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

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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 $x>x_0$, $y↦ F(x + iy )$ ...
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2answers
148 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 ...
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1answer
25 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 ...
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2answers
20 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 ...
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1answer
35 views

Improper integral (using methods in complex variables) [closed]

Let $0<a<1$. Evaluate the integral $$\int_0^\infty \frac{x^{a-1}}{1+x} dx.$$
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1answer
89 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 ...
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0answers
167 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 $a,b \in \mathbb{C}, |z| <1$, and $- \pi < \arg(z) < \pi$. Let $C$ be the right half ...
1
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1answer
24 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 ...
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1answer
39 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 $$\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| = \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|?$$
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25 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
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2answers
68 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 ...
4
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0answers
31 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} ...
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1answer
18 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 ...
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0answers
46 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 $$\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$$
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2answers
45 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 ...
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3answers
89 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}} ...
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1answer
34 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. ...
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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? $$\int_{0}^{\infty}\frac{x^4}{|x^2-n^2|^2} d x=? $$
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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 ...
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2answers
84 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 $z = ...
3
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1answer
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$\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
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3answers
43 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. ...
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2answers
258 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 ...
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23 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 ...
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0answers
56 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
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3answers
133 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 ...
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1answer
34 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}} = ...
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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!} ...
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3answers
29 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} ...
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1answer
63 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\}$
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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 ...
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17 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 ...
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2answers
35 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
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1answer
69 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 ...
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1answer
125 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? $$\oint_{|z|=2}z^3 \bar {z} e^\frac{1}{z-1} \mathrm{d}z$$
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0answers
44 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 ...
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1answer
86 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
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1answer
52 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
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2answers
96 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 ...
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26 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[ ...
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1answer
56 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 ...
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153 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
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2answers
97 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
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1answer
122 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?
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1answer
57 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
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1answer
64 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
53 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
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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
69 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
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2answers
72 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 = ...