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

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8
votes
2answers
490 views

$\int_{-\infty}^\infty \frac{\sin (t) \, dt}{t^4+1}$ must be zero and it isn't

I'm trying to evaluate the integral $$\int_{-\infty}^\infty \frac{\sin (t) \, dt}{t^4+1}$$ using residue and complex plane integration theory. Let $f(t):=\frac{\sin (t)}{t^4+1}$, $f(z):= \frac{\sin (...
3
votes
1answer
94 views

Integrate the following: $\int \frac {\log x}{\sqrt {1-x^2}}dx$

What I tried : $\int \frac {\log x}{\sqrt {1-x^2}} dx$ = $\log x \int \frac {1}{\sqrt {1-x^2}} dx$ - $\int \frac{1}{x} (\int \frac {1}{\sqrt {1-x^2}} dx) dx$ Now, $\int \frac {1}{\sqrt {1-x^2}} ...
0
votes
0answers
9 views

Divergence Theorem: Conditions for the boundary integration to vanish?

Consider the Divergence Theorem for example in two dimensions, in the upper right quadrant of Euclidean space: $$\int_0^\infty dx \int_0^\infty dy ~\vec\nabla\cdot\vec F=\oint_C ds~\vec n\cdot\vec F$$...
1
vote
1answer
80 views

Certain type of integrals $ \int_0^{\pi}d\theta\sin\theta \frac{1}{x+i\epsilon - \sqrt{y+z\cos\theta}}, $

I would like to do the following integral $$ \int_0^{\pi}d\theta\sin\theta \frac{1}{x+i\epsilon - \sqrt{y+z\cos\theta}}, $$ where the $i\epsilon$ has been added to avoid some possible divergencies. ...
0
votes
1answer
29 views

How to linearlize level curves at a saddle point

Let $f(x,y)$ be a real-valued function on a domain $D$ in $\mathbb{R}^2$, and let $(x_s, y_s)$ be a saddle point of $f(x,y)$ in $D$. That is to say, \begin{align} \frac{\partial f}{\partial x}(x_s, ...
0
votes
0answers
30 views

What kind of contour can be used for evaluating this integral?

I have the following integral: $\displaystyle\int\limits_{0}^\infty\frac{\arctan(\eta_2+x)}{(x+\eta_1)^2+\eta_3^2}\ d x$ where $\eta_1$, $\eta_2$ and $\eta_3$ are real numbers. If the integrand were ...
4
votes
2answers
39 views

Winding number of a polynomial

Consider $f(z) = c_n z^n + ... + c_1 z + c_0$, where $c_n\ne 0$. Let $C_R$ be the circle of radius $R$ centred at the origin, oriented counterclockwise. Prove that the winding number of $f\circ C_R =n ...
1
vote
1answer
37 views

Evaluating real integral by complex contour method

Please let me know where my mistake could be. I've verified the integral $$\int_{-\infty}^\infty \frac{dt}{(t^2+1)(t^2+4)}$$ to be equal to $\frac{\pi}{6}$ with a computer math system. However, I'm ...
0
votes
0answers
16 views

The Klein-Gordon equation Green's function.

How would I go about solving the following integral? \begin{equation} G_\text{ret}(x-x')=i\theta(t-t')\int\frac{d^3\mathbf{p} }{(2\pi)^32E}\left[e^{-iE(t-t')+i\mathbf{p}\cdot(\mathbf{r}-\mathbf{r}')}-...
6
votes
6answers
504 views

Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis

How do I compute $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$ What I am doing is take $$f(z)=\frac{(\log z)^2}{1+z^2}$$ and calculating $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log z)^...
1
vote
2answers
44 views

Evaluating $\int_0^{2\pi} e^{2i\theta} f(e^{i\theta})\,d\theta$ using Cauchy's Theorem

If $f(z)$ is analytic on the disk $|z| \leq 2$, evaluate $$\int_0^{2\pi} e^{2i\theta} f(e^{i\theta})\,d\theta.$$ This problem comes from Mathematical Methods in the Physical Sciences 2nd edition by ...
2
votes
0answers
32 views

Definite integral of absolute values over a simplex

I'm attempting to evaluate the following integral (note that $v_n=1-v_1-v_2-\dots -v_{n-1}$, and assume $n$ is even): $$ I_n=(n-1)! \int_{v_1=0}^1 \int_{v_2=0}^{1-v_1} \cdots \int_{v_{n-1}=0}^{1-v_1-\...
15
votes
4answers
503 views

Generalised Integral $I_n=\displaystyle \int_0^{\pi/2} \frac{x^n}{\sin ^n x} \ \mathrm{d}x, \quad n\in \mathbb{Z}^+.$

I have this integral, $$I_n=\displaystyle \int_0^{\pi/2} \frac{x^n}{\sin ^n x} \ \mathrm{d}x, \qquad n\in \mathbb{Z}^+.$$ We have the results $$ \begin{align} I_1 & = 2C, \\ I_2 &= \pi\log 2, ...
3
votes
2answers
72 views

Approaching a contour integral with singularities on each axis

How do I solve an integral like this using complex methods? $$ \int_{0}^{\infty} \frac{\ln(x)}{\left(x^2 + 2\right)\left(x^2 + 1\right)}dx.$$ I tried using two semi circles in the upper half plane ...
1
vote
2answers
46 views

Calculate $\int_{\gamma}zdz$ by using Cauchy's integral formula

How can I calculate $$\int_{\gamma}zdz~\text{with }\gamma:[0,1]\rightarrow\mathbb{C},t\mapsto te^{2\pi i t}$$ by using Cauchy's integral formula? The line $\gamma$ isn't even closed. Has anyone a hint?...
0
votes
0answers
28 views

Extend $g(z)$ holomorphically from $\mathrm{Re}(z) > 0 $ to $\mathrm{Re}(z) \geq 0$.

If I have holomorphic function $g(z)$ defined on $\mathrm{Re}(z) > 0$ and I extend in a holomorphic way to $\mathrm{Re}(z) \geq 0$, how do I get $g(0)$ purely in terms of stuff in the right half ...
2
votes
1answer
52 views

Solving an improper integral contour integral, calculated via Wolfram but in need of analytic derivation possibly

In my studies of dynamical systems I have just encountered this supposedly tough looking improper integral, which is (not really relevant for my predicament) the Melnikov function, with the integral ...
3
votes
1answer
63 views

A Contour Integral

I'm interested in computing the integral: $$ - \frac{1}{2 \pi} \int_{- \infty}^{\infty} dE \; \frac{e^{-iEt}}{E^2 - \omega^2 + i\epsilon}. $$ I have two small queries: How does one choose the ...
1
vote
0answers
40 views

Why does $\lim_{z \to c} f(z)(z-c)=0$ imply that $\int\limits_{\partial B}f(\zeta)d\zeta=0 $

I am using the book Theory of Complex Functions by Remmert. In chapter 7 there is a corollary that says for a function $f:D\to\mathbb{C}$, $D$ domain: If $f$ is bounded around $z$, then $\int\limits_{...
-2
votes
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 ...
0
votes
1answer
34 views

Solving a contour integral with Feynman prescription

My question relates to this question, where the integral $\displaystyle\int_{-\infty}^{\infty}\dfrac{e^{iax}}{x^2-b^2}dx=-\dfrac{\pi}{b}\sin(ab)$, where $a,b\gt 0$ is solved. Now, in many physics ...
0
votes
0answers
16 views

Contour integral of natural logarithm as part of inverse z-transform

I am trying to apply the inverse z-transform in order to derive a finite difference equation for the following function: $$ H(z) = \frac {1} {ln(z) + a} $$ I can get so far but I am stuck on the ...
0
votes
1answer
37 views

Application of complex analysis and contour integral in generating functions

Normally generating functions are tools of discrete mathematics and integrals deal with continuous structures. A book offered the following formula without much explanation and I'm not able to ...
1
vote
4answers
115 views

Find $\int_{0}^{\infty} \sin x^{2}\,dx$

This problem is from my textbook of complex analysis. I have attempted this as: let $$u=x^{2}$$ then $$dx=\frac{du}{2\sqrt{u}}$$ therefore $$\frac{1}{2}\int_{0}^{\infty} \frac{\sin u}{\sqrt{u}}\,du $$ ...
2
votes
4answers
114 views
4
votes
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 $\int_{0}^{2\pi}\frac{1}{(2+\cos\theta)^2}\,d\theta$. ...
5
votes
3answers
101 views

Compute $\int_{0}^{2\pi}\frac{\sin^2\theta}{5+3\cos\theta}\,d\theta$

I''m stuck in a exercise in complex analysis concerning integration of rational trigonometric functions. I have the solution but I don't understand a specific part. Here it goes: We want to find $\...
7
votes
1answer
117 views

Dumbbell Contour? $\int_0^1 \log(x)\log(1-x)dx$ via complex methods.

Having evaluated this integral via the power series and various approaches via special functions, I'm now curious if there is a direct way to compute this integral by taking a slit along $[0,1]$ and ...
0
votes
4answers
152 views

Very tricky complex integral, with poles on both sides of the real line,

I am trying to evaluate$$\int_{-\infty}^{\infty} \frac {x^2 -x^4}{1-x^6}\,dx,$$ which is an old exam problem. There is a special note on this problem that reads: Note: Your answer need not be a ...
-1
votes
3answers
34 views

Let C be the circle $|z|=1$. Evaluate $\int_{c}\frac{e^{2\pi z}}{(2z+1)^3}dz.$

Let C be the circle $|z|=1$. Evaluate $$\int_{c}\frac{e^{2\pi z}}{(2z+1)^3}dz.$$ Any idea, suggestion, advice or solution.
1
vote
1answer
52 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}}$...
1
vote
1answer
92 views

Help me to integrate $\int_{-\infty}^{\infty}{\sin^2{x}\over 1+x^2}dx=2\pi-{\pi^2\over 2}$?

$$I=\int_{-\infty}^{\infty}{\sin^2{x}\over 1+x^2}dx=2\pi-{\pi^2\over 2}$$ Apply residue theorem $$f(x)={\sin^2{x}\over 1+x^2}$$ $$\sin^2{x}={1-\cos(2x)\over 2}$$ $$I={1\over 2}\int_{-\infty}^{\...
3
votes
1answer
46 views

Is this a contour integral question?

I had this in my previous cats that I'm not sure whether it's really a complex analysis question, looks like a differential question with line integrals a bit $$\int_{(1,3)}^{(4,5)} (2y+x^2)\,dx + (...
5
votes
3answers
176 views

How do I prove $\int_{-\infty}^{\infty}{\cos(x+a)\over (x+b)^2+1}dx={\pi\over e}{\cos(a-b)}$?

How do I prove these? $$\int_{-\infty}^{\infty}{\sin(x+a)\over (x+b)^2+1}dx={\pi\over e}\color{blue}{\sin(a-b)}\tag1$$ $$\int_{-\infty}^{\infty}{\cos(x+a)\over (x+b)^2+1}dx={\pi\over e}\color{blue}{\...
1
vote
1answer
36 views

Help, why these are two different results of integral of $\sqrt{z}$ on unit circle depending the choice of Branch cut

everyone, I want test the effect of different choice of branch cut for contour, So I find a simple function, i.e. $\sqrt{z}$ with $z=re^{i\theta}$ on 1st Branch as $$I=\oint_{UnitCircle}{\sqrt{z}dz}$$ ...
0
votes
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 ...
0
votes
3answers
36 views

Simple Contour Integral

I have forgotten much of the complex analysis I once knew. How do I go about using the Cauchy Integral Formula / Residue Theorem to solve this contour integral? The region is the unit circle. $$\...
2
votes
3answers
106 views

Help, Where is wrong when I do same complex integration using two different contours

everyone! please give few hit. I want take the integral $$I=\int_{0}^{\infty}{\frac {dx}{ \sqrt{x}(1+{x}^{2})}} $$ by using the Residue Theorem. I choice two contours in complex plane with $z=r e^{i\...
2
votes
2answers
66 views

Solve this complex integral [closed]

Solve this complex integral $$\lim_{\varepsilon \rightarrow 0} \int_{-\infty}^{\infty} \frac{d\omega}{2\pi i}\frac{e^{-i\omega x}}{\omega + i\varepsilon}$$ Where $\varepsilon > 0$ and $x$ is real....
0
votes
1answer
58 views

Contour Integration, Riemann Zeta (-n)

I was reading Riemann's Zeta Function by H. Edwards, and could not understand the equation on the page 12. \begin{align*} \zeta(-n) &= \frac{\prod(n)}{2\pi i}\int_{+\infty}^{+\infty} \frac{(-x)^{-...
3
votes
1answer
72 views

How to evaluate this Fourier Transform $A\int_{-\infty}^{\infty} \frac{e^{ikx}}{(1+x^{2})^{\frac{\nu+1}{2}}}dx$

This is basically the Fourier transform of a Student´s T pdf. How do we compute it? $$A\int_{-\infty}^{\infty} \frac{e^{ikx}}{(1+x^{2})^{\frac{\nu+1}{2}}}dx$$ for $\nu$ any number greater than zero ...
0
votes
0answers
31 views

$\lim_{\rho\to0}\int_{\gamma_{\rho}}g(z)e^{iz}dz=-\pi i Res(f,a)$ with a pole $a\in\mathbb{R}$

Let $U$ be an open neighbourhood of $\overline{\mathbb{H}}=\{z\in\mathbb{C}:\Im(z)\ge0\}$ and $g:U\rightarrow\mathbb{C}$ meromorphic with a finite number of poles in $\mathbb{H}=\{z\in\mathbb{C}:\Im(z)...
1
vote
2answers
50 views

Closed form and limit of the integral of a rational function

While trying to answer this question, I wondered whether there could be a way to: (A) Find the closed form of the generalization of integrals $I$ and $J$, that is $$I_n=\int_{-\infty}^{+\infty}\frac{...
1
vote
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 ...
22
votes
5answers
1k views

Evaluating $\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$

I've been trying to evaluate the following integral from the 2011 Harvard PhD Qualifying Exam. For all $n\in\mathbb{N}^+$ in general: $$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$ ...
5
votes
3answers
1k views

Use the Residue Theorem to evaluate the integral:

$$\int_{0}^{∞} \frac{\sqrt{x}}{x^2+2x+5} dx$$ I'm thinking of using the "keyhole" contour, but I'm not sure how to proceed from there. Please help! Thanks!
4
votes
1answer
233 views

A variation of Ahmed's integral

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

Real integral using a contour integral

I am going to calculate $\int_{-\infty}^{\infty}\dfrac{x \sin \pi x}{x^2+2x+5}dx$ So I have to compute the following limit $\lim_{R \to \infty}\int_{C_1}\dfrac{z \sin \pi z}{z^2+2z+5}dz$ where $C_1$ ...
0
votes
1answer
33 views

Pdes -theoretical answer

Question.Let $Ω$ be a bounded Connected on $R^3$ with smooth boundary $\partial{Ω}$.Let $u$ be a harmonic function on $Ω$ with continuous derivatives on $Ω\cup \partial{Ω}$ prove that. $$\iint_V \ {\...