For questions about integration methods that use results from complex analysis and their applications

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1answer
51 views

Integration of 1/(square root of a third order polynomial) on the complex plane

I have to compute the integral $$ \oint_{a,b} \frac{d \lambda}{\sqrt{(\lambda-\lambda_1)(\lambda-\lambda_2)(\lambda-\lambda_3)}},$$ Here is the picture of the integration contours and cuts: The "...
1
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1answer
29 views

Integral of $\omega\wedge\overline{\omega}$ on Riemann surface

Let $X$ be a Riemann surface of genus $g$ and $\omega$ a meromorphic 1-form on it. I've read that if $\omega$ has just a simple pole in $x\in X$ (and is holomorphic on $X\setminus\{x\}$) then the ...
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1answer
42 views

Is $\int_{\gamma} \sec ^2z \ \mathrm{d}z=0$?

Let $\gamma = \gamma(0;2)$. Is $$\int_{\gamma} \sec ^2z \ \mathrm{d}z$$ equal to $0$? I'm trying to answer this question using only tools like Cauchy Theorem or the Deformation Theorem since ...
1
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0answers
36 views

Contour integral of $\sqrt[3]{z^3-1}$ on $|z| = 2$ and branches

This is an exam question that i'm trying to figure out. Apart from the title, there is a note added to the question that says that: The branch of $\sqrt[3]{}$ is the one that has $\sqrt[3]{7} \in \...
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1answer
21 views

Integral calculus - complex numbers

I need help to integrate this, i tried changing variable and it didn't work, i tried integration by parts, and it failed too. $\frac{1}{2π}∫e^{jx\omega}\frac{(1/6)}{(1/6-j\omega)}d\omega$ I need to ...
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0answers
13 views

How to compute the $PDF_{X}(x)$ of $X$ if it cannot be Fourier inverted from the characteristic function $CF_{X}(z)$?

I have a positive random variable $X>0$. I have to compute the probability density function $$PDF_{X}(x)$$ I can compute in closed-form the extended characteristic function ($z \in \mathbb{C}$) $$ ...
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0answers
24 views

calculating curvilinear integral by residue theorem

Calculate the following integral by transposing to a curve integral and then using the residue theorem: $\displaystyle \int_{0}^{2\pi}{\frac{e^{int}}{C-e^{it}}dt}, \qquad |C|\ne1, n\in \mathbb N$. ...
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0answers
17 views

An identity relating the unknown $CDF_{X}(x)$ of $X>0$ and the known characteristic function $CF_{X^2}$ of $X^2$

I have a positive random variable $X>0$. I don't know that much about its distribution and I have to compute the cumulative distribution function $$ CDF_{X}(x) = Prob(X\leq x) $$ Other definitions:...
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1answer
22 views

It's possible to generalize the Ml inequality (also call Estimation Lemma)?

The ML inequality property in complex integral says $|\int_{c}f(z)dz| \leq ML$. If I have two function in the integral, I can write the inequality: $|\int_{c}f(z)g(z)dz| \leq ML|\int_{c}g(z)dz| $ ?. ...
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1answer
39 views

Complex integration with trigonometric functions

I'm not sure how to solve this integral: $$ \int_{0}^{2\pi} [\frac{a+\cos(n\theta)}{a^2+1+2acos(n\theta)}] \ d\theta $$ SUGGGESTION: Use the function $ f(z)=\frac{1}{z^n+a} $ Solution (I developed ...
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0answers
30 views

Complex integration with $\epsilon$ and $\pi$.

I'm not sure how to solve this integral: $$ \int_\gamma \frac{dz}{(e^z+4)(z-\pi i)}, $$ where $ \gamma$ is the region $ \|z-\sqrt7i\|+\|z+\sqrt7i\|=8$. I know that region is the ellipse $\frac{x^2}...
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0answers
30 views

Asymptotic behavior of a function defined via a complex integral

I would appreciate any comment/correction about what I did for the following problem, I would be very thankful if you let me know the parts of it which may not be very precise: Let $g(z)$ be defined ...
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0answers
24 views

Evaluating line integral $I=\int_{\gamma} \frac{z^k}{(z-1)^k}dz$

$I=\int_{\gamma} \frac{z^k}{(z-1)^k}dz$ where $\gamma(t) = 2cos(t)e^{it}$ where $t\in [0,2\pi]$ My attempt : This is a path which is a boundary of $D(1,1)$ traversed twice CCW. thus we get by Cauchy'...
4
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3answers
88 views

Complex integration of $\int_{-\infty}^{\infty}\frac{dx}{(x^2+2)^3}$

$$\int_{-\infty}^{\infty}\frac{dx}{(x^2+2)^3}$$ I know that I can use a complex function $f(z)$ and I must deal with: $$\int_{-\infty}^{\infty}\frac{dz}{(z^2+2)^3}$$ So I need the roots of $z$ $$...
4
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2answers
83 views

Compute $\int_{0}^{2\pi}\frac{1}{(2+\cos\theta)^2}\,d\theta$

I''m stuck in a exercise in complex analysis concerning integration of rational trigonometric functions. Here it goes: We want to evaluate $\int_{0}^{2\pi}\frac{1}{(2+\cos\theta)^2}\,d\theta$. ...
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1answer
53 views

Contour Integration problem solve [closed]

Let C be the circle $z=2+e^{i\theta}$, where $0\le\theta\le2\pi$, Evaluate $$\int_{c}\frac{\sin z}{z^2+2z}dz.$$ i need delicate explanation to understand. i really tried to solve this problem but my ...
4
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2answers
122 views

Compute $\int\limits_Q^1\sqrt{(1-x^2)(1-\frac{Q^2}{x^2})}\mathrm{d}x$

I am trying to compute the integral $$\int\limits_Q^1\sqrt{(1-x^2)(1-\frac{Q^2}{x^2})}\mathrm{d}x$$ where $0\leq Q <1$ is a real number. I tried to substitute $x=\cos y,$ but this didn't bring ...
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4answers
64 views

Integrate $e^{-x}\cos x$ with respect to $x$ by complexifying the integral?

I am supposed to find the integral by complexifying it and noticing that $e^{-x}\cos x$ is the real part of $e^{(-1+i)x}$. However I don't see how you can figure that out before knowing the expression ...
4
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1answer
52 views

Conditions on a complex measure to be real

Let $(X,\mathcal{S}, \mu)$ be a measure space with $X$ a locally compact Hausdorff space, $\mathcal{S}$ the Borel subsets of $X$ and $\mu$ a complex measure. Suppose that $$ \int_X f \ d\mu \in \...
1
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1answer
53 views

Calculating $\int_0^{\infty } \frac{\ln (x)}{\sqrt{x} \left(a^2+x^2\right)^2} \, \mathrm{d}x$ using contour integration

I can do this integral using the keyhole contour the answer is:$$\int_0^{\infty } \frac{\ln (x)}{\sqrt{x} \left(a^2+x^2\right)^2} \, \mathrm{d}x = -\frac{\pi (-6 \ln (a)+3 \pi +4)}{8 \sqrt{2}a^{7/2}}$...
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0answers
39 views

contour integral z/conjugate(z)

I am trying to calculate: $$\int_C \frac{z}{\bar{z}}dz$$ where C is the upper semicircles of the circles centred in (0,0) of radii 1,2 joined at their intersections with the real axis by the real ...
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0answers
53 views

Xi Function on Critical Strip - Mellin Transform

Story I'm trying to prove the following identity $$\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cos(xt) dt = \frac{1}{2} \pi (e^{\frac{1}{2}x} - 2e^{-\frac{1}{2}x} \psi(e^{-2x}))$$ where $$\psi(...
1
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1answer
45 views

Contour integral $\int_{-\infty}^{\infty}e^{-iax}/(-b+\cos(x))\mathrm dx$ with $a>0$ and $0<b<1$

The integral is $$\text{PV}\int_{-\infty}^{\infty}\frac{e^{-iax}}{(-b+\cos(x))}\, dx$$ with $a>0$ and $0<b<1$. This integral stems from the Fourier transform of a Green's function in ...
1
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3answers
136 views

Evaluate $\int_{-\infty}^{\infty} \frac{(1-\cos { y } )}{\mid{y}\mid^{1+\alpha}}dy$ [closed]

How do I evaluate the following integral? $$\int_{-\infty}^{\infty} \frac{(1-\cos { y } )}{\mid{y}\mid^{1+\alpha}}dy=\frac{\pi}{\Gamma(1+\alpha)\sin(\frac{\pi\alpha}{2})}$$ Thank you in advance. ...
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1answer
43 views

Contour Integral of $\frac {1}{1+z^2}$ over $\delta B(0,2)$

My next question about contour integrals is: Is it true that: $$\int_{\delta B(0,2)} \frac{1}{1+z^2}dz = \int_{\delta B(0,2)} \frac{\frac{z}{1+z^2}}{z} dz = \left[ 2\pi i \frac{z}{1+z^2}\right]_{z=...
0
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1answer
34 views

Contour integral for $\bar{z}^3$

Consider the curve $C: [0,\frac{\pi}{2}] \to \mathbb{C},\,C(t) = 2\exp(-it)$. Is it true that $$ \begin{align*} \int_C \bar{z}^3\,\mathrm dz &= \int_0^\frac{\pi}{2} \overline{(2 \exp(-it))}^3 (-2i\...
1
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2answers
41 views

Complex curve integral $\frac{1}{1+z^2}$

I want to calculate $\int_\gamma\frac{1}{1+z^2}\,\mathrm dz$ where $\gamma = \delta B(i,1)$, circle with radius $1$ around $i$. So i have $\gamma(t) = i+\exp(it),\,t \in [0,2\pi]$ with $$\int_\gamma \...
1
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0answers
25 views

Basics on Schwarz -Christoffel Integral

I've just began to study the Schwarz-Christoffel integral, but I'm having trouble to understand some very basic points. For example, take $S:\mathbb{H}\to \mathbb{C}$ (where $\mathbb{H}:=\{z\in \...
0
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0answers
30 views

Complex Analysis Problem (Argument principle or Rouché's Theorem ?)

My problem: Let f be analytic in $\overline{B(0;R)}$ with $f(0) = 0$, $f'(0)\ne0$ and $f(z)\ne0$ for $0 < |z| \le R$. Put $\rho = \min\limits_{|z|=R} |f(z)| > 0$. Define $N: B(0; \rho) \...
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5answers
99 views

Evaluate $\int_0^{\infty} \frac{\log(x)dx}{x^2+a^2}$ using contour integration

This question is Exercise 10 of Chapter 3 of Stein and Shakarchi's Complex Analysis. Show that if $a>0$, then $\int_0^{\infty} \frac{\log(x)dx}{x^2+a^2}=\frac{\pi \log(a)}{2a}.$ The hint is ...
3
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3answers
91 views

Mean Value Property to show that entire function is a constant

Let $f(z)$ be an entire function so that, $$ \int \frac{|f(z)|}{1 + |z|^3} dA(z) < \infty$$ where the integral is taken over the entire complex plane. Show that $f$ is a constant. I believe ...
4
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1answer
233 views

A variation of Ahmed's integral

Given that the closed form exist, evaluate the following Integral: $$\displaystyle \int\limits_{0}^{1} \frac{(x^2+4)\sin^{-1}x}{x^4-12x^2+16} \, dx $$
2
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2answers
88 views

Evaluating $\int_c\frac{1}{\sin\frac{1}{z}}\text{d}z$ over $C= \{z\big\vert|z|=\frac{1}{5}\}$

Evaluate $$\int\limits_{|z|=\frac{1}{5}} \frac{1}{\sin\frac{1}{z}}\text{d}z$$ My attempt: I know that this function has non isolated singularity at $0$, and simple poles at $\frac{1}{n \pi}$. ...
1
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1answer
29 views

Evaluation of complex integral

$$\int_{\text{c}}\frac{\sin \pi z^2 + \cos \pi z^2}{(z+1)(z+2)}$$ Where $\text{C}$ is the circle $|z| =3$ I'm a little confused about how to do this. Should this be done the normal way ? How do I ...
1
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1answer
34 views

Radius of convergence and the existence of antiderivative

I think I have some misunderstandings regarding some basic concepts. First, the question I'm dealing with is the following: Let $f$ be analytic in $\{z ;|z|>1 \}$, and $\int_{|z|=2}f(z)dz=0$. ...
6
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1answer
119 views

Prove that a complex-valued entire function is identically zero.

Suppose $f$ is entire and $$\iint_\mathbb{C}|f(z)|^2dxdy < \infty$$Prove that $f\equiv 0.$ So far I have: Suppose $f$ is bounded. Then $f$ is constant by virtue of Liouville and so the ...
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0answers
33 views

Complex integration f [duplicate]

Prove that $$\int_{0}^{\infty} \frac{dt}{1+t^{n}}=\frac{\pi}{n}\csc\frac{\pi}{n}$$ Please help,I don't have a clue
2
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0answers
28 views

How do I visualise the contour integral of a complex function? [duplicate]

I've just learnt about the contour integral of a complex function, but I'm having trouble figuring out what it is calculating visually. I understand it is somewhat analogous to the line integral for ...
0
votes
1answer
63 views

Determine this real integral with the Residue-theorem.

$$\int_{-\infty}^{\infty}{\frac{\sin x}{x^4-6x^2+10}\,\mathrm dx}$$ I get that when I evaluate the $\frac{\sin x}{x}$ one, I work with $\frac{e^{ix} - e^{-ix}}{2ix}$, I create a huge semicircle and a ...
1
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1answer
42 views

Complex integration $\Rightarrow$ delta-distribution?

My physics textbook states that $$ \int_{-\infty}^{\infty} e^{i(p-p')x/\hbar}dx = 2\pi \hbar \,\delta(p-p')$$ Whereas $\delta(p-p')$ is the delta-distribution. I see that for $p=p'$ the integral ...
1
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1answer
29 views

Antiderivative of $\lvert z \rvert^2$

How does one determine (or show) that the complex function $f(z)=\lvert z \rvert^2$ does not have an antiderivative? (I'm assuming this because contour integrals along two different curves with the ...
1
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3answers
83 views

principal value of $\int_{-\infty}^{\infty}\frac{\sin^2(x)}{x^2}\mathrm{d}x$

I know the answer is $\pi$ there is a proof here. Now looking to my textbook (textbook image) the result should be $0$. Using the last equation on the right hand page we have: $$ i\pi(\sin^2(x))'|_{x=...
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0answers
38 views

General Form of Theta Functions from Functional Equations

From Elliptic Curves: Function Theory, Geometry and Arithmetic by McKean and Moll: Exercise 3.1.2. Discuss the general solution of the two identities (a) $f(x+2)=f(x)$ and (b) $f(x+2\omega)=e^{ax+b}f(...
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1answer
28 views

Complex integration on a closed curve

Find $\oint_C \frac{dz}{z-2}$ on the square $C$ with vertices $\pm2\pm 2i$ . As there is a pole at $z=2$, I removed it by taking a semicircle of small radius $r$ about $2$ and the integral on the ...
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0answers
12 views

Complex integration on a curve having singularity.

Find $\oint_C \frac{dz}{z-2}$ on the square $C$ with vertices $\pm2\pm 2i$ . As there is a pole at $z=2$ , I removed it by taking a semicircle of small radius $r$ about $2$ and the integral on the ...
0
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3answers
70 views

Evaluating the real integral $\int_{0}^{2\pi}\frac{1}{2+\sin\theta}d\theta$ using complex analysis

I thought it's value would be zero, since the complex integrand: $$\Im\left(\int_{C}\frac{1}{2+e^{i\theta}}d\theta\right)$$ Where $C$ is the unit disc, is nonsingular. Also $e^{iz}\ne -2$ for any $z$...
3
votes
3answers
82 views

Evaluating the integral of $\frac{\cos(x) - e^{-x}}{x}$ using contour integration

I am trying to evaluate the value of $$\int_0^\infty\frac{\cos(x) - e^{-x}}{x}dx$$. I am assuming I am supposed to use contour integration, as I was required just before to calculate the value of $$\...
0
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2answers
44 views

Winding number of closed curves

Let $c_1,c_2$ be closed curves in $\mathbb C^{\times}$ and we define $c(t):=\frac{c_1(t)}{c_2(t)}$. Proof the following for the winding number $win(c,0)=win(c_1,0)-win(c_2,0)$. I have no idea to ...
0
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0answers
18 views

Cycle homology to closed curve

Let $c$ be a cycle in $\mathbb C^{\times}$ and $c_n:[0,1]\to \mathbb C^{\times},c_n(t)=e^{2\pi int}$. Show that $c$ and $c_n$ are homologous. They are homologous if the winding number $win(c-c_n,z)=...
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0answers
25 views

Every cycle in a domain $D$ is null homolog

Let $D$ be a domain, where every cycle is null homolog and $f$ be a biholomorphism. Proof that every cycle $c$ in $f(D)$ is nullhomolog. Let $c$ be a cylce in D, it is null homolog, if $$\frac{1}{2\...