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

learn more… | top users | synonyms

2
votes
2answers
82 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 ...
1
vote
0answers
26 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 ...
1
vote
1answer
281 views

Integrating using the residue theorem

How would I calculate the following integral? $$\int_{-\infty}^\infty \frac{1}{(x^2 + 1)(x^4+4)^2} dx$$ Part (a) says define Laurent's theorem for the Laurent series expansion and give the definition ...
0
votes
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 ...
5
votes
1answer
172 views

Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration $$ \int_{0}^{1}{\ln\left(\,x\,\right)\over \,\sqrt{\vphantom{\large A}\,1 - x^{2}\,}}\,{\rm d}x $$ I am not sure as to how to work with the branch ...
3
votes
1answer
76 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 $2\pi i*Res_{(+bi)}$ ...
1
vote
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 ...
1
vote
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 ...
1
vote
1answer
43 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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
3answers
228 views

Cauchy's argument principle, trouble working simple contour integral

I'm trying to teach myself Cauchy's argument principle by doing a simple example, but apparently I'm missing something, because every time I try to do the contour integral I get 0. Cauchy's argument ...
8
votes
6answers
174 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 ...
2
votes
0answers
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 $m$ ...
3
votes
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 ...
4
votes
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 ...
2
votes
0answers
61 views

Integrating $\int_0^1 dx\,\ln(x-a)/(x-b)$ paying attention to cuts.

I am trying to compute the following integral, for complex $a$ and $b$ $$ \int^1_0 dx \frac{\ln(x-a)}{x-b} $$ by turning it into something in terms of dilogarithms. But for certain values of $a$ and ...
2
votes
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 ...
5
votes
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 ...
0
votes
0answers
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: ...
1
vote
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 ...
12
votes
4answers
493 views

Closed form of $I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx$

Does the integral below have a closed-form: $$I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx,$$ where $\tan^{-1} (\cdot)$ is inverse tangent function. ...
4
votes
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..
4
votes
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 ...
1
vote
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 ...
3
votes
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 ...
0
votes
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)
1
vote
1answer
133 views

inverse Laplace transfor by using maple or matlab

I want to use inverse Laplace transform to F function by using maple or matlab. However I cannot get any answer. I know the answer from table but I want to use one of softwares. from table: ...
1
vote
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 $\frac{-1}{2\sqrt{2}i}$, ...
1
vote
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 ...
0
votes
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 $\mathbb{C}$\ ...
1
vote
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 ...
1
vote
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, $$\int^\infty_0e^{-(s-ia)t}dt\color{red}{=}\frac{1}{s-ia}\int^\infty_0e^{-t}dt=\frac{1}{s-ia}$$ ...
1
vote
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 ...
4
votes
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 && ...
2
votes
1answer
58 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 ...
1
vote
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)& \ \ ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
0answers
31 views

How to relate two integration contour?

How one can relate two integration contour? For example if I have an integration contour like $\int_{-a}^{a}f(x)dx$ here let say a=infinity. How I can say that the integral $\int_{2}^{3}f(x)dx$ is a ...
5
votes
6answers
136 views

Find $\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$

How can we find the integral: $$\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$$ I tried to find and got it to be $\cfrac{\pi}{\sqrt2}$. Am I correct? Please help me with an ...
1
vote
0answers
47 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
votes
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 ...
3
votes
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 ...
0
votes
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 ...
10
votes
2answers
275 views

Evaluating $\int_0^\infty \frac{\cos(ax)-e^{-ax}}{x \left(x^4+b^4 \right)}dx$

How can we evaluate $$\int_0^\infty \frac{\cos(ax)-e^{-ax}}{x \left(x^4+b^4\right)}dx \quad a,b>0$$ using Complex Analysis? This problem was given in a Complex Analysis book which I was reading. ...
2
votes
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} ...
9
votes
4answers
196 views

Evaluate $\int_0^\infty \frac{(\ln x)^2}{x^2+4} \ dx$ using complex analysis.

Evaluate $\int_0^\infty \frac{(\ln x)^2}{x^2+4} \ dx$. This is the last question in our review for complex analysis. Hints were available upon request, but being the student I am, I waited until the ...
1
vote
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 ...