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

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How to compute the definite integrals of special functions?

How can these integrals be solved: $${1\over \pi} \int_{0}^{\infty}\left({{\sqrt{x}(a-bx)}\over {x^{3}+(a-bx)^{2}}}\right)\cos(\sqrt{\alpha x}) \exp(-xt)\,\mathrm{d} x, $$ and $${1\over \pi} ...
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41 views

How to Solve this Improper Integral with six poles?

I'm trying to solve the following integral, where $a>0$, $b>0$, $y\in\mathbb{R}$ and $z\in\mathbb{R}$ are given constants: $$ \int_{-\infty}^{0} \left[ ...
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1answer
48 views

ML-inequality: How to show that $e^{i2x} = e^{i2z}$ when evaluating $\int_{-\infty}^\infty \frac {\cos^2 (x)}{x^2 + 1} dx$

I am to solve the following integral: $$\int_{-\infty}^\infty \frac {\cos^2 (x)}{x^2 + 1} dx$$ We use contour integration in combination with residue calculus, so for $R > 1$ ($R$ is the radius ...
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2answers
102 views

Showing $ \int_0^{2 \pi } \frac{dt}{a^2 \cos^2 t + b^2 \sin^2 t} = \frac{2 \pi}{ab}$ [duplicate]

The question: Let $\gamma$ be a contour such that $0 \in I(\gamma),$ where $I$ is the interior of the contour. Show that $$\int_\gamma z^n \, \text{d}z = \begin{cases} 2\pi i & \text{if } n = ...
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1answer
59 views

Numerical or analytical or exisistence: Inverse Laplace Transform

Edit 1: With the hint of Ron, we can simplify the question to : $$\bar{f}(s)=\frac{1}{(s^2+1)\arctan s }$$ So what about this function's inverse Laplace Transform? Or can anyone tell me that the ...
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1answer
54 views

Contour integral of $\int_0^\infty \log(x) e^{-x} dx$

Is it possible to resolve this integral using integral contour? What should be the contour? \begin{equation} \int_0^\infty \log(x) e^{-x} dx = -\gamma \approx -0.577216 \end{equation} where $\gamma$ ...
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3answers
46 views

Higher order poles, how high?

Clarification: I claim, for function $g(z)$, analytic and nonzero at $z=0$, if I have the function $f(z)=g(z)/z^n$ there is no use in trying to find poles of order smaller than n. And I would be ...
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21 views

Recovering cosh(ax) from it's fourier transform

Let's say $f(x)=\cosh(ax)$, where $a$ is a complex number and $x$ is real. Then the fourier transform is $F(\omega)=\sqrt \frac{\pi}{2} \delta(\omega-ia)+\sqrt \frac{\pi}{2} \delta(\omega+ia)$. So ...
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Does the dirac delta function have a residue?

I came to this question by looking at the fourier transform of a hyperbolic cosine. Let's say $f(x)=\cosh(ax)$, where $a$ is a complex number and $x$ is real. Then the fourier transform is ...
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definite integral of form G(cos(x), sin(x)) by complex integration

Given: the following integral: $$ \int_0^{2\pi} \frac{\mathrm{sin}(3x)}{5-3\mathrm{cos}(x)}\,\mathrm{d}x = 0 $$ Prove it by using complex integration and the residue theorem. But I do something wrong ...
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39 views

Contour integral with a different contour

This question is from the post: Contour integral with branch cut. My question is: if we choose the key hole contour with branch cut on the positive x axis, it seems that we have an addtional term: ...
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61 views

Solution of $\int_0^{\pi} \frac{ y \cos y}{s^2+y^2} dy$

Is there a solution for the following integral (even in terms of Bessel or Struve functions)? $$ \int_0^{\pi} \frac{ y \cos(y)}{s^2+y^2} \,dy $$
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1answer
43 views

$\int_c \frac{8-z}{z(4-z)} dz$

I want to calculate the following contour integral: $$\int_c \frac{8-z}{z(4-z)} dz$$ where $C$ is the circle of radius $7$, center $0$, negatively oriented. Do I have to do this the long and ...
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3answers
141 views

integral $\int_0^\pi \frac{\cos(2t)}{a^2\sin^2(t)+b^2\cos^2(t)}dt$

I want to compute this integral $$\int_0^\pi \frac{\cos(2t)}{a^2\sin^2(t)+b^2\cos^2(t)}dt$$ where $0<b \leq a$. I have this results $$I_1=-\frac{ab}{2\pi}\int_0^\pi ...
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1answer
80 views

Non trivial integral with the Bose-Einstein distribution and Cosine function

Do you have any idea how to solve this integral? $$\int\limits_0^\infty {\frac{{\cos \left( mx \right)}}{{x + {x_0}}}\left( {1 + n\left( x \right)} \right)} - \int\limits_0^\infty {\frac{{\cos ...
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Branch cut jump of $\log z$

$$\int _ C \log z\,dz$$ where $C$ is a full circle in positive direction with radius $R$. I substitute $z=Re^{it}$, $dz=Rie^{it}dt$ $$\int _ 0 ^{2\pi} \log (Re^{it})Rie^{it}\,dt$$ $$\int _ 0 ^{2\pi} ...
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1answer
47 views

Complex analysis integration with logs

$$\int_C \operatorname{Log}\left(1-\frac 1 z \right)\,dz$$ where $C$ is the circle $|z|=2$ I don't even know how you would begin doing this. I understand $\operatorname{Log}(z)=\ln|z|+i\arg(z)$, ...
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1answer
57 views

Why can't branch cut pass through poles?

In the wiki article, Example (IV) – branch cuts. Why can't we can't we choose the contour so that the branch cut is on the negative x axis. If we choose this, the two residual is out of the contour, ...
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2answers
28 views

Surface area of revolution of curve

I am wondering why this particular integration is being found difficult to solve. Would appreciate any help I can get. the graph is $y = x^3$ and the limits are $0 \leq y \leq 1$
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2answers
53 views

ML-inequality: Why does this hold $|e^{-3y+3ix}| = e^{-3y}$ during numerator estimation of $f(z) = \frac{e^{3iz}}{z^2 + 1}$ [duplicate]

Given the following related to an ML-inequality for $R > 1$: Estimation of the numerator from the function $f(z)$ is supposed to develop as follows: I'm wondering why and how exactly the ...
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3answers
76 views

Real Methods to Evaluate $2 \int_{-1}^{1}x^2 \sqrt{1-x^2}dx$

I was recently contacted by a friend to find the values of the two following integrals by any means. $$ I=2\int_{-1}^{1}x^2 \sqrt{1-x^2}dx$$ $$ J=\int_{-1}^{1}(1-x^2) \sqrt{1-x^2}dx$$ The first ...
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Effective Branch cut

I have problems understanding how to get to the "effective Brach cut" in the top answer of this post: Dog Bone Contour Integral ? The answer says that one has a Branch cut for ...
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2answers
37 views

Integral on the real line between 0 and infinity using contour integration

For part (a) I have that the singularity is at $(1+i)/root2$ and it is a simple pole? For part (b) I have that the residue at $f(z)$ at that point is $-(1+i)/4root2$ For part (c) I used the ML ...
3
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2answers
88 views

Integrals on the real line using contour integration

I know I am supposed to split it up like this and gamma(R) tends to zero and the other tends to my integral as R tends to infinity? I compute the residue at $2i$ which I think is $sin(2i)/2$ ? ...
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2answers
41 views

Contour integral, f(z)=$ze^{z^2}$

For part $(a)$ is the answer just $0$? Using Cauchy-Goursat theorem? For part $(b)$ I am confused. Do I use ? It seems very complicated. Am I missing a trick?
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1answer
59 views

A real integral (may be requires contour integration)?

The integral I have in mind is $$\int^\infty_0 x^{r}(x + \lambda)^{-1}dx$$ where $r \in (-1, 0)$, and $\lambda$ is a non-negative constant. I apologize if this is really easy and I am missing some ...
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1answer
17 views

Question about a certain step in Rudin's General Cauchy Theorem proof

I am having trouble seeing a certain claim that Rudin makes in proving his "Global Cauchy's Theorem": $\textbf{Cauchy's Theorem.}$ Suppose $f$ is holomorphic in $\Omega$, which is an open set in ...
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36 views

Contour integration, cos(z)sin(z)

Evaluate \begin{equation*} \int_{\Gamma}\cos(z)\sin(z)dz,~\Gamma:\gamma(t):=\pi t+(1-t)i,~0\leq t\leq 1. \end{equation*} I think I should do it using this \begin{equation*} ...
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2answers
68 views

$\int \frac{\exp (z)(\sin(3z)}{(z^2-2)(z^2)} dz$ on $|z|=1$

So I need to calculate \begin{equation*} \int \frac{\exp(z) \sin(3z)}{(z^2-2)z^2} \, dz~\text{on}~|z|=1. \end{equation*} So I have found the singularities and residues and observed that the ...
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Contour integral from first principles

What does it mean by 'evaluate from first principles? Does it mean use ? For part (a) do I parametrise as $\gamma(t)=a+2e^{it}$ with $t$ between $0$ and $2\pi$? Doing this I end up with the ...
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Counting poles that are shared between $f$ and $g$

Suppose I have a meromorphic function $f(z)$ with poles at $f_i$ and $\mathcal{Res}(f,f_i)=1$, and $g(z)$ with poles at $g_i$ and $\mathcal{Res}(g,g_i)=1$. I would like to construct a function ...
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1answer
49 views

Contour integral $|z-i|=1/9$

Calculate \begin{equation*} \int_{\Gamma}\frac{1}{z^4+16}dz, \end{equation*} where $\Gamma :|z-i|=\frac{1}{9}$. I have asked I similar question to this but I still do not understand.... when I find ...
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1answer
62 views

Calculate contour integral

Calculate $\displaystyle \int_\Gamma \frac 1{z^4 + 81}$ where $\Gamma: |z+i| = \frac 34$ Can somebody help me with this question please or give me a hint on how to get started, as I have never ...
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2answers
78 views

The value of $\sum_{n=-\infty}^{n=\infty}\frac{1}{n^2-z^2}$ on $\mathbb{C}\setminus\mathbb{Z}$

Show, for $z\not\in\mathbb{Z}$, that $$\sum_{n=-\infty}^{n=\infty}\frac{1}{n^2-z^2} = \frac{-\pi}{z\tan(\pi z)}$$ Hint: You may assume that there exists $C$ such that $|\pi\cot(\pi w)|\leq ...
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2answers
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Computing $\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$.

I would like to compute the following integral : $$\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$$ using Residue theorem. I took the contour corresponding to half of the "donuts" ...
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0answers
15 views

contour integral of complex function

How would one compute the contour integral of along the wedge shape contour for the function $f = z^{-3/2} = \dfrac{1}{r^{3/2}}\dfrac{1}{\cos(\dfrac{3\theta}{2})+i\sin(\dfrac{3\theta}{2})}$ or ...
4
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1answer
61 views

How to use a contour integral to evaluate $\frac{2}{\pi }\int\limits_0^\infty {x{e^{ - {x^2}t}}\sin ax} {\rm{ }}dx$?

So I faced this question in our textbook: Using a contour integral, evaluate the improper integral $$\frac{2}{\pi }\int\limits_0^\infty {x{e^{ - {x^2}t}}\sin ax} {\rm{ }}dx$$ I don't need the ...
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1answer
87 views

Evaluate integral over complex path numerically to show that $C_\infty$ is equivalent to $-I$

I would like to evaluate $$C_\infty = \int_{R = -a}^{R = a} H_0^{(1)}(z) e^{-izt} dz $$ where $H_0^{(1)}(z)$ is the Hankel function of the first kind, $a \rightarrow \infty$, and $$ z = R - i ...
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1answer
59 views

Application of Complex Variables

Considering the integral : $$\int_{-\infty}^{+\infty}\left(\dfrac{\sin\alpha z}{\alpha z}\right)^2 \dfrac{\pi}{\sin{\pi} z}dz,\quad \alpha \lt \dfrac{\pi}{2}$$ around a circle of large radius, prove ...
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3answers
48 views

Integration of complex functions with trig functions: $\int_0^{2 \pi} \frac{ d\theta}{5-\cos( \theta )}$

$\int_0^{2 \pi} \frac{ d\theta}{5-\cos( \theta )}$ How should I integrate this? Using the exponantial identities of trig? Any hints will be great... Thank you!
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2answers
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What does “let $f\left(z\right)=\bar{z}$” mean in this context

I'm reading a paper/handout on contour integrals and Cauchy's Theorem which says in an example Let $f\left(z\right)=\bar{z}$. $\cdots$ Then \begin{align} ...
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1answer
57 views

Weird (?): Use Cauchy's Integral formula to calculate $\int _{|z|=3} \frac{z}{z^2-\pi^2}dz$

Weird (?): Use Cauchy's Integral formula to calculate $\large \int _{|z|=3} \frac{z}{z^2-\pi^2}dz$. But the function $\frac{z}{z^2-\pi^2}$ is holomorphic on all of $|z|=3$. Am I missing something? ...
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1answer
52 views

How i can find the fourier transform of $\frac{\sinh(ax)}{\sinh(\pi x)}$ where,$ |a| < \pi$

Using a rectangular contour in the complex plane, bypassing the poles at $z=0$ and $z=i$, i got $$\int_{-\infty}^{+\infty}\frac{\sinh(ax)e^{ikx}}{\sinh(\pi x)}dx - ...
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71 views

Integration Around Part of a Branch Cut

I am studying the integral, given by a Laplace transform, $$\int_0^\infty\!e^{-\alpha x}\sinh^{-2/3}x\left(1+\frac 12\sinh^2x\right)^{-1/6}\left(1-\beta\sinh^{4/3}x\right)^{1/2}\,\mathrm dx$$ From ...
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39 views

Where am I using the condition that $\gamma$ does not contain the origin

I am trying to evaluate $\displaystyle \int_\gamma z^n dz$ where $\gamma$ is a circle not containing the origin with positive orientation ($n\in \mathbb{N}$). I first calculated $\displaystyle ...
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1answer
69 views

Symbolic contour integral evaluation

Can anyone help with the evaluation of the following contour integral : $$\oint\limits_C \phi(x,y)\,dx+\psi(x,y)\,dy.$$ Where the contour $C$ is given by: What I am looking for is how to split ...
1
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1answer
28 views

Limit as $r$ tends to zero of integral $\int_C \frac{e^{iz}-1}z \mathrm dz$

Let $\mathcal C$ be a semi-circle of center $O(0,0)$ and radius $R$, such that $y \ge 0$. Find the limit as $R$ tends to zero of: $$\int_{\mathcal C} \frac{e^{iz}-1}z \mathrm dz$$ How can I find ...
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2answers
60 views

Inverse Laplace transform of s/s-1

Finding the inverse laplace transform: $$L^{-1}\left\{\frac{s}{s-1}\right\}$$ I wrote: $$L^{-1}\left\{\frac{s}{s-1}\right\}=L^{-1}\left\{\frac{1}{s-1}\right\} + L^{-1}\{1\}=L^{-1}\{1\} + e^{t}$$ And ...
1
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0answers
29 views

Integral of function has different values depending on contour?

What can I say if the integral of my function has different values depending on contour? If my function were analytic on a domain it would evaluate to $0$ right? My contours give values like $\pi ...
3
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3answers
238 views

How to know if an integral is well defined regardless of path taken.

I can calculate \begin{equation*} \int_0^i ze^{z^2} dz=\frac{1}{2e}-\frac12, \end{equation*} but why can I calculate this irrelevant to the path taken? Is this since it is analytic everywhere - if ...