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

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2
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2answers
65 views

Evaluating an integral using Cauchy's Integral Formula

I am having a little bit of trouble with the following: $$\int_{\gamma}\frac{z^2-1}{z^2+1}dz$$ where $\gamma$ is a circle of radius $2$ centered at 0. I am trying to separate this or simplify it into ...
4
votes
2answers
129 views

Complicated contour integral to be solved.

Anyone can help to solve the following integral? $$I=\int_{0}^{\infty} dp p^{-1}e^{-2p^{2}M^{-2}}\sin(pr)\frac{M^2}{M^2+p^2}$$ at this stage I am able to write the integral as ...
1
vote
2answers
97 views

Various evalutions of $\int_0^\infty \sin x \sin \sqrt{x} \,dx$

I'm looking for various ways to evaluate the integral: $$\int_0^\infty \sin x\sin \sqrt{x}\,dx$$ I'm mainly interested in complex analysis. I can think of a wedge -shaped contour of angle $\pi/4$ but ...
0
votes
1answer
61 views

cauchy int formula, function not holomorphic

Use Cauchy's integral formula to evaluate the following integral, $$\int \limits_{\Gamma} \frac{\sin(\pi z^2)+\cos(\pi z^2)}{(z-1)(z-2)}dz$$where the contour $\Gamma$ is parameterised by $\gamma : ...
0
votes
2answers
68 views

unobvious cauchy integral formula

Use Cauchy's integral formula to compute the following: $$\int \limits_{\Gamma} \frac{\cos(z)+i\sin(z)}{(z^2+36)(z+2)}dz$$ where $\Gamma$ is the circle of centre $0$ and radius $3$ traversed in the ...
3
votes
1answer
143 views

Show that $ \int^1_0 x^3 \sqrt{x} \sqrt{1-x} dx = \frac{\pi}{5!} \frac{1.3.5.7}{2^5} $

I'm trying to show the following. $$ \int^1_0 x^3 \sqrt{x} \sqrt{1-x} dx = \frac{\pi}{5!} \frac{1\cdot3\cdot5\cdot7}{2^5} $$ This is a problem regarding contour integration. My complex analysis ...
1
vote
2answers
122 views

Compute the integral $\int_C (z^2-1)^\frac{1}{2} dz$ where $R>1$

Let C be the circle of Radius $R>1$, centered at the origin, in the complex plane. Compute the integral $\int_C (z^2-1)^\frac{1}{2} dz$ where we employ a branch of the integrand defined by a ...
1
vote
1answer
68 views

contour integrals parametrising and solving

Use Cauchy's integral formula to compute the following: $$\int \limits_{\Gamma} \frac{e^{-z}}{z-1}dz$$ where $\Gamma$ is the square with parallel sides to the axes, centre $i$ and side length $5$ ...
2
votes
0answers
45 views

Integration imaginary and real part with branch cut

I have some problems with this integral $$ I=\int_{0}^{1}z(1-z)log(1-z(1-z)\frac{q^2}{m^2})dz $$ I see $z(1-z)$ get max value at $\frac{1}{4}$ and if $q^2>4m^2$ log function will be negative and ...
1
vote
3answers
112 views

How do I integrate $\int_{0}^{\infty}\frac{\cos(ax)-\cos(bx)}{x^2}\text{d}x$?

How do I integrate $\int_{0}^{\infty}\frac{\cos(ax)-\cos(bx)}{x^2}\text{d}x$, for positive and real $a,b$? I know the contour that I have to use is a semicircle with a small semicircle cut out near ...
1
vote
2answers
79 views

Integration using Cauchy's Theorem

I am attempting to evaluate the integral $$\int_C\left(z+\frac{1}{z}\right)^{2n}\frac{dz}{z}$$ where C is the unit circle centered at the origin. Using parameterized $z=e^{i\theta}$ and got that ...
0
votes
1answer
18 views

Complex Integration parametrisation

I'm trying to integrate $\int_\gamma (z^2-2)dz$ where $\gamma$ is a spiral that loops 3 times and ends at (3,0) on the Argand diagram. I have found the parametric equations for this contour to be ...
0
votes
1answer
20 views

Complex contour integration of a branch (Not even sure what it's asking)

$f(z)$ is the branch $z^{-1 + i} = e^{(-1 +i)\ln{z}}$ such that $|z| > 0$ and $0 < arg(z) < 2\pi$. I'm to integrate $f(z)$ over the contour $e^{i\theta}$ (just the unit circle). ... I have ...
1
vote
0answers
16 views

Find the criteria on a variable

What would the criteria on the variable $v$ be such that $f\left( t\right) $ is always negative . $$f\left( t\right) =\int_{\mathbb{R}^{+}}\frac{\cos \left( xt\right) }{x^{v}}% dx=\frac{\Gamma \left( ...
0
votes
0answers
14 views

Parametrize the contour that consists of

Parametrize the contour depicted below that consists of a line segment and a circular arc (the circle is centered at the origin). Parametrize the pieces only. I could not get the image but I will ...
0
votes
0answers
22 views

Prove that z(t) and z~(t) are admissible parametrizations of the same smooth curve

Prove that $$z(t)=t+it^2, 0\leq t \leq1 $$ and $$\tilde{z}(t)=tan\Gamma+itan\Gamma, 0 \leq \Gamma \leq \frac{\pi}{4}$$ are admissible parametrizatiions of the same smooth curve. Do the above ...
0
votes
2answers
20 views

Use this parametrization to compute the following integral.

Let $$\Gamma$$ be the circumference centered at 1-i of radius 5 and transversed once in the counterclockwise direction. Parametrize the contour $$\Gamma$$. Use this parametrization to compute the ...
0
votes
1answer
35 views

Parametrize the contours of integration where Gamma is arc of the circle of radius…

Parametrize the contours of integration and write the integrals in terms of the parametrizations. Do not calculate them. $$\int\frac{\bar(z)}{z^3}dz$$ where $$\Gamma$$ is the arc of the circle of ...
2
votes
1answer
36 views

Contour integration over a spiral

Evaluate $$\int_{\gamma} (z^2-2) \mathrm{d}z$$ where $\gamma$ is the following curve: Use two methods: direct calculation via a parametrization of $\gamma$, and the fundamental theorem. ...
0
votes
1answer
48 views

Parametrize the contours of integration

I am having a difficult time figuring this problem out: Parametrize the contours of integration and write the integrals in terms of the parametrizations. $$\int_{\Gamma} (3\bar{z}^2+2z^3)\,dz$$ ...
1
vote
0answers
65 views

Imaginary part of An Squre Root Integration

I am looking for a particular form of an integral which some simplified version of it has the following form $$ \Im\int_{0}^{\infty} \frac{\sqrt{1+u^4-u^6}}{u^5}du. $$ Could someone gives some idea ...
0
votes
0answers
22 views

Double checking if contours are correct

Since $$ |z| = 1 $$ is the unit circle centered at (0,0) which is used as a contour for a lot of integration problems, would $$ |z - i| = 1, |z + 3| = 1 $$ simply be translations of the unit ...
1
vote
1answer
41 views

Complex integration using parametrization

Let $C$ be the circle $|z-z_0| = r$ traversed counter-clockwise, and let $\alpha$ ne any nonzero real number. Parameterize $C$ by $z=z_0+re^{i\theta}$, with $-\pi < \theta < \pi$, and compute ...
0
votes
0answers
48 views

Well-defined of complex line integral

Let $C : [a,b] \rightarrow \mathbb{C}$ be a continuous path. Then $C$ is a piecewise differentiable path if there exists a partition of $[a,b]$, $a = t_0 < t_1 < ... < t_n = b$ such that $$ ...
1
vote
5answers
87 views

Contour integration of cosine of a complex number

I am trying to find the value of $$ -\frac{1}{\pi}\int_{-\pi/2}^{\pi/2} \cos\left(be^{i\theta}\right) \mathrm{d}\theta,$$ where $b$ is a real number. Any helps will be appreciated!
2
votes
1answer
211 views

Calculating Inverse Laplace Transform of stretched exponential

I am trying to solve a Laplace transform problem that has gotten way over my head in terms of complex analysis knowledge. I would like to solve the Inverse Laplace Transform $(s\rightarrow t)$ of ...
0
votes
0answers
10 views

Equality of two integral representations

I have two integral representations given by a contour integral: $$ I_1(x,y) = \oint f_1(x,y,t) dt, \\ I_2(x,y) = \oint f_2(x,y,t) dt $$ for which one needs to prove that they're equal. Both ...
1
vote
2answers
94 views

Example of contour integration

Could someone help me evaluate the following integral with contour integration ? $$\int_{0}^{2\pi}\frac{d\theta}{(a+b\cos\theta)^2}.$$ Constraints are: $a>b>0$.
0
votes
2answers
111 views

Residue theorem: When a singularity gives infinite to the residue

What if one of the singularity gives infinity to the residue. Consider this contour; $$X=\int_{\gamma} e^{i(\frac{z^{2}+1}{2z})}\frac{{(z^{2}-1)}^4}{2z^2(z-i)^{3}(z+i)^{3}}dz$$ I have ...
0
votes
2answers
58 views

Prove that $\oint _{|z|=R} (f-g)' dz = 0$ (Residue Theorem)

I know that $f$ and $g$ have a pole or order $k$ in $z=0$. $f-g$ is holomorph in $\infty$. I need to prove that: $$\oint_{|z|=R} (f-g)' dz = 0$$ Any help? Note: $f$ and $g$ only have a singularity ...
2
votes
1answer
152 views

Residue theorem:When a singularity on the circle (not inside the circle)

This is the integration I am trying to solve $$\int_{0}^{\pi} \sin^{2}(\theta)\sec^{3}(\theta)d\theta$$ putting $$z=e^{i\theta}$$ $$\int_{\gamma} \frac{-2{(z^{2}-1)}^2}{i(z-i)^{3}(z+i)^{3}}d\theta$$ ...
3
votes
2answers
88 views

Definite integral (in the complex plane?)

I want to prove that $$\int_{0}^{\infty} \frac{dx}{1+x^b} = \frac{\pi}{b \sin(\pi/b)} \ ,$$ where $b\in (1,\infty)$. I thought about doing it in the complex plane since the integrand is a ...
3
votes
2answers
61 views

Inverse Laplace Transform of $\frac{s}{(s-a)^{3/2}}$

Find the inverse laplace of: $\frac{s}{(s-a)^{3/2}}$ I tried working through this using partial fractions and convolution but I can't seem to get a requitible answer. How would I go about solving ...
2
votes
1answer
84 views

Prove $\int_{[a,b]}f=\int_{[a,c]}f+\int_{[c,b]}f$

Let $a,b\in\mathbb C$ and $c\in[a,b]$. Let $f$ be continuous on $[a,b]$. Use the definition to show that \begin{equation} \int_{[a,b]}f=\int_{[a,c]}f+\int_{[c,b]}f \end{equation} Note: You should ...
2
votes
2answers
75 views

About the “mixed” form of Gauss and Fresnel integrals

How to integrate the "mixed" form of Gauss and Fresnel integrals as following? $$\int_{-\infty}^{+\infty} {e^{-x^2-ia(x+b)^2} dx} $$ where $a \in R, b \in R$. [EDIT] As Claude Leibovici pointed ...
1
vote
0answers
33 views

Bessel functions and contour integrals

I have a Bessel function $ x^{2}J''+xJ' + (x^{2}+m^{2})J=0 $ Supposing $ J(x) = x^{m}j(x) $ the equation can be reduced to $$ x(j'' + j) + (2m+1)j'=0 $$ My question is, how do i show that $$ ...
0
votes
0answers
16 views

Analyticity of Mellin Barnes integral

How to decide the analyticity of Mellin-Barnes integral? In particular, When Fox's H-function is analytic? Is the condition for existence, analytic and condition for convergence both have the same ...
0
votes
0answers
11 views

Can any one tell m ewith one example, how to evaluate a double Mellin Barnes integral?

What is meant by asymptotic expansion of Gamma function? i.e. $ |\Gamma(z)| = |\Gamma(x+iy)| \approx \sqrt{2 \pi} \left|y\right|^{\left(x - \frac{1}{2}\right)} e^{-\pi \frac{|y|}{2}}, \quad ...
0
votes
1answer
33 views

Contour integration on semicircle as R -> infinity

$$f(z)=\frac{e^{iz}-1-iz}{z^3}$$ What is the value of $$\int_{C} f(z) dz$$ if C is the arc of the semicircle with radius $R\to \infty$ ,going counterclockwise from point (R,0) to (-R,0) Attempt: I ...
4
votes
2answers
217 views

Using complex analysis to evaluate $\int_0^\infty\frac{(\ln x)^3}{1+x^2}d x$

Here is my attempt: Let $R>1>r$ and $C$ be the closed curve in $\mathbb{C}$ consists of the following pieces: $$C_1=\{Re^{it}: t\in(0,\pi)\},\quad C_2=[r,R],\quad C_3=\{re^{it}: ...
1
vote
3answers
89 views

Contour Integration of Line Segments

I am trying to use contour integration to find the integral of: $$ \int_\gamma ydz $$ where we have the union of line segments from $0$ to $i$ and then to $i+2$. I simply do not understand how to ...
0
votes
0answers
61 views

Contour integral with two branch cuts

I'm trying to solve this integral: \begin{equation} \int_0^\infty d\omega \,\frac{\left(\left(\omega ^2+1\right) \cos (\delta )-2 \omega \right) \log ...
1
vote
1answer
29 views

Integrate cos(z) over a quarter of an ellipse.

The complex form of the equation for an ellipse with foci at 1 and -1 is $|z-1|+|z+1|=\sqrt{8}$. a) Find the values of $a$ and $b$ such that $x^2/a^2+y^2/b^2=1$ describe the same ellipse. b) Let $C$ ...
1
vote
2answers
63 views

$\sum_{n=0}^{\infty}\binom{2n}{n}\frac{1}{2^{2n}}$

I want to examine the convergence of the series $$\sum_{n=0}^{\infty}\binom{2n}{n}\frac{1}{2^{2n}}$$ In case it converges I want to evaluate it. I tried the D' Alembert theorem but it was ...
2
votes
0answers
31 views

Contour integral with signum function

I need to solve the following integral $$\int\limits_{\left| {s - a} \right| = \delta } {\frac{{{\mathop{\rm sgn}} (is)}}{{{e^{2\pi irs}}}}ds}$$ Where the contour is the semicircle in the upper ...
5
votes
1answer
91 views

A difficult one-variable exponential integral

I am trying to work out a closed form for the integral \begin{equation} \int_{0}^{1} \frac{1}{\sqrt{s(1-s)}} \exp\left(-\left(\frac{a}{s} + \frac{b}{1-s}\right) \right) \,ds \end{equation} where ...
2
votes
1answer
52 views

Is the integration $\int_{C_R}e^{ikz}dz=0\ (\text{if}\ k>0)$ correct?

When I read P.W. Milonni's book "Fast light, slow light, and left-handed light", I encounter this problem. In chapter 2 of the book, the author introduces the integral ...
1
vote
3answers
36 views

What is the length of the contour $γ(t)=5e^{it}$ for $t$ in the interval $[0,2\pi]$?

Let $C$ be the contour $γ(t)=5e^{it}$ for $t$ in the interval $[0,2\pi]$. What is the length of $C$? Would the length of $C$ be $5$ or $10$? I think $r=5$ so I am not sure whether that would be the ...
1
vote
1answer
73 views

Complex Analysis (Contour Integration)

Given complex numbers $z_1$ and $z_2$, let $[z_1, z_2]$ denote the straight line segment path from $z_1$ to $z_2$. Recall that we can parametrize this by $x(t) = z_1 + t(z_2 - z_1)$ for $t \in ...
2
votes
2answers
120 views

Calculating an integral (using methods from complex analysis) (hints only please)

From Rudin's book, we are to calculate $\int_\mathbb{R} \Big(\frac{\sin x}{x}\Big)^2 e^{itx}dx$ where $i$ is the imaginary number and $t\in\mathbb{R}$. I'm looking for a hint on how to get started. I ...