute $$\int_{0}^{\infty} \frac{x^2 + a}{x^6 + a^3}dx, \quad a>0$$ but have no contour to work with. Does anyone have ideas on how to compute this integral?
0
votes
1answer
40 views

Analytic inverse of $f(z) \neq 0, f(0) = 0, f'(z) \neq 0 $ within minimum modulus on boundary.

Suppose $f(z)$ is analytic on closed disk of radius $r$ and $f(0)=0$, $f'(z) \neq 0$. Show that $f$ has an analytic inverse on $\{|z| \leq m\}$ where $m$ is the minimum of $|f(z)|$ on $\{|z| = r\}$. ...
0
votes
0answers
18 views

How to determine contours by looking at the exponential integrands?

I know that we determine the contours in contour integrals by looking at the exponential integrand (assuming there is indeed an exponential integrand in the given integral) but I don't know how. For ...
1
vote
1answer
37 views

Help With Bromwich Inversion Formula Proof

To prove(copied from handwritten notes so possibly wrong): Bromwich Inversion Formula. Fix $x_0∈ℝ $. If $F$ is complex analytic on $\{z:\Re z > x_0\} $ and for every $x>x_0$, $y↦ F(x + iy )$ ...
8
votes
2answers
117 views

Dog Bone Contour Integral

Would someone please help me understand how to integrate $$ \ \int_0^1 (x^2-1)^{-1/2}dx\, ? $$ This is a homework problem from Marsden Basic Complex Analysis. The text book suggested using a "dog ...
1
vote
1answer
19 views

Let $C=\partial D_1(\mathbf i/2)$, compute $\int_C\frac{dz}{z^2+1}$

Let $C=\partial D_1(\mathbf i/2)$, compute $\int_C\frac{dz}{z^2+1}$ $C=\partial D_1(\mathbf i/2)$ is the boundary of the disc with center $\mathbf i/2$ and radius $1$, then $\mathbf i$ is ...
0
votes
2answers
13 views

Finding the value of $I=\int_C \overline{z} dz$ along $|z|=2$ from $z=-2i$ to $z=2i$

I want to find the value of the integral $$I=\int_C \overline{z} dz$$ When $C$ is the right hand half of the circle $|z|=2$ from $z=-2i$ to $z=2i$ Refer to beautifully made picture: Now I am new to ...
0
votes
1answer
29 views

Improper integral (using methods in complex variables) [on hold]

Let $0<a<1$. Evaluate the integral $$\int_0^\infty \frac{x^{a-1}}{1+x} dx.$$
3
votes
1answer
54 views

Method of Steepest descents integral

I am looking to evaluate the following asymptotic integral: Find the leading term of asymptotics as $\lambda\to\infty$ $I(\lambda)=\int_0^1\cos(\lambda x^3)dx$ Using method of steepest descents ...
3
votes
0answers
61 views

Arguing that a complex function tends to zero fast enough to ensure that $\lim_{N \to \infty} \int_{C} f(s) \, ds = 0$

Consider the complex function $$ f(s) = \frac{\Gamma(a+s)}{\Gamma(b+s)}\frac{z^{s}}{\sin (\pi s)}$$ where $|z| <1$ and $- \pi < \arg(z) < \pi$. Let $C$ be the right half of the circle $|z|= ...
1
vote
1answer
21 views

Solve an Integral one open and one complex limit

If we pose an integral $2ie^{-\zeta^2}\int\limits_{-\infty}^{i\zeta} e^{-x^2} dx$ where $\zeta$ is a complex number. For imaginary $\zeta$ this comes down to the error function. I would appreciate ...
1
vote
1answer
37 views

Why is $\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| \leq \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|$?

Why is $$\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| \leq \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|$$ and not $$\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| = \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|?$$
0
votes
0answers
20 views

Inverse $z$ transform - contour integration

Here is my task: Find inverse $z$ transform of $$X(z)=\frac{1}{2-3z}$$if $$|z|>\frac{2}{3}$$ using definition formula. I found that $$x(n)=\dfrac{1}{3}\left (\dfrac{2}{3}\right ...
4
votes
2answers
45 views

Evaluating contour integral without using Residue Theorem

Find the value of the integration without using Cauchy integral formula/Residue theorem: $\int_{C}\cfrac{dz}{z^2+1}$ where C is a simple closed contour oriented in counter clockwise ...
3
votes
0answers
28 views

Integral representation of Bessel function K

Does someone have an idea how to connect the following function (appearing in the quantization of a real scalar field in a uniformely accelerated frame) : $$ K(x,y) = \int_{0}^{\infty} \frac{dt}{t} ...
0
votes
1answer
17 views

Evaluating this contour integral

Let $R$ be the rectangle with vertices at $-1$, $1$, $1+2i$, $-1+2i$. Compute $$\int_{\partial R} \frac{(z^2 +i)\sin(z)}{z^2+1}dz$$where the boundary of $R$ is traversed counterclockwise. Here is ...
0
votes
0answers
45 views

Applying contour integration to $\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$

Is it possible to apply contour integration to find the value of following integral $$\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$$
0
votes
2answers
36 views

Compute the following integral, where $C$ is the circle $|z|=3$

Evaluate:$$\int_{C} (1 + z + z^2)(e^\frac{1}{z}+e^\frac{1}{z-1}+e^\frac{1}{z-2}) dz $$ where $ C$ is a circle $|z|=3$ and $z \ \epsilon \ \mathbb{C}$ The function that is being integrated has ...
1
vote
3answers
81 views

Evaluating an integral using Gamma function [closed]

For $r \in (0,2)$, I would like to evaluate the integral $$\frac{2}{r} \int_0^{\infty} \frac{\sin(u)}{u^r} du.$$ The answer should be $$\frac{\pi \cdot \mathrm{cosec}{\frac{r\pi}{2}} ...
1
vote
1answer
33 views

Show that there exists an entire function $h$ such that $\lim_{n\to\infty}{h(nz)}=0$ for all $z\ne0$

Show that there exists an entire function $h$ such that $\lim_{n\to\infty}{h(nz)}=0$ for all $z\ne0$. The following construction is in Walter Rudin's Real and Complex Analysis Chapter 16, Exercise 11. ...
0
votes
1answer
52 views

How can I solve this integral with complex number?

$n$ here is a complex number such that $n=n_r+in_i$ How can I solve this integral? $$\int_{0}^{\infty}\frac{x^4}{|x^2-n^2|^2} d x=? $$
3
votes
2answers
34 views

Computing $\int_{\gamma}e^zdz$, where $\gamma$ is a particular semicircle

How can I compute $\int_{\gamma}e^zdz$, if $\gamma$ is the semicircular arc depicted below? So, $\gamma=3e^{i\theta(t)}$, with $0\le\theta(t)\le\pi$, and then ...
1
vote
2answers
81 views

$\int\limits_{\gamma} \frac{z}{(z-1)(z-2)}dz$, $\gamma(\theta) = re^{i\theta}$, $2 < r < \infty$

For $0 < r < 2$, we can use Cauchy's integral formula and choose our holomorphic function to be $f(z) = \frac{z}{z - 2}$ since $z = 1$ is the only pole, but if $r > 2$, then both poles $z = ...
3
votes
1answer
43 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 ...
2
votes
3answers
30 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. ...
6
votes
2answers
247 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
20 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 ...
2
votes
0answers
53 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
124 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
27 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
40 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
27 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
1answer
60 views

integral of $ \int_{\gamma}e^{1/z}dz$ [closed]

How do you find the integral of $$ \int_{\gamma}e^{1/z}dz$$ in the domain $ D= \{Re z >0\}$
0
votes
0answers
35 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
15 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
31 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
67 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
105 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
40 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
78 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
48 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
93 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
0answers
24 views

Calculate the integral $f(z)=\frac{e^{iz}}{z(z-\pi/2)^2}$ over $|z+1|=2$

Calculate the integral $f(z)=\frac{e^{iz}}{z(z-\pi/2)^2}$ over $|z+1|=2$. Since the singularity at $z=0$ is in the given contour, I integrated using Cauchy's theorem to get $$2\pi i \left[ ...
-1
votes
0answers
39 views

Calculate the integral of $f(z)= 2Re(z) + 3Im(z)$ over the contour $|z|=4$

I know to parametrize $z$ but really need step by step help with contour integration.
1
vote
1answer
54 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
149 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
79 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
91 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
56 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
62 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 $2\pi r$.