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### The definition of Residue

In Wikipedia the definition of a residue of a function $f$ in a point $a$ is a unique value $R$ such that $f(z)-\frac{R}{z-a}$ has an anti derivative in a punctured disk $0<|z-a|<\delta$. How is ...
Evaluate\begin{align} \int_{-\infty}^{\infty} \frac{\sin(x)}{x^2+4x+5} \, dx \end{align} I set $f(z) = \frac{1}{z^2 + 4z + 5} = \frac{1}{(z-z_0)(z-\bar{z_o})}$, where $z_0 = -2+i$ and \bar{z_0} ... 1answer 29 views ### Using Jordan's Lemma for residue integration Evaluate \begin{align} \int_{0}^{\infty} \frac{\cos(ax)}{x^2+1} \, dx \end{align} So far, I setf(z) = \frac{1}{z^2+1}$, and the singularity point are$z = \pm i. But I am using the top ... 3answers 76 views ### Using residues to evaluate an improper integral Use residues to evaluate the improper integral \begin{align} \int_{0}^{\infty} \frac{1}{(x^2+1)^2} \, dx \end{align} First, I saidf(z) = \frac{1}{(z^2+1)^2}$. My only question so far is how ... 1answer 24 views ### Finding a residue I am given$f(z) = \frac{\text{Log} z}{(z^2 + 1)^2}$. The singularity point occurs at$z = i$, with a pole order of$m = 2$. How can I go about finding$\text{Res}_{z=i} f(z)$? The correct answer ... 0answers 23 views ### Polynomial Expansion in a Complex Improper Integral I want to evaluate the integral below by using the Residue theorem. $$\int_{-\infty}^{\infty}\frac{\exp(-iwt)w}{(w-ic)\sqrt{w^2-aw+b}}dw$$ There are branch cut points due to the square rooted term ... 2answers 52 views ### Calculate the integral$\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz$I am looking to solve $$\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz,$$ where$\varGamma$is the contour$|z|=4\pi/3$. We have been asked first to consider$e^{z}=1$and$e^{z}=-1$which I get to be ... 0answers 22 views ### Integration of a complex function having square rooted denominator How to evaluate this integral ? $$\int_{-\infty}^{\infty}\frac{1}{(w-ia)\sqrt{w^2-ibw+b^2c}}dw$$ where$a$,$b$and$c$are real and greater than zero. Does the residue theorem work on it? Please ... 2answers 103 views ### Integral$\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$Consider $$\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$$ Is it equal to$0$? Why ? Any hint ? 1answer 74 views ### Using Cauchy integral formula to calculate$\int_\gamma \frac{\cos{z}}{z^n}$Let$\gamma(\vartheta)=\mathrm{e}^{i\vartheta},\,\vartheta\in[0,2\pi]$, and consider the integral $$I(n)=\int_\gamma \frac{\cos{z}}{z^n},$$ where$n\in \{0,2,4,6,...\}$. Is there any way to prove ... 0answers 47 views ### Can we use a sum of residues to develop an asymptotic expansion for this unknown function? In the course of solving a particular physical problem, I have derived a relationship between two unknown functions: $$f(s) = \frac{s \sinh{\frac{\pi s}{2}}}{2 \pi i \beta} \int_{-c- i ... 0answers 15 views ### A question on Residues and a family of functions I have the following:$$\alpha(t)=\int_{\left|z\right|=1}z^{k}\frac{f^{\prime}_{t}(z)}{f_{t}(z)}\,dz$$where f_{t}(z) is a family of entire functions depending on t \in \Delta and k \geq 0. By ... 3answers 71 views ### Integral \int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x I want to evaluate the following integral via complex analysis$$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a >0$$Which function/ contour should I consider ? 1answer 43 views ### Evaluating a few complex integrals on the unit circle I'm stuck on a few of these, but I have most of the details worked out: (i) \int_{|r|=1}(z^2-4)^{-1}\,dt=\int_{0}^{2\pi}ie^{i\theta}(e^{2i\theta}-2)^{-1}\,d\theta (ii) ... 0answers 34 views ### Integrating \oint_\Gamma \cos(\log|z|)\cosh(\text{Arg}(z))\text{Arg}(z)e^{is(z-1)}dz using residue calculus. I'm trying to use the residue calculus to evaluate$$\oint_\Gamma \cos(\log|z|)\cosh(\text{Arg}(z))\text{Arg}(z)e^{is(z-1)}dz,$$where s>0, and where \text{Arg} is the principal argument, ... 3answers 85 views ### Confusion about a way to compute a residue at a pole Suppose I have a function of the form$$f(x)=\frac{1}{(x-a)(x-b)^2(x-c)^3}$$Clearly, I have a simple pole at a, and poles of order 2,3 at b,c, respectively. By definition, the residue at ... 1answer 43 views ### \sum_{i=1}^{n}\operatorname{Res}(f,z_{i}) + \operatorname{Res}(f,\infty) = 0 Let f \in H(\mathbb{C}- \{ z_{1}, \dots, z_{n} \}). I need a proof of the fact that$$\sum_{i=1}^{n}\operatorname{Res}(f,z_{i}) + \operatorname{Res}(f,\infty) = 0.$$Where can I find it ? 1answer 48 views ### Find the residues of f(z) = \left( \frac{z-1}{z+1}\right)^{\frac{1}{2}}\frac{1+z} {1+z^{2}} Consider the function$$f(z) = \left( \frac{z-1}{z+1}\right)^{\!\frac{1}{2}}\frac{1+z} {1+z^{2}}$$I want to calculate the residues of f in \{+i,-i\}. Using the usual techniques, we have that ... 2answers 52 views ### Using residue to find a complex integral Given the following:$$\int_{\varGamma_R} {z\,dz\over e^{2\pi iz^2}-1}, \ \ \ \varGamma_R=\{z\in \Bbb C:|z|=R\},\quad n<R^2<n+1,\,n\in\Bbb N.$$I want to use the residue for this, but I can't ... 1answer 142 views ### Integral Using Harmonic Functions Evaluate the integral:$$\int^{2 \pi}_0 \dfrac{\cos^2 \theta}{|2e^{i\theta}-z|^2} \, d \theta \qquad \mbox {when} \, |z| \neq 2.$$Now, I thought about trying to change this to look like a Poisson ... 1answer 45 views ### Quarternionic Analysis What is/are the current understanding/opinions about Quarternionic Analysis as a generalization of Complex Analysis with respect to a "Quarternionic Residue Calculus" (if such a thing exists)? i.e. ... 0answers 195 views ### Integrating \int_0^\infty\frac{\log (1+z^2)}{e^z-1}dz using residue calculus. I've been looking at how to integrate the following definite integral using the residue calculus, but can't seem to get my thoughts together. I know the \log term is a multivalued function and the ... 2answers 61 views ### \int_{0}^{\infty}\frac{\cos2\pi x}{x^4+x^2+1}dx=-\frac{\pi}{2\sqrt{3}}\mathrm{e}^{-\pi\sqrt{3}} Can somebody help me out with the following integral? Prove that: \int_{0}^{\infty}\frac{cos2\pi x}{x^4+x^2+1}dx=\frac{-\pi}{2\sqrt{3}}e^{-\pi\sqrt{3}} I have already determined the ... 0answers 62 views ### Pole on path of integration. Upon evaluating$$\int_0^\infty \frac{1}{1+z^5}dz$$using the Residue Theorem, why isn't the pole at z=e^{\pi i} taken under consideration in the summation of the residues? We were taught that ... 4answers 180 views ### Calculating \int_0^\infty \frac {\sin^2x}{x^2}dx using the Residue Theorem. I am trying to compute the following integral using the Residue Theorem but am quite stuck:$$\int_0^\infty \frac{\sin^2x}{x^2}dx$$I have tried applying Jordan's lemma, having written \sin(x) as ... 1answer 41 views ### Integral \int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt I want to solve the integral$$\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$$Which function and contour should I consider ? 2answers 92 views ### \int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx with residue calculus I'm trying to compute \displaystyle \int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx, (0<a<1) Let f denote the integrand. I'm using the rectangular contour given by the following curves: ... 1answer 65 views ### Residue of a simple pole. Why are they different? We'll show you two way of calculation of the Residue in consideration.$$f(z) = \frac{z\sin(z)}{1-\cos(z)}$$I'm interested to calculate the residues in 2\pi and -2\pi. I choose one of ... 1answer 58 views ### Find the order of the poles of \dfrac{z}{\cos z} I know that they are simple poles, but how can you find this? The usual equation that I have for finding the order of poles which is,$$ \displaystyle\lim_{z\to z_0} (z-z_0)^{n}f(z), $$and seeing ... 1answer 85 views ### Would like help with a contour integral. Disclaimer: the knowledge I have about contour integration is solely from the book "Mathematical Methods in the Physical Sciences" by Mary L. Boas. I am trying to understand how the following ... 3answers 95 views ### Question Residues -integral at Complex Analysis How can i find the integral below , which transformation should i do ? İ think i need to get sin and cos but i can't see$$\int\limits^{+\infty}_{-\infty} \frac{ \exp\left({ax}\right)} ... 1answer 83 views ### Choice of branches for contour integration. Suppose I have the following function of a complex variable $$f(z)=\log(z)(z^2+1)^{1/2}.$$ Wolfram Alpha tells me the branch cuts of$f(z)$are$z\leq 0$(presumably for the logarithmic term), and ... 2answers 35 views ### Residue of a a complex quotient I have the following Laurent expansion corresponding to the function: $$f(z)=\frac{z+2}{z^2-4iz-3}$$ $$f(z)=\left(-1+\frac{1}{2}i\right)\sum_{n=1}^\infty ... 2answers 151 views ### How to show \int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a})? (a\ge0)$$\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a}), \ a\ge0$$I tried to solve but came up with \pi(2-e^{-a}) . Could you tell me where did I do the mistake? if x=z then ... 1answer 28 views ### Carry out integral by using Cauchy's theorem I have kind of a silly question, which probably has an easy answer which I should know myself, but here goes. Say we want to integrate$$ \int_{-\infty}^\infty dx \frac{1}{(x^2 + 1)(x - 1 - i)}. $$If ... 2answers 119 views ### How to compute the integral \int_0^\infty\frac{x}{e^x+1}dx using the Residue theorem. How to compute the integral \int_0^\infty\frac{x}{e^x+1}dx using the Residue theorem, just as the title says. I have used rectangles, circles to do, but without any progress. By changing variable ... 0answers 58 views ### How to calculate this residue how to calculate this residue$$Res(\frac{Ln z}{z(z+1)},0).$$Is it \infty? And if this could not be calculated, then how to calculate$$\int_0^\infty \frac{x}{e^x+1}dx$$by changing variables ... 2answers 62 views ### Intuition behind the residue at infinity [duplicate] The residue at infinity is given by:$$\underset{z_0=\infty}{\operatorname{Res}}f(z)=\frac{1}{2\pi i}\int_{C_0} f(z)dz$$Where f is an analytic function except at finite number of singular points ... 2answers 79 views ### calculation of \int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx , where n\in \mathbb{N} Calculation of \displaystyle \int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx , where n\in \mathbb{N} \bf{My\; Try}:: Using \displaystyle \cos (x) = \frac{e^{ix}+e^{-ix}}{2}, we get ... 1answer 50 views ### Residue Theorem for trigonometric integrals. I am working on the following statement. Let Q = Q(x,y): \mathbb R^2 \to \mathbb R be a rational function, which is continuous on the unit circle S_1(0). Let furthermore f: \mathbb C \to ... 2answers 101 views ### How to show the residue of an analytic function's derivative is equal to zero? Let r>0 . for f: \Bbb D_r(0)-{0}\mapsto \Bbb C analytic function show that Res(f';0)=0 we know by residue therom ∫_Cf'(z)dz=2iπRes(f',0) What property of analytic functions will we use? ... 1answer 42 views ### Computing the residues for 1 /( z^2\sin(z)) I am trying to find the residues for the function 1 / (z^2\sin (z)). By expanding the function around the singularites I managed to find \text{Res}(f;0) = 1/3!. There is also a singularity at z = ... 1answer 93 views ### To calculate residue of the function f(z) = \frac{z^2 + \sin z}{\cos z - 1}. I was trying to find the residue of the function$$f(z) = \frac{z^2 + \sin z}{\cos z - 1}.$$Here is the my attempt: The given function has a pole of order two at z = 2n\pi. So, we use the ... 1answer 91 views ### \int_{0}^{+\infty}\frac{sinx}{x^{k}(1+x^{2})}dx \ via residue calculus I want to evaluate with calculus of residues$$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $$k \in \mathbb{N}, k \geq 1 If k = 1 we have$$\int_{0}^{+\infty}\frac{\sin ... 2answers 70 views ### Residue theory complex $$\int_{-\infty}^{\infty}\frac{\cos x}{x^4+5x^2+4}dx$$ Give full justification of your answer, including appropriate bounds for the contributions from all portions of your contour! I am not ... 1answer 30 views ### Discrepancy in counting the number of poles in complex function when refactoring If I have a function that looks like this: $$f(z) = \frac{(z-i)^2}{\sin^2z}$$ and I want to find its poles within the unit circle contour,$|z| = 1$, it seems from this equation that there is a pole ... 0answers 83 views ### Number of zeros equal number of linearly independent analytic functions I'm trying to read this paper and I'm stuck on a particular point. The authors are constructing an analytic function$f(z)$which have to satisfy the following boundary conditions: ... 1answer 55 views ### Evaluating real improper integral by residues I've been trying to solve this integral and have been getting nowhere: $$\int_0^\infty \frac{dx}{(1+x^2)x^a} \;,\; 0<a<1$$ The solution says that $$\int_0^\infty \frac{dx}{(1+x^2)x^a} = ... 1answer 194 views ### Calculate \displaystyle \int_0^\infty \frac{\ln x}{1 + x^4} \mathrm{d}x using residue calculus I need to evaluate this integral using calculus of residues:$$\int_0^\infty\frac{\ln(x)}{1+x^4}\mathrm{d}x$$I know I need to consider$\displaystyle ...
Following is the question that i am trying to solve: "Consider a second order linear ODE $x\dfrac{d^{2}y}{dx^{2}} + x\dfrac{dy}{dx}$ + (3 - 2x)y = 0 A) Find the solution employing Laplace ...