# Tagged Questions

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

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### Contour integral with trigonometric functions

Does the following integral, where $n$ is an integer, have an analytic solution? $I_1 = \int_{0}^{2 \pi} \frac{\sin(n x) \sin(x) \cos(x)}{\cos(x)^2 + a} \mathrm{d}x$ I tried writing $\sin(n x)$ as ...
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### Evaluate the integral $\int_0^\infty \frac {x^{1/2} dx}{x^2 + 1}$ using method of residues

I am trying to evaluate the integral $\int_0^\infty \frac {x^{1/2} dx}{x^2 + 1}$ using method of residues. I can solve this very easily without the $x^{1/2}$ on top, but I do not know what to do when ...
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### Complex Integral Over an Ellipse

Suppose we had a function defined as $$S(\zeta) = \zeta - \sqrt{\zeta^2 - c^2}$$ and we wish to evaluate $$\oint_\gamma \frac{S(\zeta)}{z-\zeta}\, d\zeta,$$ where $\gamma$ is the (positively ...
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### 2 ways for $\int_{|z|=r} x^2dz$

As the title says, I have to compute $$\int_{|z|=r} x^2dz$$ for the circle traversed anti-clockwise in 2 different ways. If I use a parametrisation I quickly get to 0 - which might be wrong though ...
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### An inequality from complex contour integral on page 81 of Stein and Shakarchi's Complex Analysis

On page 81 of Stein and Shakarchi's Complex Analysis, there is an inequality, \left\lvert \int_{A_{R}} f \right\rvert \leq \int_{0}^{2\pi}\left\lvert\frac{e^{a(R+it)}}{1+e^{R+it}} ...
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I have been asked to compute the integral $$\int_{|z|=r}x^2dz$$ for the circle traversed anticlockwise, without parametrisation, by observing that $$x = ... 1answer 34 views ### Almost Gamma function with imaginary exponential: substitution/contour trick In a physics course, this integral showed up: \int_0^\infty dp\, p^2\, e^{ibp}\; , \quad b \in \mathbb{R} My prof proceeded with the seemingly insane substitution x=-ib to yield \Gamma(3)x^{-3} ... 1answer 53 views ### Evaluate \int_{-1}^{1}\mathop{dx}\frac{1}{(x^2+a^2)^k}? My original question was what is the condition for the following integral to converge:$$\int_{-1}^{1}\mathop{dx}\left[\frac{1}{x^{2k}}-\frac{1}{(x^2+a^2)^k}\right].$$I know k>0. By motivation ... 0answers 91 views ### On \sum a^n \tan(n\theta) It is well known that$$\sum_{n=0}^{\infty} a^n \cos(n\theta) = \frac{1-a\cos(\theta)}{1-2a\cos(\theta)+a^2}\sum_{n=0}^{\infty} a^n \sin(n\theta) = \frac{a\sin(\theta)}{1-2a\cos(\theta)+a^2}$$... 2answers 73 views ### Why doesn't this contour work for complex integration? I am asked to integrate:$$\int_0^\infty \frac{1}{(1+x)x^{1/3}} dx$$Complexification of this integral leads to:$$f(z)=\frac{1}{(1+z)z^{1/3}}$$singularities: z=0 and z=-1. So I thought, let's ... 0answers 28 views ### Jordan's lemma for different exponential functions/integration directions Okay, lets say we have this complex integral \int_{-\infty}^{\infty} \frac{e^{-iaz}}{(z+i)(z-i)} where a>0. If I were to specify a semicircular contour in the lower half of the real line going ... 3answers 77 views ### Conversion into contour integral and poles Say I have this integral \int_{-\infty}^\infty \frac{x^2}{x^6+1} dx . Now I know that it has six poles according to this denominator which are the six roots for -1. The question is after I split ... 1answer 40 views ### How to show the existence of an entire function I have been working on this problem for quite sometime. For part (i), I obtained the Taylor series for 4\sin(z) - \sin(4z). At z = -\pi, the Taylor series is: 4\sum_{n=0}^{n} \frac{(z + ... 1answer 17 views ### Evaluating a contour-integral. Consider the ellipse C given by x^2 + y^2/4 = 1. How to evaluate$$\int_C x^2 \, \nu(d(x,y))$$where \nu is the Lebesgue length measure on C? I am not sure if this can be computed like a ... 2answers 66 views ### What is the use of Dirichlet Integral? [closed] How can I find the value of$$\large\int_0^\infty\left(\dfrac{\sin x}x\right)^5dx$$using Contour Integrals? I attempted it using Integration by Parts and got the an got the answer. I have studied ... 0answers 85 views ### Contour Integral of Square root Function. Branch Cuts I am doing a physics problem and have come across a contour integral that I just don't know how to solve. I do not have the complex analysis background and I am wondering if anyone can explain how to ... 0answers 44 views ### Can I split this integral to a sum over three contours? I have the following integral$$ Z = \frac{1}{2\pi i} \int dx \, \frac{1}{(x-a_1)(x-a_2)(x-a_3)}\times \frac{1}{(x+\epsilon - a_1)(x + \epsilon - a_2)(x+ \epsilon - a_3)} $$and this integral has ... 0answers 60 views ### Find the different values of this integral when all paths of integration are possible This is the question.. I only know how to do the question from infinity to infinity.. enter image description here Find the different values of this integral when all paths of integration are ... 1answer 76 views ### How do I evaluate the following Integral The integral is$$\int_{0}^{\infty}dx\frac{1}{\sqrt{1+a(1+x^2)^m+b(1+x^2)^{m-2}x^2}},$$where m, a and b are real numbers such that the integral is definitely convergent. Any ideas on how to solve ... 2answers 85 views ### Why should I take this Contour for I = \int_0^{\infty} \frac{dx}{1+x^3} ? (Analytic Continuation) When discussing analytic continuation, my lecturer used the following example,$$ I = \int_0^{\infty} \frac{dx}{1+x^3} $$I have in my notes that the contour was taken as below. I must admit I was ... 0answers 35 views ### Is this integral automatically zero? If I integrate \int e^{iz}\,dz for z complex, along the positive real line, then is the imaginary part of the integral i\int \sin(x)\,dx automatically equal to zero (integration only along the ... 0answers 22 views ### Inverse Laplace transform of the form F(s) = \frac{s^{m}}{(1+a \cdot s)^{n}(1+b \cdot s)^{h}} I am trying to solve the inverse Laplace transform of the form $$F(s) = \frac{s^{m}}{(1+a \cdot s)^{n}(1+b \cdot s)^{h}}$$ where, a and b are known constants, m, n, ... 2answers 107 views ### How do I evaluate the following integral \int_{-\infty}^\infty\mathop{dx} \frac{x^n}{(x^2+a^2)^m}? I am interested in the following integral$$\int_{-\infty}^\infty\mathop{dx} \frac{x^n}{(x^2+a^2)^m},$$given that m>n/2 (this is just what I wrote so that the integral converges. If this is not ... 1answer 73 views ### Use contour integration to compute the Fourier transform, The problem statement is: Use contour integration to determine the Fourier transform, \large \hat f(ξ)=∫_{-\infty}^{\infty}f(x)e^{−iξx}dx, of \large f(x)=\frac{1}{2−2x−x^2}. Some issues that I ... 0answers 33 views ### In P.V. contour integration in complex analysis, using a wedge vs. a keyhole contour When is it clearly better -- perhaps even necessary -- to use a keyhole contour, instead of a wedge contour? The wedge contour minimizes computation of residues, as we can choose it so that it ... 0answers 31 views ### show that \int_{C_R} \frac{z e^{iz}}{1+z^2}dz tends to zero Show that \int_{C_R} \frac{z e^{iz}}{1+z^2}dz where C_R is the half circle in the upper half plane with radius R tends to 0 as R goes to infinity. My professor showed me something with ... 2answers 73 views ### Compute \int _0^\infty\frac{x \sin x}{1+x^2}dx with the residue theorem Compute \int _0^\infty\frac{x \sin x}{1+x^2}dx with the residue theorem Ok so I have done a couple of these but I'm stuck on this one. I want to use$$ \int_0^\infty \frac{ze^{iz}}{1+z^2}dz $$... 0answers 25 views ### On the right half-plane, what is an upper bound for \frac{1}{\log(z+2)}? I am trying to estimate some factors in my integrand in complex integration, and I think the upper bound for \frac {1}{log(z+2)} on the semicircle in the right half plane is just \frac ... 2answers 65 views ### Contour integral of \log(z)/(z+a)^2 around z=0 My question is primarily conceptual: Consider a function f(z) which has a branch cut from z=0 to z=\infty along the positive Re(z) axis. If I wish to integrate it along a small, clockwise circle ... 2answers 134 views ### Complex integration using singularities I'm working on Ablowitz and Fokas' Complex Variables. On section 3.5 on singularities, problem 2 reads: Evaluate the integral of f(z) over the unit circle centered at the origin: a) f(z)=z/(z^2 + ... 0answers 26 views ### The use of Cauchy Goursat Theorem for a contour I have been attempting this question for some time. My reason is as follow: The contour C (−i, −2 − i, −2 −4i, −4i) is a rectangle on the left of the Imaginary axis (vertical axis). For any z ... 1answer 47 views ### Is a binomial expansion of the denominator valid here? I am trying to prove this integral tends towards zero (a variation of the Hankel contour problem). I am not sure if my approximation is valid here, My integral round a circular pole at the origin is, ... 2answers 92 views ### Evaluating the improper integral \int_{0}^{\infty} \frac{x^3}{e^{x}-1} dx [duplicate] I read somewhere that$$ \int_{0}^{\infty} \frac{x^3}{e^{x} - 1} = \frac{\pi^4}{15}$$Does anyone see a way to prove this? My first idea was doing a contour integration and use the residue theorem, ... 2answers 85 views ### Cauchy Integral Formula (When f(z) is not analytic everywhere inside C) enter image description here For the question in the photo, I have thought about it for a long time. My idea is as follow: C (|z| = 2) is a positively oriented simple connected contour. The point ... 1answer 55 views ### Inverse Laplace transform of \frac{\exp(\frac{\lambda s}{1 - 2s})}{(1 - 2s)^{k/2}} (MGF of noncentral chi-squared distribution) I am trying to use the countour integral to calculate the inverse Laplace transform of the function$$F(s) = \frac{\exp(\frac{\lambda s}{1 - 2s})}{(1 - 2s)^{k/2}} \hspace{1cm}\mathrm{for} \hspace{1cm} ...
I want to ask the inverse Laplace transform of the following function: F(s) = \frac{1}{s \cdot (1 + a \cdot s)^{m} \cdot (1 + b \cdot s)^{m-k}} \cdot \Bigl[\exp{(\frac{- c \cdot s}{ 1 + b \cdot s } ...
### How to Integrate$\int_C\frac{Log~z}{(z-i)^2}$ where $C:|z-i|=1$?
Let $C:|z-i|=\alpha$,where $0<\alpha<1$ I need to evaluate this expression $~\int_C\frac{Log~z}{(z-i)^2}$ what I tried to do was representing $Log~z$ as a Laurent series on ...