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

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1answer
68 views

The Poisson Integral is harmonic

We have proved that for $h(e^{\mathcal{i}\theta})$ continuous on the unit circle, the Poisson Integral of $h$ defined by ...
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2answers
158 views

$\int_0^\infty \frac{x^2}{(x^2-4)(x^2-9)}\,\text dx$

I am trying to compute the following contour integration but am quite stuck I have to evaluate it analytically, by extending it to the complex plane and solving an appropriate integral involving a ...
2
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2answers
117 views

Contour Integral of $\log(z)/(1+z^a)$ where $a\gt1$

I am asked to prove that: $$ \int_{0}^{+\infty}\frac{\log z}{1+z^{\alpha}}\,dz = -\frac{\pi^2}{\alpha^2}\cdot\frac{\cos\frac{\pi}{\alpha}}{\sin^2\frac{\pi}{\alpha}},$$ provided that $\alpha > 1$, ...
4
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1answer
160 views

A bessel function integral

$$\int_{-\infty}^{\infty} dy \frac{J_1 \left ( \pi\sqrt{x^2+y^2} \right )}{\sqrt{x^2+y^2}} = \frac{2 \sin{\pi x}}{\pi x} $$ How do I show this?
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1answer
58 views

Change the order of integrals:$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$

$$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$$ From this it is obvious that $x\in[0,1],y\in[0,1-x],z\in[0,x+y]$. For it asks for the order to be in $$\int dz\int dx\int f(x,y,z)dy$$ . My method ...
1
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1answer
75 views

Show the length of a contour, given by traversing once round a circle radius r, is 2πr

I have tried this problem using the definition for length of a contour $$ L(\gamma) = \int |\gamma'(t)| dt $$ Along the contour $\gamma =Z +re^{it}$ But I cannot get it to work out at $2\pi r$.
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2answers
63 views

$\int_0^{2\pi} e^{\cos(\phi)}\cos(\phi - \sin(\phi)) d\phi$ via contour integration

Can anyone help me calculating this integral using contour integration? $\int_0^{2\pi} e^{\cos(\phi)}\cos(\phi - \sin(\phi)) d\phi$ I've used the subctraction formula of the cosine: $$\cos(\phi - ...
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0answers
24 views

Compute the integrals $\int_{C_1}f(z)dz$ and $\int_{C_2}f(z)\,dz$.

Given two smooth contours, $C_1$ and $C_2$, that respectively lie on the upper and lower half plane compute $\int_{C_1}f(z)dz$ and $\int_{C_2}f(z)\,dz$. Let $a$ be a fixed real positive number. ...
3
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1answer
85 views

Regarding branch cuts and contour integration

I am trying to compute the following integral through the use contour integration. $$ \int_0^1 \frac{dx}{\sqrt{x^2-1}} $$ So, I am considering the same integrand but from $-1$ to $1$, then doing the ...
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2answers
78 views

Compute $\int^{2 \pi} _0 \frac{1}{a + \sin \theta} d\theta$

I want to compute $\displaystyle \int^{2 \pi} _0 \frac{1}{a + \sin \theta} d\theta$, with $a > 0$, where we may use the Cauchy Integral Formula. The following hint is given: Write $sin \theta = ...
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1answer
42 views

Using Cauchy Integral Formula: $\small\displaystyle \int_c \frac{e^{-z^2}}{z^2}dz$

I am going over the solutions to previous problems in order to prepare for a test. I am having a hard time understanding even basic applications of Cauchy's Integral Formula. For example, I have ...
3
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0answers
56 views

Integration over a variety

If $ M $ is a differentiable manifold equipped with an Atlas $ \mathcal{A} = ( U_i , \varphi_i )_{ i \in I} $, we can then calculate the integral of a differential form $ \omega $ over $ M $ with the ...
1
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1answer
40 views

Calculate $\cos(z)/(z^2-\pi^2)$ using Cauchy integral formula on region |z|=4

I want to verify if my reasoning and answer is correct here. Since $\pi$ and $-\pi$ are both contained within the circle centered at 0 with radius 4, we can use the Cauchy integral formula to deal ...
2
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1answer
24 views

Calculate $\sin(z)/(z+i)$ using Cauchy Integral Formula on region $|z+i|=3$

I just want to know what I'm doing wrong here. So we have a singularity at $z=-i$ but this is inside the region of circle centered at $-i$ with radius 3. Hence by Cauchy Integral Formula we have ...
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1answer
19 views

Circle difference in contour integral

Let's say I am integrating a function over $|z| = 1$ and $|z-1| = 1$, is there any difference? I think the answer for both cases will be same, as in both cases, $$ z = \exp^{i\Theta} $$ and $$ dz = i ...
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1answer
31 views

Evaluating $\int_\Gamma \frac{2z^2-z+1}{(z-1)^2(z+1)}dz$ along the contour that is shaped by the figure-8 centered at $z=-1$ and $z=1$.

In my answer key, it says this equals $0$, but I get $4 \pi i$. Here's why: $$ \int_\Gamma \frac{2z^2-z+1}{(z-1)^2(z+1)}dz = \int_\Gamma\biggl[\frac{1}{(z-1)^2}+\frac{1}{z-1}+\frac{1}{z+1}\biggr]dz ...
1
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1answer
21 views

Confusion regarding contour integral solution

In Schaum's complex variable book, there is an exercise in contour integration: $$ \int \overline{z}^{2} dz $$ over $|z|=1$. The answer seems to be $0$, but when I integrate like this using contour ...
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1answer
45 views

Find the line integral of $1/(z^2+4)^2$ over region $\gamma$

I have to find: $$I=\oint_{\gamma}\frac{dz}{(z^2+4)^2}.$$ $\gamma$ in this case is a circular curve defined by $|z-i|=2$, which is a circle centered at $i$ with radius $2$. It is clear that the ...
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0answers
28 views

Find Line integral of $e^{-z} /{z-\pi/2}$ on a region $\gamma$

Let $\gamma$ be the diamond connecting points $x=2, -2$ and $y=2, -2$. and its oriented positively (counter-clockwise, I believe?). I'm not so sure if we can use the Cauchy integral formula here and ...
6
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0answers
156 views

Complex contour integral: How does the stationary point method used in this case?

I was reading a paper which has the following integral in order to do the inverse Laplace transformation: $$ I=\frac{1}{2\pi i}\int_{-i\infty+\gamma}^{i\infty+\gamma} ...
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0answers
101 views

Using the “appropriate” formula

I am asked to solve $$\int_{C}\frac{1}{z+i}dz$$ where $C$ is parametrized $z(t) = 2+e^{it}$ for $t \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ by finding the antiderivative $F(z)$ of $f(z)$ and then ...
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2answers
43 views

Paremetrising the Contour

I'm trying to paremetrise the Contour of a unit circle descibed anti clockwise. This is so I can integrate $$ \int_{|z| = 1} \frac{e^z}{4z^4} dz $$ Now normally $z(t)=e^{it}$ for $t\in [0,2\pi]$ is ...
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1answer
39 views

Explaining information on Contour

I wish to compute the following line integral $$\int_{C}(x-iy)dz$$ where $z(t)=(e^t- 1,t)$, $t \in [0,2]$ $dz = dx + idy = (e^t - 1 + i)dt$ We then have $\int_{C}(e^{t} - 1 - it)(e^t - 1 + i)dt$ I'm ...
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1answer
37 views

Complex integration confusion

I wish to compute $\int_{C}(x^2 - iy^2)dz$, where $C := \{z\mid |z|=1\}$ is positively oriented. I am a bit confused on what $dz$ actually is. I know I have $\int_{C}x^2dz - i\int_C y^2dz$, but I ...
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0answers
32 views

Integrating along a contour

I wish to compute $\int_{C} \frac{dx}{x^2 + y^2} -2xydy+ i\int_{C}(xdx - ydy)$, where $C$ is the contour is parametrized by $z(t) = (\cos(t),\sin(t))$ for $t \in [0, 2\pi]$. To compute this, I should ...
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0answers
86 views

Fundamental Theorem of Calculus for Complex Numbers

Lets say we have the integral:$$\int_\gamma\frac{1}{z}+z^2dz$$ and we would like to apply the Fundamental Theorem of Calculus here for complex numbers. Now, we let $\gamma$ be any curve connecting ...
4
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1answer
44 views

Simple Question About Contour Integration

If you are integrating $$\int_\gamma y^2\,dz$$ Where $\gamma$ is the line segment from $1$ to $i$. You parameterize the line $$x(t)=1-t$$ $$y(t)=t$$ $$\implies z(t)=1-t+it$$ Now, if you want to use ...
1
vote
1answer
38 views

Complex integration parametric form

Evaluate$\int_{\gamma(0;1)} \frac{\cos z}{z}dz$. Write in parametric form and deduce that$$\int^{2\pi}_0 cos(\cos\theta)\cosh(\sin\theta)d\theta=2\pi$$ By Cauchy's integral formula, ...
0
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1answer
49 views

how to calculate $\frac{1}{2\pi i} \int_{\gamma} \frac{2z}{(z-1)^ 4(z-3)}$

How to calculate $\frac{1}{2\pi i} \int_{\gamma} \frac{2z}{(z-1)^ 4(z-3)}dz$ When $\gamma = C_+(0,4)$ and where $\gamma = C_-(0,2)$. I need to use the residuformula which states that is f is ...
0
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1answer
37 views

Computing a contour integral of a function that is not analytic inside the contour

I'm wondering if there is another way to calculate the contour integral of $\int(\tan(z/2)/(z-1))$ in the square w/ sides $Re(z)=+/-2$, $Im(z)=+/- 2$ other than using the residue theorem. The cauchy ...
2
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0answers
41 views

compute the complex-valued integral for the branch cut

Let $C$ be the circle of radius $2$ centered at origin. Let $f(z)$ be the branch cut of the function $z^{2−i}$ on the domain $−π < θ < π$. Compute the integral $$ \int_C f(z) dz$$ My attempt: ...
1
vote
1answer
46 views

Finite integral with removable singularity

I wanted to integrate $\frac{(exp(-x) -1)^2}{x}$ from $x=0$ to $x=a$ where $a$ is finite. Since the integrand, viz., $\frac{(exp(-x) -1)^2}{x}$ has a removable singularity at $x=0$ , I can take the ...
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3answers
45 views

Applying Cauchy's Integral Theorem to $\int_{C_R} z^n \ dz$

First, Cauchy's Integral Theorem: If $f$ is a continuous function on $U$ admitting a holomorphic primitive $g$, and $\gamma$ is a closed path in $U$, then \begin{equation} \int_\gamma f = 0 ...
2
votes
1answer
41 views

Splitting complex contour integrals into real and complex parts

The question I am stuck on is : By considering the contour integral $\int_{C(0,1)} \frac{1}{z^2+4z+1}$ (where $C(0,1)$ is the unit circle) show that $$\int_0^{2\pi} ...
7
votes
4answers
263 views

Integral by residue - “dog bone”

Let $I=\int_{-1}^{1}\frac{x^2 dx}{\sqrt[3]{(1-x)(1+x)^2}}$. I used complex function $f(z)=\frac{z^2}{\sqrt[3]{(z-1)(z+1)^2}}$, which we can define such that it is holomorphic on ...
1
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0answers
46 views

Complex integral over sphere in polar coordinates

I have trouble evaluating the integral: $$\int_{B(0,\frac{3R}{|h|})} \frac{1}{(r e^{2i a}-e^{i a})}dr da$$ In fact I just need to estimate it from above in terms of $|h|log (\frac{1}{|h|})$, where ...
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5answers
212 views

Show that $\int_1^{\infty } \frac{(\ln x)^2}{x^2+x+1} \, dx = \frac{8 \pi ^3}{81 \sqrt{3}}$

I have found myself faced with evaluating the following integral: $$\int_1^{\infty } \frac{(\ln x)^2}{x^2+x+1} \, dx. $$ Mathematica gives a closed form of $8 \pi ^3/(81 \sqrt{3})$, but I have no ...
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1answer
73 views

Prove that $f=u+iv$ is differentiable if and only if $\lim_{r→0} \frac{1}{πr^2 } \int_{C(z_0,r)}f(z)dz=0$

Suppose that $u,v$ are real-valued function that having continuous partial derivative of first order in the neighborhood of $z_0=x_0+iy_0$ . Prove that $f=u+iv$ is differentiable if and only if ...
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1answer
39 views

Evaluate $∫_γ \frac{z^2+1}{z(z^2+4)} dz$ Where $γ(t)=re^{it}$ with $0≤t≤2π$ for all possible value of $r$, $0<r<2$ and $2<r<∞$

Evaluate $∫_γ \frac{z^2+1}{z(z^2+4)} dz$ Where $γ(t)=re^{it}$ with $0≤t≤2π$ for all possible value of $r$, $0<r<2$ and $2<r<∞$ Theorem: Let $f: G \to \mathbb C$ be analytic, suppose ...
3
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3answers
222 views

Evaluating $\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx$ using complex analysis

I'm taking a complex analysis course and doing some practice computing residues & evaluating integrals. I pulled out an old book called "The Cauchy Method of Residues: Theory and Applications, ...
3
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1answer
56 views

Show that $\lim_{r\to 0} \frac{1}{r^2}\int_{C_{r}}f(z)dz = 2 \pi i\frac{\partial f}{\partial \bar{z}}(z_0)$

Well, after spending hours on this problem, I'm still stuck, so I thought I'd turn to you guys. The problem statement is as follows. Let $f$ be a complex-valued function that is $C^1$ in the disk $|z ...
0
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1answer
56 views

Simple Complex analysis integration

If we let $\gamma$ be the circline path from $ 0$ to $1$, how do we list all possible values of $$\int_{\gamma} z^3dz$$ One of which, I think, could be over the real axis, s.t. $$\int_{\gamma} ...
1
vote
1answer
67 views

$PV \int_{ia-\infty}^{ia+\infty}\frac{e^{ikt}}{\sqrt{t^2-b^2}}dt$

How would I calculate \begin{align} PV \int_{ia-\infty}^{ia+\infty}\frac{e^{ikt}}{\sqrt{t^2-b^2}}dt \end{align} The square root should be dealt with as is most appropriate, e.g. by taking a branch ...
6
votes
1answer
172 views

Evaluate Integral with $e^{ut}\ \Gamma (u)^{2}$

I am trying to integrate this integral: $$f(x)=\frac{1}{2\pi j}\int_{c-j\infty}^{c+j\infty}x^{-s}\sigma ^{ms-m}\left [ \frac{\Gamma \left ( \frac{s}{\beta} \right )}{\Gamma \left ( \frac{1}{\beta} ...
1
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1answer
48 views

Spectral representation of an analytic function

I have a question about the spectral representation of an analytic function $G$ on a Riemann surface (specifically, the complex plane with a finite amount of cuts), i.e. the representation of the form ...
2
votes
2answers
64 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
128 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
96 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 ...