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

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1answer
6 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
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1answer
17 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
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1answer
25 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
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0answers
50 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 ...
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0answers
19 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
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1answer
30 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 ...
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0answers
29 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 $$ ...
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5answers
71 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
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1answer
35 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 ...
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0answers
4 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 ...
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2answers
78 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: $a>b>0$.
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4answers
43 views

Example of contour integration [closed]

Could someone help me evaluate the following integral with contour integration ? $$\int_{-\infty}^{\infty}\frac{dx}{(x^2+1)(x-2i)(x-3i)(x-4i)}$$
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2answers
89 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
52 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
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1answer
85 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
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2answers
68 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
47 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
80 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
56 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 ...
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0answers
17 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 $$ ...
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0answers
10 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 ...
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0answers
10 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
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1answer
23 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 ...
3
votes
2answers
93 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
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3answers
39 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
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0answers
13 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
18 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
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2answers
58 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
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0answers
23 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
76 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
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1answer
48 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 ...
0
votes
3answers
32 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
45 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
95 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 ...
1
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0answers
29 views

Evaluate Complex Line Integral

Evaluate $\int _C f$ where $f(z)=x^2+iy^2$ and where $C$ is given by $z(t)=t^2+it^2, 0\leq t \leq 1$. I tried reading an example in the book, using the formula $\int_C ...
1
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1answer
89 views

Computing Complex Integral to Determine Analytic Continuation of $f(z) = \int_0^\infty {{e^{-zt}} \over {1 + t^2}} dt$

My question is the following: Find the analytic continuation of the function $f(z)$ defined by $$ f(z) = \int_0^\infty {{e^{-zt}} \over {1 + t^2}} dt, \ \ \vert \arg(z) \vert < {1 \over ...
4
votes
2answers
98 views

Improper integrals with singularities on the REAL AXIS (Complex Variable)

I'm having some troubles when I try to solve improper integrals exercises that have singularities on the real axis. I have made a lot of exercises where singularities are inside a semicircle in the ...
2
votes
1answer
36 views

Contour integral over a segment

Let $S$ denote the segment that connects the points $O(0, 0), \; A(1, 1)$. I want to evaluate the integral: $\displaystyle \int_{S} z^2 \, dz$. The segment can be parametrized as $\gamma(t)=(t, t), ...
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1answer
18 views

integration of an open curve about isolated singularities

I know if I integrate a circular arc of an angle $\theta < 2\pi$ about an isolated singularity of the complex funciton I would get a fraction $\frac{\theta}{2\pi}$ of the residue of that ...
1
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1answer
39 views

contour integral in a region where the function doesn't have any poles

What is the value of the following contour integral? The contour is a circle with radius $0.5$ around $z=i$ point: $|z-i|<\frac{1}{2}$ $$\oint_C\frac{dz}{2-\sin z}$$ I think it is $0$ because ...
0
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3answers
32 views

How To Find The Length Of An Irregular Arc

How would you find out the length of an irregular arc. e.g. An arc with a base length of $10$cm and a height of $5$cm - what would be the length of that arc? Is there a specific formula I could use?
4
votes
3answers
106 views

Evaluation of integral $\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$

I'm trying to evaluate the following integral: $$\mathcal{J}=\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$$ Well there are $3$ poles , one lying on the real line the other on ...
0
votes
1answer
56 views

The Laplace transform of the Heaviside function

I am studying complex analysis but, because I'm an engineer, I have a lot of doubts. I'm going to present my doubts and it would be nice if someone helps me to see things clearly. Let's start with ...
2
votes
2answers
99 views

Principal value of Fourier Integral

I have tried to find the principal value of $$\int_{-\infty}^\infty {\sin(2x)\over x^3}\,dx.$$ As $ {\sin(2x)\over x^3}$ is an even function, its integral may not be zero in the given limits. I ...
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1answer
48 views

Proof some 2 D Fourier transforms

Here are several Fourier transforms I used, I would like to prove those identity. I took some times to figure out how they are derived, I tried the residue theorem and other methods, but I failed, ...
3
votes
0answers
29 views

Did I apply correctly the Lebesgue dominated convergence theorem?

Let's concentrate on $$\int_0^\pi e^{iRe^{i\theta}} i d\theta$$ If $R \to \infty$, this integrand converges pointwise to $0$; plus, the modulus of the function is $= e^{-R\sin\theta} \le ...
2
votes
0answers
37 views

How to compute the covariance matrix of a random variable uniformly distributed in an ellipsoid

Suppose that x is a random variable uniformly distributed in an ellipsoid \begin{equation} x^{T}Mx\leq\delta, \end{equation} where $x\in \mathbb{R}^{n}$. Clearly, the mean of $x$ is zero. The ...
11
votes
4answers
190 views

How to compute $\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$

Could you explain to me, with details, how to compute this integral, find its principal value? $$\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$$ $f(z) =\frac{\sqrt{z}}{z^2-1} = \frac{z}{z^{1/2} ...
4
votes
1answer
31 views

Integral principal value with $\cos$ and $x^2$

Could you tell me how to solve this integral? $$\int_0^{\infty} \frac{\cos x -1}{x^2}dx$$ I think I should focus on this integral $$\int_{\Gamma} \frac{e^{iz}-1}{z^2+ \varepsilon^2}$$ where ...
2
votes
1answer
34 views

Integral with denominator raised to n-th power, residues

I don't know how to calculate this integral: $$\int_{-\infty}^{\infty} \frac{d x}{(1+x^2)^{n+1}}$$ If we denote by $\Gamma$ a curve = semicircle centered at $0$ with radius $R$ + segment $[\ R, R]$, ...