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

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36 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
42 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(...
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1answer
41 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 ...
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1answer
222 views

A variation of Ahmed's integral

Given that the closed form exist, evaluate the following Integral: $$\displaystyle \int\limits_{0}^{1} \frac{(x^2+4)\sin^{-1}x}{x^4-12x^2+16} \, dx $$
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3answers
134 views

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

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
37 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=...
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2answers
39 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 \...
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1answer
33 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\...
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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 \...
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0answers
27 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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3answers
86 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 ...
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5answers
95 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 ...
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1answer
28 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 ...
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1answer
105 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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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 $\frac{1}{n \pi}$. ...
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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 $\int_{|z|=2}f(z)dz=0$. ...
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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
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0answers
27 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 ...
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1answer
61 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 ...
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1answer
41 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 ...
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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 ...
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3answers
77 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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1answer
73 views

Does this function belong to $H^1(\mathbb D)$?

$\mathbb D$ is the unitary disk centered at $0$. Does the following function belong to $H^1(\mathbb D)$? .$$f_\epsilon(z) = \frac{1}{(1-z)\left(\frac{1}{z}\log\frac{1}{1-z}\right)^{1+\epsilon}}, z\in\...
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0answers
34 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
645 views

Proof of Laurent series co-efficients in Complex Residue

Am trying to see if there is any proof available for coefficients in Laurent series with regards to Residue in Complex Integration. The laurent series for a complex function is given by $$ f(z) = \...
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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 $z$...
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1answer
27 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
11 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 ...
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1answer
924 views

How do you integrate Gaussian integral with contour integration method?

How do you integrate $$\int^{\infty}_{-\infty} e^{-x^2} dx$$ with contour integration method? I do not even know how to setup the problem.
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3answers
81 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 $$\...
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2answers
39 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 ...
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0answers
16 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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44 views

If $I=\int_{0}^R \frac {1}{1+x^{2n}}dx$, why$ \int_{0}^R \frac {1}{1+(x e^{i\frac{\pi}{n}})^{2n}}d(xe^{i\frac {\pi}{n}})=e^{i\frac {\pi}{n} }I$?

Just read a proof in the textbook Basic Complex Analysis, and there is one point that I do not understand: Let $I=\int_{0}^R \frac {1}{1+x^{2n}}dx$, where $x$ is real. We have the following: $$ \...
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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\...
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2answers
44 views

Contour integration with logarithms

I'm having trouble calculating the below integral to get the right answer: $$\frac{1}{2\pi i}\int_\gamma \frac{3}{z-2}\; dz$$ where $\gamma$ is parametrised by $\gamma(t)=3e^{it}, t\in [0,2\pi]$. So ...
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1answer
30 views

Recommendations for tutorials specifically devoted to real integration using contour integral techniques.

Complex analysis, and in particular contour integrals and the residue theory have proved a very powerful tool in computing a large class of real function integrals which would be quite troublesome to ...
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3answers
153 views

Need help with $\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$

I'm having trouble trying to evaluate this definite integral. Mathematica didn't help much. $$\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$$ where $a$, $b$ $\in \Bbb R^+$. Is it possible ...
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4answers
58 views

Computing the residue of a rational function

The real integral I am trying to compute with residues/contour integration is $\int_{-\infty}^{\infty}\frac{x^2}{(x^2+a^2)^3} \,dx$ For $a$ positive and by using the complex integral $$\int_{C_R}\...
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0answers
41 views

Exchange series and integral in a complex context.

Let $a_{n},b_{n},c_{n}$ complex sequences and let $k>0$ a real parameter. Assuming that $$\sum_{n\geq1}\sum_{m\geq1}\left|\frac{a_{m}b_{n}}{c_{m+n+k}}\right|<\infty\tag{1} $$ if $k>1/2 $ ...
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1answer
26 views

Complex line integral over a square

Evaluate following complex line integral.Let $c=\{z|\max\{|\text{Re}(z)|,|\text{Im}(z)|\}=1\}$ be the square with orientation $+1$. Calculate $$\int_c \frac{z\ dz}{\cos(z)-1}$$ with $f(z)=\frac{z}{\...
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0answers
24 views

Find the complex integral over a path

I have to integrate the complex function $$z+1/z$$ which is parameterized by $\gamma(t), 0 \le t\le 1$ and satisfies $Im\gamma(t) > 0$, $\gamma(0) = -4+i$ and $\gamma(1) = 6+2i$. Can I assume the ...
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1answer
48 views

Solution to nonlinear ODE with square root

How do I solve the following equation? $\dot{x}=\sqrt{x^{2}-\frac{2}{3}x^{3}}$ with $x(0)=0$? I'm guessing I have to work with $dt=\frac{dx}{\sqrt{x^{2}-\frac{2}{3}x^{3}}}$ and integrate in [0,t'] $\...
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1answer
18 views

How do you compute a complex exterior derivative?

The context is deriving cauchy riemann equations using green's/stoke's theorem. The function is the complex function $f(x,y)=u(x,y)+iv(x,y)$ with associated one form $u(x,y)dx+iv(x,y)dy$. Here is my ...
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0answers
54 views

Showing that if $\int_C f(z) \, dz=0$ for every circle $C$ in $\mathbb{C}$, then $f$ is holomorphic.

I understand that this is the difficult direction of Morera's proof, applied to disks, rather than triangles. However, the trick of defining $$F(z)=\int_{\gamma(t)}f(w)\,dw, \text{ with } \gamma(t)=...
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0answers
18 views

How to find integrals of the form $\int_{-r}^{+r}\frac{\gamma^{2}-1-y^{2}}{y-z\gamma} dy$

I'm trying to solve the above integral which will give a dispersion relation. Note: $\gamma$ and y are real but z is a complex variable.
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0answers
25 views

Integral problem with branch point from Physics

The question come from a Summation like this $${ \sum _{ { z=i\omega }_{ n } } { \frac { -\alpha E\pi }{ 4{ z }^{ 3 }\sqrt { -\alpha -z } } } }$$ I can use Cauchy theorem to transform it to a ...
3
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2answers
48 views

Closed contour integral: $\int_{\mathbb{c}}\frac{ z}{2z^{2}+1} dz$ where the contour is the unit circle

first and foremost please excuse my English. given $∫_c \frac{{z}}{2z^{2}+1}dz$ where the contour is the unit circle. so c = $e^{it}$ from 0 to $2\pi$. since the contour is the unit circle we can ...
6
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1answer
159 views

The complex function $\log(1+e^{iz})$

G.H. Hardy states the following: The function of the complex variable $z$ $$ e^{i p z} \log(1 \pm e^{iz})\frac{1}{z^{2} \pm \theta^{2}} = f(z), \ (-1 < p <1, \theta >0),$$ ...
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2answers
59 views

Using residues to compute complex integrals

Let $\phi:\mathbb{C}\backslash\{\pm i,\pm 2i\}\rightarrow\mathbb{C}$ with $\phi(z)=\frac{e^{iz}}{(z^2+1)^2(z^2+4)}$. How can I find $$\int_{-\infty}^\infty\frac{\cos x}{(x^2+1)^2(x^2+4)}dx$$ ...
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2answers
36 views

Prove the integral of $Re(z)$ around a simple closed curve is imaginary

If $θ$ is any simple closed curve in $\mathbb{C}$, prove that $\int_{θ}Re(z)dz$ is pure imaginary I have shown if $θ$ is the unit circle then the answer is $\frac{i}{2}$ but I'm struggling with a ...