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

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28 views

Find the Fourier transform of $u(x) = \frac{x \cos(2x)}{(1+x^2)^2}$

Find the Fourier transform of $$u(x) = \frac{x \cos(2x)}{(1+x^2)^2}$$ My work Okay so we want $$\int_\mathbb R \frac{e^{-ixt}x\cos(2x)}{(1+x^2)^2}dx$$ Of course we want to apply the residue ...
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1answer
35 views

Intuition/Understanding of “Infinite” Countour Integrals

I'm trying to clarify some thoughts on contour integration. If I have an integral $\int_{c-i\infty}^{c+i\infty} f(z) dz$, where $f(z)$ has finitely many poles in the complex plane...is this ...
3
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1answer
77 views

Evaluating $\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$

Find $$\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$$ We see that the only poles are at $x=\pm bi$. Integrating over the semicircular contour implies that it is equal to $2\pi i*Res_{(+bi)}$ ...
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2answers
43 views

Integral of $((x^2+1)((x-1)^2+1))^{-1}$

Find $$\int_{-\infty}^{\infty}\frac{1}{(x^2+1)(2-2x+x^2)}dx$$ So I am going to integrate this using a semicircular contour. Is it safe to say that on the curved part, the integral vanishes? because ...
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0answers
18 views

Contour Integral of sin(z)/(z^2-z)

Find the integral $\int_{\lambda}\frac{\sin(z)}{z(z-1)}$ where $\lambda(t) = 10e^{it},t\in[0,2\pi]$ We notice that there are poles at $z = 0$ and $z=1$. So we can use residue theorem but I am ...
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0answers
36 views

Geometric interpretation of Cauchy-Goursat Theorem?

This theorem seems almost magical. The algebraic derivation doesn't really provide any insight into why it works. So could someone give me a geometric interpretation of it? This: Geometrical ...
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1answer
44 views

Path integral in the complex plane

Evaluate $\int_Tz\,\mathrm dz$ and $\int_T\overline z\,\mathrm dz$ where $T$ is the triangle with vertices $0,1,-i$ oriented clockwise. I am trying to solve this question, but I'm unsure how to ...
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22 views

How to integrate $\int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}$?

In Lancaster & Blundell's QFT book they show that \begin{equation}A:= \int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}\end{equation} returns a nonzero value for $x$, $t$ and $m$ ...
2
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1answer
47 views

using contour integrals

Let $ \gamma (t)= e^{it} $ where $0 \leq t \leq 2 \pi.$ Evaluate $\int_{\gamma}$ $e^{z}$ $dz$ . Use the result to show that $\int_{0}^{2\pi} e^{\cos(t)}\cos(t+ \sin(t)) dt = 0$. I have worked out ...
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19 views

integration, anti- derivative, complex [duplicate]

Let $\gamma(w,R)$ denote the circular contour $t\mapsto w+Re^{it}$ where $0\lt t\lt2\pi$. Evaluate $$\int_\gamma\dfrac1{1+z^2}dz$$ when $\gamma$ is: ...
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0answers
24 views

Complex integration, limits, arctan

$\left.\frac12i\;\text{Log}\frac{1-(-i+e^{it})}{1+(-1+e^{it})}\right|_0^{2\pi}=\frac12i\left(\log\left|\frac{i}{i}\right|+i\arg 1-\log|1|-i\arg1+2\pi ik\right)$ could someone explain how this is ...
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4answers
84 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 ...
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6answers
175 views

Evaluate $\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx$ (solution verification)

I tried to find the integral $$I=\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx \tag1$$ I substituted $x=t^2, 2tdt=dx$ and chose $\log x$ and $\sqrt{x}$ to be principal values. We have ...
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0answers
19 views

Integral of Bessel function with Gaussian over a quadratic

I need help with the following integral: $$ \int_{0}^{\infty} \frac{J_0(ax)xe^{-bx^2}}{1-cx^2}dx $$ Where $ J_0(x) $ is a Bessel function of the first kind (of zero order). I've looked up a few ...
4
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1answer
71 views

Evaluating $\int_0^{\infty} \frac{\sqrt{x}}{x^2+2x+5} dx$ using complex analysis

how do I compute $$\int_0^{\infty} \frac{\sqrt{x}}{x^2+2x+5} dx$$ with complex analysis? I feel like im calculating the residue wrong and I cant get to the answer correctly. I tried to branch cut ...
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0answers
25 views

circular contour integral with complex numbers [closed]

Let gamma(w,R) denote the circular contour t maps to w + Re^it where 0 < t < 2Pi. Evaluate the integral of 1/1+z^2 when gamma is gamma(i; 1)
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1answer
85 views

Circular contour integration.

solving one of the 5 options would be much appreciated as this will give me an idea on how to solve the rest. Let $\gamma(w,R)$ denote the circular contour $t\mapsto w+Re^{it}$ where $0\lt ...
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0answers
29 views

Contour integration and the square root branch cut

Consider the following equation $$ \int_0^\infty f(\sqrt{x(x-a)}) dx $$ For $a>0$ real and some analytic function $f(z)$ which dies off sufficiently fast for $\Re[z]>0$ and $\Im[z]>0$ so ...
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2answers
95 views

Integral $\int_0^{2\pi}\frac{dx}{2+\cos{x}}$

How do I integrate this? $$\int_0^{2\pi}\frac{dx}{2+\cos{x}}, x\in\mathbb{R}$$ I know the substitution method from real analysis, $t=\tan{\frac{x}{2}}$, but since this problem is in a set of ...
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3answers
81 views

Integral with branch cut ( Problem while calculating residue)

While calculating this integral $\int_{-1}^{1}\frac{dx}{\sqrt{1-x^2}(1+x^2)}$ , I am really struggling to calculate the residue at (-i), I am getting the value of residue as $\frac{-1}{2\sqrt{2}i}$, ...
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1answer
44 views

Complex analysis $\int^{2\pi}_0 \frac{d \theta}{(2-\sin \theta)^2}$

how do I compute $$\int^{2\pi}_0 \frac{d \theta}{(2-\sin \theta)^2}$$ I tried substituting $z=e^{i\theta}$ but it just got very messy..
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1answer
21 views

Using Cauchy Integral Formula (Excersice from BCA-Marsden)

I'm trying to evaluate the following integral: $ \int_{\gamma} \frac{z^2 -1}{z^2 +1}dz$ where $\gamma$ is the radius 2 circle centered at $(0,0)$. This function is holomorphic in $\mathbb{C}$\ ...
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3answers
75 views

Geometrical Interpretation of the Cauchy-Goursat Theorem?

The Cauchy-Goursat theorem is really non-intuitive and is very astounding. Can someone geometrically explain to me why its true? I'm specifically talking about this version of the theorem: For ...
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0answers
18 views

Winding numbers are continuous: The proof was too easy

There's a question in my complex analysis book: Let $G$ be a region and let $\gamma_0$ and $\gamma_1$ be two closed smooth curves in $G$. Suppose $\gamma_0\sim\gamma_1$ and $\Gamma$ is a homotopy ...
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0answers
11 views

How to justify this complex substitution using contour integration

I tried to solve the laplace transform of $\cos(at)$ and $\sin(at)$ using Euler's formula. That is, $$\int^\infty_0e^{-(s-ia)t}dt\color{red}{=}\frac{1}{s-ia}\int^\infty_0e^{-t}dt=\frac{1}{s-ia}$$ ...
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2answers
121 views

Why is the pole generally outside the contour loop when its outside the contour loop in 2D?

The following contour integral is path dependent with the following results \begin{align} \oint_C\dfrac{dz}{z} = \begin{Bmatrix} 2\pi i && \text{when $z=0$ is inside C} \\ 0 && ...
3
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1answer
36 views

Determining the value of an integral using complex methods

I need to find the value of the following integral using complex analysis: $$\int_{-\infty}^{\infty}\frac{\sin(k_1\ x)+\sin(k_2\ x)}{x^2-a^2}\ dx$$ where $k_1, k_2, a$ are real coefficients. The ...
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0answers
49 views

Saddle Points in Complex Plane of trig function

I am trying to analytically Fourier transform a set of functions that have the form $f(k) e^{-\rho~\psi(k)}$ where the general $f(k)$ is some linear combination of trig functions without poles, ...
9
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1answer
111 views

An integral $\int^\infty_0\frac{\tanh{x}}{x(1-2\cosh{2x})^2}{\rm d}x$

I would like to enquire about the possible methods of computing the following integral $$\color{blue}{\int^\infty_0\frac{\tanh{x}}{x(1-2\cosh{2x})^2}{\rm d}x=\ ?}$$ A possible way I see of doing this ...
4
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1answer
67 views

Contour integral using residue

Assume that $f(z) \in \{\sqrt{2z^2 + 1}\}$ $,f(0) = 1$ We have a cut: $\gamma = \{|z| = \frac{1}{\sqrt2}, Re(z) \geqslant 0 \}$ $\oint\limits_{|z|=1} \frac{z dz}{(z+2)(f(z) + 3)}$ I found ...
3
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3answers
26 views

Finding the $n$th Taylor coefficient of $g(z)=\frac{z}{(z-b)^2}$ centered at $a$ (where $a=2-\sqrt{3}$ and $b=2+\sqrt{3}$?

I've introduced $a$ and $b$ in order to simplify the notation : $a=2-\sqrt{3}$ and $b=2+\sqrt{3}$. I'm trying to compute the Taylor Series for $g(z)=\frac{z}{(z-b)^2}$ centered at $a$. I denote the ...
2
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2answers
41 views

Integral $\int_{\pi/2+\delta}^{3\pi/2-\delta} x^{R \cos \varphi} d \varphi$ bounded

This is probably a silly question, or maybe I am missing a very simple slick trick, but I am trying to see how the following integral is bounded in terms of $\delta$: \begin{equation} ...
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1answer
49 views

Finding closed paths $\gamma(a,r)$ such $\int_{\gamma(a,r)} \frac{5z^2-8}{z^3-2z^2}$ takes value $-2i\pi$ or $18i\pi$?

Finding closed paths $\gamma(a,r)$ such $\displaystyle \int_{\gamma(a,r)} \frac{5z^2-8}{z^3-2z^2}$ takes value $-2i\pi$ or $18i\pi$? From this question it is already know that $\displaystyle ...
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1answer
49 views

Prove $\int_0^\pi\sin^{2n}t dt$ without using Residue Theorem

How may one prove something similar as in here but from $0$ to $\pi$ and without using the Residue Theorem? I was told to consider the contour integral $$\int_{|z|=1}(z-\frac{1}{z})^{2n}dz/z $$ and ...
2
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1answer
50 views

Complex path integral $\int_{\gamma(0,1)} \frac{5z^²-8}{z^3-2z^2}dz$?

I'm trying to practice by computing $$\int_{\gamma(0,1)} \frac{5z^²-8}{z^3-2z^2}dz$$ I've first tried to expand in partial fractions and then set $z=e^{i\theta}$, but while it first simplifies quite ...
2
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1answer
29 views

How to calculate the residue of the fourier transform?

I have been struggling calculating the Fourier transform of $f(x)=\frac{x}{(x^2+1)^2}$. I tried to calculate $f(t)=\int\frac{x}{(x^2+1)^2}e^{-ixt}\,dx$ directly by integration by parts, but it is not ...
2
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2answers
56 views

Inverse Laplace Transform of e$^{-c \sqrt{s}}/(\sqrt{s}(a - s))$

I am trying to find the Inverse Laplace of the following function: $$ F(s) = \frac{\mathrm{e}^{-x b \sqrt{s}}}{ b (a - s)\sqrt{s}} $$ I really don't know where to start on this one as I have only ...
4
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0answers
59 views

Contour integration of $\int_{-\infty}^{\infty}\frac {\sin^3 x}{x^3} dx$: where are the singularities?

I have just begun to study complex analysis and I'm trying to calculate $$ \int_{- \infty}^{\infty} \frac {\sin^3 x}{x^3} dx $$ with the "help" of an exercisebook. I have followed these steps: ...
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5answers
62 views

Evaluating a contour integral where C is a square

I've been working problems all day so maybe I'm just confusing myself but in oder to do this, I have to the take the integral along each contour $C_1-C_4$ My issue is how to convert to parametric ...
2
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1answer
59 views

Evaluate the integral of the given contour

I'm being asked to evaluate $\int \frac{1}{z^3(z^2+1)}dz$, where C is the circle $\lvert z-1 \rvert=\frac32$ I started by determining the zeroes, which are $0, -i, \,i$ Then I applied the Cauchy ...
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1answer
29 views

Complex definite integral $\int_{0}^{\pi}\frac{ire^{it}}{2-2ire^{it}}dt$

I am trying to evaluate the integral $$\int\limits_{0}^{\pi}\dfrac{ire^{it}}{2-2ire^{it}}dt$$ but I don't know how to proceed.
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1answer
46 views

Evaluating contour integrals along given C's

Ok, so I have the following problem that I am working on. It says to evaluate $$\int \frac{z}{(z-1)(z-2)}dz$$ where C are given by \begin{align} a)& \ \ C:\lvert z \rvert=\frac12\\ b)& \ \ ...
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1answer
48 views

Find the value of the integral on the contour C

Ok, so I'm trying to figure out this problem. It asks to find the value of the contour integral $\dfrac{e^z}{z^2(z-\pi i)}$ on the contour $C$ shown in the following figure I believe that in order ...
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2answers
63 views

Looking for intuïtive explanation why contour integral of $\frac{dz}{z} $equals $2\pi i$ in complex analysis

$$\oint \frac{dz}z = 2\pi i$$ I've seen the derivation of it using the parametrisation. Since this result is used all the time in my complex analysis course, i'd like to understand this ...
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1answer
22 views

Contour integral of convergent power series

Given that $\frac{e^z}{z^k} = z^{-k} + z^{1-k} + \frac{z^{2-k}}{2!} + \frac{z^{3-k}}{3!} + ...$ converges uniformly on any set $\{z \in C: r \leq |z| \leq Z\}$ (where $0 < r < R$), show that for ...
5
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1answer
60 views

using complex or real analysis solve $\int_{0}^{\pi/2}\frac{x^m}{\sin x}dx$

closed form for $$\int_{0}^{\frac{\pi}{2}}\frac{x^m}{\sin x}\ dx$$ I slove it for some m but in general i failed. I tried by part , by substitution,by using $\sin x =\frac{e^{ix}-e^{-ix}}{2i}$ . I ...
0
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0answers
24 views

How to show that integration contours are related?

I have one geometry below in which the integration contours are shown with red and blue line. How I can show that the contour in blue line i.e (B to C) is with in the integration contour in red ...
1
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1answer
63 views

Inequality complex integral with $|f|\le 1$.

Let $f:\mathbb C\longrightarrow \mathbb R$ be a continuous function such that $\,\lvert\, f(z)\rvert\le 1$ for all $z\in S^1\subset \mathbb C$. Prove that $$\left| \int_{\lvert z\rvert=1} ...
-2
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0answers
19 views

Contour Integral in Higher Dimensions

As far as I can tell, when one contour integrates, he does so around a closed contour of dimension one. (Is this right?) Can one contour integrate a function using a contour of higher dimension ie a ...
0
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0answers
30 views

Integration contour relationship.

We have the two integration contours as shown below, How we can prove that the integration contour B is the subset of the integration contour A? Also note that the figures does not represent the ...