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

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46 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. $$\int_{\Gamma} (3\bar{z}^2+2z^3)\,dz$$ ...
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0answers
63 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
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 ...
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1answer
40 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
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 $$ ...
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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!
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1answer
190 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
7 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
90 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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2answers
108 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 ...
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2answers
57 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
141 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 $$z=e^{i\theta}$$ $$\int_{\gamma} \frac{-2{(z^{2}-1)}^2}{i(z-i)^{3}(z+i)^{3}}d\theta$$ ...
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2answers
86 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
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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
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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 ...
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2answers
72 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
31 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
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 ...
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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 ...
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1answer
32 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 ...
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2answers
208 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}: ...
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3answers
79 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 ...
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0answers
54 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 ...
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1answer
27 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 $C$ ...
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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
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0answers
30 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
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1answer
87 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
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 ...
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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
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1answer
68 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
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2answers
119 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 ...
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0answers
34 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 ...
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1answer
107 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 ...
5
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2answers
195 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
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1answer
40 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
31 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
44 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 ...
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3answers
53 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?
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1answer
57 views

How can I calculate the integral $ \int_{\left| z \right| = r} \frac{dz}{(z-a)^n(z-b)^n} $ [closed]

How can I calculate the integral? $$ \int_{\left| z \right| = r} \frac{dz}{(z-a)^n(z-b)^n} $$ for $ \left| a \right| < r < \left| b \right|$ and $ m, n > 1$ I tried to use the cauchy ...
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3answers
138 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 ...
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1answer
76 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
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2answers
108 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
49 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, ...
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41 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
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0answers
46 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 ...
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4answers
206 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} ...
5
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1answer
38 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
42 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 $[\ R, R]$, ...
2
votes
2answers
132 views

Pole on a contour. Problem with integration

I have a problem with calculation of the complex integral $$\int_{|z|=1}\frac{z^2+3z+2i}{(z+4)(z-1)}dz$$ Apparently integrand has a pole in $1$ lying on our circle. What can I do? I cant use Cauchy ...
5
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0answers
57 views

Integral with contours

I want to evaluate the integral $\displaystyle \int_0^\infty \dfrac{\ln x}{e^x+1}\,{\rm d} x$ using contour integration. At first I though using a rectangular. Problem is that I cannot establish the ...