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

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Stokes' Theorem and Surfaces

Stokes' Theorem states the following: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot nd \textbf{S}\end{equation*} for a given C that is the ...
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1answer
21 views

Showing that $\tan(\pi z) = z$ has exactly three solutions in the strip $|\Re(z)| < 1$

We can't use Rouche's theorem here directly, so we have to apply the argument principle. If $f(z) = \tan(\pi z) - z$ , then $f'(z) = \pi \sec^2(\pi z) - 1$. Choose the rectangle $\Gamma$ with ...
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0answers
25 views

What advanced methods in contour integration are there?

It is well known how to evaluate a definite integral like $$ \int_{0}^\infty dx\, R(x), $$ where $R$ is a rational function, using contour integration around a semicircle or a keyhole. Most complex ...
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1answer
28 views

$\int_{|z| = 2} \frac{1}{f(z)(1+f(z))^2} dz$ where $f(z) = z^{1/2}$ with branch such that $\Re f(z) \geq 0$

As the title states, the definite integral in question is $$\int_{|z| = 2} \frac{1}{f(z)(1+f(z))^2} dz,$$ where $f(z) = z^{1/2}$ with branch cut such that $\Re f(z) \geq 0$, i.e., the cut is the ...
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3answers
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calculation of $\int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx $, where $n\in \mathbb{N}$

Calculation of $\displaystyle \int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx $, where $n\in \mathbb{N}$ $\bf{My\; Try}::$ Using $\displaystyle \cos (x) = \frac{e^{ix}+e^{-ix}}{2}$, we get ...
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3answers
78 views

Guidance or advice with $I=\int_0^{2\pi}\frac{1}{4+\cos t}dt$

Let $$ \begin{align} I=\int_0^{2\pi}\frac{1}{4+\cos t}dt \end{align} $$ I would like to evaluate this integral using cauchhy's Integral formula, I understand that I have to convert this into a form ...
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2answers
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Complex Numbers - Finding Roots

Hi there I was wondering if someone could help me? I am struggling to find the roots of the polynomial $z^4+2z+3=0$ It is not a quadratic so can't use the quadratic formula so am not quite sure ...
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2answers
54 views

How should I calculate $\displaystyle\int_{-\infty}^\infty\exp\left\{-\frac{1}{2}(x-it)^2\right\}dx$?

I've read that the residue theorem would help to calculate $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\exp\left\{-\frac{1}{2}(x-it)^2\right\}}_{=:f(x)}dx$$ Since $f$ is an entire function ...
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2answers
49 views

Is it possible to use complex logarithm to integrate $1/(z+i)$ along a path?

Evaluate the following on the path $\gamma_1$ with endpoints $[-1,1+i]$ $$ \begin{align} I_1=\frac{i}{2}\int_{\gamma_1} \frac{1}{z+i}dz -\frac{i}{2}\int_{\gamma_1}\frac{1}{z-i}dz \end{align} $$ Am I ...
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0answers
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Problem with calculating winding number in sum of curves

Let $$ \begin{align} \gamma= \gamma_1 +\gamma_2+\gamma_3,\\ \gamma_1(t)=e^{it}, t\in[0,2\pi] \\ \gamma_2(t)=-1+2e^{-2it}, t\in [0,2\pi]\\ \gamma_3(t)=1-i+e^{it},t\in [\frac{\pi}{2},\frac{9\pi}{2}] ...
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1answer
25 views

winding number of $\gamma$ and point exterior to $\gamma$

$$ \begin{align} n(\gamma,z_0)=\frac{1}{2\pi i}\int_\gamma\frac{1}{z-z_0}dz . \end{align} $$ Is it safe to say that $n(\gamma,z)=0,\forall z \in \mathbb{C}\backslash \gamma^*$
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1answer
25 views

Contour Integrals evaluation verification

$$ \begin{align} \gamma(t)=2cost + isint, t\in[0,2\pi] \end{align} $$ Could someone verify my thinking and my results for the following Integrals and if I have properly justified my thoughts: $$ ...
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1answer
45 views

Is this Integral Calculation correct? $\int_{|z|=1}(z^2+2z)^{-1}dz=\pi i$

$$ \begin{align} \int_{|z|=1}(z^2+2z)^{-1}dz=\int_{|z|=1}\frac{1}{z(z+2)}dz= \\ =\int_{|z|=1}\frac{1}{2z}dz+\int_{|z|=1}\frac{1}{2(z+2)}dz= \\ =\int_{|z|=1}\frac{1}{2z}dz+ 0= \\ ...
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3answers
64 views

Can $\frac1{z^2}$ be integrated on $|z+i|=\frac32$ using Cauchy's theorem?

$$ \begin{align} \int_{|z+i|=\frac{3}{2}}\frac{1}{z^2}dz=0 \end{align} $$ Is it safe to say the Integral is $0$ due to cauchy's Theorem? Does this apply for any $z_0$ that lies inside the circle ...
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2answers
261 views

Integral $ \int_0^\infty \frac{\ln(1+\sigma x)\ln(1+\omega x^2)}{x^3}dx$

Hello there I am trying to calculate $$ \int_0^\infty \frac{\ln(1+\sigma x)\ln(1+\omega x^2)}{x^3}dx $$ NOT using mathematica, matlab, etc. We are given that $\sigma, \omega$ are complex. Note, the ...
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1answer
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Problem in showing that contours $\gamma_2$ is equivalent to $ \gamma $

Let $\gamma_2(t)= e^{-it^2}, t\in[0,\sqrt{2\pi}]$ and $\gamma(t)=e^{2\pi it}, t\in[0,1]$ Show that $\gamma_2 \sim \gamma $. I think that for the latter to be true $\gamma_2$ should be ...
3
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1answer
45 views

Calculation of a Contour Integral

$$ \begin{align} \int_{|z|=1}(4-z^2)^{-1/2}dz \end{align} $$ The exercise hints at the usage of $e^{2Logz}$ , although any solution or methodology for such integrals would be welcome.
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1answer
29 views

Geometric explanation of a contour's image

$$\gamma(t)= t^2 + i\, t^4 , \quad t\in [ -1, 1]$$ What is the geometric explanation of the image of the above contour? Intuitively , I think it's ellipsoid-like, but I don't know how to put it in a ...
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1answer
33 views

Prove that $ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$

Let $C_r$ be the circle centered on $0$ with radius $r$ and $t\in \mathbb{R}$. How to show that $$ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$$
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1answer
81 views

Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$

I want to solve the following integrals analytically \begin{aligned} I_1 = & \int\limits_0^{\infty}\frac{e^{a/x^2}}{1-e^{b/x^2}}e^{-x^2}dx \\ I_2 = & ...
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1answer
30 views

Contour Integration Confusion

I am trying to find the value of $\displaystyle\int_0^\infty\frac{(\log x)^2}{1 + x^2}\,dx$ using contour integration. My approach: I have calculated the residue at z = $i$ and have shown that ...
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2answers
43 views

Contour expression explanation

$$ \begin{align} & \int_\gamma zdz \end{align} $$ $$\\ \gamma = [e,1]+[1,-1+\sqrt3] $$ contour $\gamma$ is defined as above and I can't understand it. Could someone please explain it to me and ...
4
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2answers
258 views

Evaluation of the integral $\int_0^\infty \left(\frac{\pi^2}{4}-x^2\right)^{-2}\cdot\frac{\pi^2}{4}\cos^2 x\,dx$

What are the steps to evaluate the following definite integral? (Answer provided) $$\int_0^\infty {{\pi^2\over 4}\cos^2x\over\left({\pi^2\over4}-x^2 \right)^2} dx={\pi\over 4}$$?
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3answers
211 views

Contour integration to compute $\int_0^\infty \frac{\sin ax}{e^{2\pi x}-1}\,\mathrm dx$

How to show: $$\int_{0}^{\infty}\frac{\sin ax}{e^{2\pi x}-1}dx=\frac{1}{4}\frac{e^{a}+1}{e^{a}-1}-\frac{1}{2a}$$ integrating $\dfrac{e^{aiz}}{e^{2\pi z}-1}$ round a contour formed by the rectangle ...
3
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2answers
278 views

Contour integration of $\int_{-\infty}^{\infty} \frac{1-b+x^2}{(1-b+x^2)^2 + 4bx^2}dx = \pi$

Given $0< b <1$, derive the equality: $\displaystyle \int_{-\infty}^{\infty} \frac{1-b+x^2}{(1-b+x^2)^2 + 4bx^2}dx = \pi$ by integrating the function $(1+z^2)^{-1}$ around the rectangle with ...
10
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4answers
293 views

Integrate $ \int_0^\infty \frac{ \ln^2(1+x)}{x^{3/2}} dx=8\pi \ln 2$

I am trying to evaluate this integral. $$ I=\int_0^\infty \frac{ \ln^2(1+x)}{x^{3/2}} dx=8\pi \ln 2 $$ Note $$ \ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n}, \ |x| < 1. $$ I was trying to do ...
2
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1answer
17 views

Estimate of line integral of O(x^n) function

Let $f$ be an analytic function in some sector in the complex plane behaving as $$f(z)=\mathcal O(z^n)$$ for some $n$ as $z\to\infty$. Can one prove in general that line integrals of $f$ (in this ...
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3answers
160 views

Scary contour integral, but is also an integral representation for $\Gamma$-function

This problem is supposed to be from an old Acta Mathematica volume I circa 1880's, and is attributed to Bourguet. By using a parabola with its focus on the origin as a contour, show that: ...
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integration with 4 branch point

I come across a problem of contour integration with 4 branch point. The problem came down to be equivalent to integrate $\sqrt{(x^2-1)^n(x^2-2)^m}$ over the imaginary axis. So, there are two branch ...
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4answers
255 views

Contour Integral: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$

I want to compute: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$ for natural $n>1$ using Residue Calculus. I am thinking of using some kind of a keyhole or bone contour that could go around the ...
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0answers
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$ \int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz $ on a closed curve of index $N$ with respect to the point $1$.

Let $\gamma$ be a closed curve in the right half plane that has index $N$ with respect to the point $1$. Find $$ \int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz $$ This is a problem from an old ...
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2answers
217 views

Improper integral involving $e^x$

How do you show that $$\int_{-\infty }^{+\infty }{\frac{{{x}^{2}}\, \text{d}x}{\left( \beta +{{\text{e}}^{x}} \right)\left( 1-{{\text{e}}^{-x}} \right)}}=\frac{\left( {{\pi }^{2}}+{{\ln }^{2}}\beta ...
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1answer
39 views

Writing the floor function as a contour integral

The function $f(z)=\frac{\pi}{\sin \pi z}$ has simple poles of residue 1 at the integers. Hence, by the residue theorem, I consider the interesting idea of drawing a (perhaps rectangular, for example) ...
2
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1answer
123 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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4answers
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Evaluating $\int_{0}^{\infty} \frac{x^{3}- \sin^{3}(x)}{x^{5}} \ dx $ using contour integration

EDIT: Instead of expressing the integral as the imaginary part of another integral, I instead expanded $\sin^{3}(x)$ in terms of complex exponentials and I don't run into problems anymore. ...
3
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3answers
402 views

Complex analysis: contour integration

Evaluate by contour integral: $$\int_0^1{ dx\over (x^2-x^3)^\frac 13}$$ Should I go for some kind of substitution so that the limit changes to $0$ to $\pi/2$?
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1answer
64 views

Complex Contour Integration - Complex Analysis

I'm just practising for my upcoming exam, and I've come across a question I'm having a bit of difficulty with. I've been asked to show the following; $$\int_{0}^{\infty} \frac{dz}{\cosh(z)} = ...
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1answer
133 views

Calculating Riemann zeta function of a complex number given the complex contour integral

Can you please demonstrate how one would calculate the Riemann Zeta function of any complex number, given that the Riemann Zeta function is equal to the following (shown in ...
4
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1answer
316 views

Integrate $ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx $

Hi I am trying to come up with a closed form expression for $$ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx,\quad p\geq 0. $$ I am interested in this general case in terms of p. For small p, we can ...
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2answers
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Tricky contour integral resulting from the integration of $\sin ax / (x^2+b^2)$ over the positive halfline

I am trying to evaluate the definite integral $$\int_0^\infty \frac{\sin ax\ dx}{x^2+b^2}$$ where $a,b>0$. This is a problem on an assignment for a class in complex variables. I understand the ...
3
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1answer
330 views

Contour Integration: $\int_0^\infty\frac{1}{x^a(1-x)}\,dx$ for $0<a<1$.

I've been trying to calculate $$\int_0^\infty\frac{1}{x^a(1-x)}\,dx\quad\text{with }0<a<1.$$I haven't had much luck. I tried taking the branch cut with of the positive reals and estimating that ...
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0answers
49 views

Proper way to set up “Pac-Man” contour integral

I'm trying to evaluate $$ \int_0^\infty \frac{x^a}{1+x} \: dx, \: -1<a<0 $$ using contour integrals. Actually, I have found the correct answer using a "Pac-Man" contour and residues. My only ...
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3answers
196 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, ...
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0answers
59 views

What does this complex contour integral represent?

How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results." The ...
10
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3answers
419 views

Integrating $\int_0^\infty \frac{\log x}{(1+x)^3}\,dx$ using residues

I am trying to use residues to compute $$\int_0^\infty\frac{\log x}{(1+x)^3}\,dx.$$My first attempt involved trying to take a circular contour with the branch cut being the positive real axis, but ...
2
votes
1answer
56 views

Calculating $\int_0^\pi \sin^2t\;dt$ using the residue theorem

I want to use the residue theorem to calculate $$I:=\int_0^\pi \sin^2t\;dt$$ Since $\sin^2$ is an even function, we've got $$I=\frac{1}{2}\int_0^{2\pi}\sin^2t\;dt$$ The solution of this exercise ...
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0answers
46 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 $a$ ...
1
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2answers
37 views

Show elementarily that $\lim_{R\to\infty}\int_{\Gamma_1} \frac{e^{iz}}{z} = 0$

Context: I am trying to show that $\int_0^\infty x^{-1}\sin x dx = \frac{\pi}{2}$ using complex analysis, by first integrating $\oint_{\Gamma} z^{-1}e^{iz}$, where $\Gamma$ is a closed contour ...
1
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1answer
59 views

Contour Integration Limitations?

So, I'm trying to evaluate the following integral by complex contour integration ONLY: $$\int_0^\infty{\frac{x^\alpha}{x(x+1)}} dx$$ where alpha is real and not an integer. Obviously, we need to use ...
2
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1answer
62 views

Computing a very messy contour integral

I'm hoping that someone might be able to help me with the following problem. I'll walk through my current work and indicate where I'm stuck. Compute the contour integral: $$ \oint_{|z-1-i| = ...