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

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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 ...
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1answer
25 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 ...
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0answers
38 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: ...
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1answer
29 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
37 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
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1answer
27 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} ...
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4answers
185 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 ...
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0answers
32 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 ...
9
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5answers
190 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
71 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
38 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 ...
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3answers
96 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
50 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
49 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
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1answer
63 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
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1answer
167 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} ...
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1answer
38 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 ...
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2answers
53 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
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2answers
117 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 ...
0
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1answer
34 views

Contour Integral Evaluation [closed]

Evaluate $$\int_0^{2\pi}\frac{dt}{a+b\sin(t)}$$ Assuming that $a,b$ are real and $a>|b|$. How do I do this? I am very stuck.
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2answers
90 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
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1answer
56 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 : ...
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2answers
62 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 ...
3
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1answer
133 views

Show that $ \int^1_0 x^3 \sqrt{x} \sqrt{1-x} dx = \frac{\pi}{5!} \frac{1.3.5.7}{2^5} $

I'm trying to show the following. $$ \int^1_0 x^3 \sqrt{x} \sqrt{1-x} dx = \frac{\pi}{5!} \frac{1\cdot3\cdot5\cdot7}{2^5} $$ This is a problem regarding contour integration. My complex analysis ...
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2answers
96 views

Compute the integral $\int_C (z^2-1)^\frac{1}{2} dz$ where $R>1$

Let C be the circle of Radius $R>1$, centered at the origin, in the complex plane. Compute the integral $\int_C (z^2-1)^\frac{1}{2} dz$ where we employ a branch of the integrand defined by a ...
1
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1answer
62 views

contour integrals parametrising and solving

Use Cauchy's integral formula to compute the following: $$\int \limits_{\Gamma} \frac{e^{-z}}{z-1}dz$$ where $\Gamma$ is the square with parallel sides to the axes, centre $i$ and side length $5$ ...
2
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0answers
29 views

Integration imaginary and real part with branch cut

I have some problems with this integral $$ I=\int_{0}^{1}z(1-z)log(1-z(1-z)\frac{q^2}{m^2})dz $$ I see $z(1-z)$ get max value at $\frac{1}{4}$ and if $q^2>4m^2$ log function will be negative and ...
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3answers
104 views

How do I integrate $\int_{0}^{\infty}\frac{\cos(ax)-\cos(bx)}{x^2}\text{d}x$?

How do I integrate $\int_{0}^{\infty}\frac{\cos(ax)-\cos(bx)}{x^2}\text{d}x$, for positive and real $a,b$? I know the contour that I have to use is a semicircle with a small semicircle cut out near ...
1
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2answers
64 views

Integration using Cauchy's Theorem

I am attempting to evaluate the integral $$\int_C\left(z+\frac{1}{z}\right)^{2n}\frac{dz}{z}$$ where C is the unit circle centered at the origin. Using parameterized $z=e^{i\theta}$ and got that ...
0
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1answer
17 views

Complex Integration parametrisation

I'm trying to integrate $\int_\gamma (z^2-2)dz$ where $\gamma$ is a spiral that loops 3 times and ends at (3,0) on the Argand diagram. I have found the parametric equations for this contour to be ...
0
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1answer
19 views

Complex contour integration of a branch (Not even sure what it's asking)

$f(z)$ is the branch $z^{-1 + i} = e^{(-1 +i)\ln{z}}$ such that $|z| > 0$ and $0 < arg(z) < 2\pi$. I'm to integrate $f(z)$ over the contour $e^{i\theta}$ (just the unit circle). ... I have ...
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0answers
14 views

Find the criteria on a variable

What would the criteria on the variable $v$ be such that $f\left( t\right) $ is always negative . $$f\left( t\right) =\int_{\mathbb{R}^{+}}\frac{\cos \left( xt\right) }{x^{v}}% dx=\frac{\Gamma \left( ...
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0answers
11 views

Parametrize the contour that consists of

Parametrize the contour depicted below that consists of a line segment and a circular arc (the circle is centered at the origin). Parametrize the pieces only. I could not get the image but I will ...
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0answers
19 views

Prove that z(t) and z~(t) are admissible parametrizations of the same smooth curve

Prove that $$z(t)=t+it^2, 0\leq t \leq1 $$ and $$\tilde{z}(t)=tan\Gamma+itan\Gamma, 0 \leq \Gamma \leq \frac{\pi}{4}$$ are admissible parametrizatiions of the same smooth curve. Do the above ...
0
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2answers
18 views

Use this parametrization to compute the following integral.

Let $$\Gamma$$ be the circumference centered at 1-i of radius 5 and transversed once in the counterclockwise direction. Parametrize the contour $$\Gamma$$. Use this parametrization to compute the ...
0
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1answer
29 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 ...
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1answer
32 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. ...
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1answer
39 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
62 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 ...
0
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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 ...
1
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1answer
39 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
33 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
80 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
116 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 ...
0
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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
83 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
99 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
55 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
votes
1answer
108 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
votes
2answers
69 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 ...