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

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Locations of singularities of a function with respect to given contours

Show that $\int_{C_1}f=\int_{C_2}f$, where $C_1:|z|=1$, $C_2:|z|=2$, and $f(z)=\frac{2z+1}{\sin z}$. Hint: Locate the singularities of $f$ in each case and indicate their location with respect ...
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1answer
37 views

Complex Integrations and Contours

Show that $$\int_C \frac{2 z^2-5}{(z^2+1)(z^2+4)} dz \le \frac{\pi R (2 R^2+5)}{(R^2-1)(R^2-4)} $$ Let $C$ be the upper half of the circle $z=R$ for any $R>2$. Do I need to actually find the ...
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0answers
53 views

Prove: $\int_C \frac{dz}{z^2+1} =0$ on the annulus $6\lt |z| \lt 8$

Let $D$ be the annulus $6\lt |z| \lt 8$ and let $C$ be any simple closed contour inside $D$. Show that there holds: $$\int_C \frac{dz}{z^2+1} =0$$ This has two singular points, $z=\pm i$, these are ...
4
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1answer
42 views

$\int_{\gamma}f(z)\log\left(\frac{z+1}{z-1}\right)dz = 2\pi i\int_{x=-1}^{x=1}f(x)dx$ on an ellipse

I am a self-studier and this is a problem from a course I've been doing. I would appreciate help showing: $$\int_{\gamma}f(z)\log\left(\frac{z+1}{z-1}\right)dz = 2\pi i\int_{x=-1}^{x=1}f(x)dx$$ ...
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2answers
51 views

Closed form for $\int_0^\infty\frac{1}{(1+x^2)^s}\,dx$ when $s\in (0.5,\infty)\setminus\mathbb{N}$

I know that the improper integral $$ \int_0^\infty\frac{1}{(1+x^2)^s}\,dx $$ is convergent for $s>0.5$ and divergent otherwise. Furthermore, it has a closed form for $s \in \mathbb{N}$ (this can ...
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2answers
44 views

Calculating residue $\int_C \frac{8-z}{z(4-z)}dz$

I want to calculate the following: $$\int_C \frac{8-z}{z(4-z)}dz$$ $C$ is a circle of radius $7$, centered at the origin,negative oriented. I want to do this via finding the residues at $z=0,4$. I ...
3
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4answers
103 views

Integral using contour integration [duplicate]

Here is the integral I want to evaluate: $$\int_{0}^{2\pi} \frac{dx}{a+b \cos x }, \quad a>b >0$$ Apparently there are limitations as to what values $a, b$ are supposed to take but let us not ...
3
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1answer
77 views

Steepest descent method with movable maximum

Suppose we want to find the asymptotic behavior as $n \rightarrow \infty$ of the integral $$\int_C \frac{dz}{z} \frac{e^z}{z^n}=\int_C \frac{dz}{z} \exp(z-n \ln z)$$ where $C$ is some contour in the ...
3
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1answer
37 views

Polynomial Inequality via Contour Integration

Problem. Let $P(z)=\sum_{k=1}^{n}a_{k}z^{k}$ be a polynomial which is real on the real axis. Prove the inequality ...
3
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3answers
74 views

Evaluating past exam problem: $\int_C \frac{\sin z}{(z+1)^7} \mathrm{d}z$

I want to evaluate the following: $$\int_C \frac{\sin z}{(z+1)^7} \mathrm{d}z$$ Where $C$ is the circle of radius $5$, centre $0$, positively oriented. Now this has one root at $z=-1$. Now I should ...
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0answers
22 views

real integral using residue theorem

Edit before posting: my result didn't match with the solution, found the error while posting, figured I would post it anyway because someone else might find it useful I'm trying to solve: ...
3
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1answer
83 views

How to evaluate such integral with pole structure?

Let's have integral: $$ I = \int \limits_{-\infty}^{\infty} \frac{e^{-\frac{x^{2}}{2}}}{x - a - i0} $$ How to evaluate it? I tried to do following: $$ \frac{1}{x -a - i0} = \int ...
3
votes
2answers
89 views

How to integrate $e^{-\cos(\theta)}\cos(\theta + \sin(\theta))$

I am struggling to find a way to evaluate the following real integral: $$\int_{0}^{2\pi}e^{-\cos(\theta)}\cos(\theta + \sin(\theta))\:\mathrm{d}\theta$$ The exercise started by asking me to ...
4
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3answers
114 views

Evaluating $\int_{-\infty}^{\infty}\frac{\cos x}{e^x + e^{-x}}$ using the Residue Theorem

I consider the complexification $$f(z)=\frac{e^{iz}}{e^z+e^{-z}}$$ Poles of $f$: $\text{Denominator}=e^{-z}(e^{2z}+1)=0\Rightarrow e^{2z}=-1=e^{i(\pi + 2\pi k)}\Rightarrow z=\frac{i\pi(1+ 2k)}{2}$, ...
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0answers
24 views

Question about an boundary integral equation with a jump in the boundary

I have the following problem: $$\Delta u = 0\;in\;\Omega$$ with several boundary conditions. Applying Green's second identity the representation formula can be derived: ...
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2answers
182 views

A (basic?) contour integration problem

I am trying to prove the following using complex analysis: $$\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}}{a^{2}+n^{2}}=\frac{\pi}{a\sinh(a\pi)}$$ I am told to use the following function: ...
6
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1answer
73 views

Evaluating $\int_0 ^{\infty}\frac{dx}{x^{1/3}(1+x)}$ using Complex Analysis

I am trying to use the residue theorem to evaluate $$I=\int_0 ^{\infty}\frac{dx}{x^{1/3}(1+x)}$$ I'll explain my difficulty in finding a contour, then I explain my difficulty in finding a new contour ...
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1answer
31 views

removable singularity

Let $C$ be the positively oriented boundary of the square with vertices $(1,0)$, $(1,-i)$, $(-1,-i)$ and $(-1,0)$. If $$ f(z)=\frac{\sin(z)}{z}, $$ then clearly $f$ has a removable singularity on ...
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0answers
21 views

Change of variables in contour integrals

Lets say I had a keyhole contour around a branch cut on +x axis. Is the path of integration supposed to remain the same? for the value of the integral to remain the same, am I supposed to go around ...
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0answers
130 views

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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0answers
44 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[ ...
1
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1answer
53 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 ...
3
votes
2answers
106 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 = ...
2
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1answer
66 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 ...
2
votes
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 $\gamma$ ...
1
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3answers
48 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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0answers
23 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 ...
2
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0answers
47 views

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 ...
1
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2answers
46 views

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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0answers
41 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: ...
3
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0answers
62 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)? $$ \int_0^{\pi} \frac{ y \cos(y)}{s^2+y^2} \,dy $$
1
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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 ...
3
votes
3answers
144 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 ...
2
votes
1answer
101 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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0answers
17 views

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} ...
4
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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 $\operatorname{Log}(z)=\ln|z|+i\arg(z)$, ...
1
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1answer
62 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
57 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 ...
2
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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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0answers
16 views

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
40 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
92 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
42 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?
2
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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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0answers
39 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*} ...
3
votes
2answers
71 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 ...
0
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0answers
33 views

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 ...
1
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0answers
28 views

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 ...