Questions tagged [residue-calculus]

Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory.

Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory. The method consists of using a closed contour on the complex plane to evaluate complex or real integrals.

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Replacement $\int_{0}^{2\pi }\frac{1}{a+\sin(3t)}dt $

I can not understand why a replacement I tried in resolution with the residual method of the goniometric integrals leads to a wrong result. $$\int_{0}^{2\pi }\frac{1}{a+\sin(3t)}dt $$ $$a\in \mathbb{R} |a|<1$$ I…
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Finding value of a complex integral with residues by intuition

I have $f(z)=z^3-1$. Clearly there the three complex roots of unity, $$1, -\frac{1}{2}+\frac{i\sqrt{3}}{2}, -\frac{1}{2}-\frac{i\sqrt{3}}{2}$$ and so for simplifity I will just denote these as $1, \omega, $ and $\omega^2$. If I need to find,…
AveryJessup
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Integral with $\cos$ function

I know how to compute $\displaystyle\int_{0}^\infty \dfrac{\sin x}{x(x^2+1)}dx$. In fact, we can compute $\displaystyle\int_{-\infty}^\infty \dfrac{\sin x}{x(x^2+1)}dx$ and just use the fact that $f(x)=\dfrac{\sin x}{x(x^2+1)}$ is an even function.…
Cachorro
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residue theorem over the simples zeros of riemann zeta

given the circle integral $$ \oint \frac{\zeta (2s)}{\zeta (s)} $$ if taken only over the reidues due to the simple zeros -2,-3-6 etc i get the sum $$ \sum_{n=1}^{\infty} \frac{\zeta (-4n)}{\zeta '( -2n)} 2\pi i $$ which is 0 but what did i make…
Jose Garcia
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$\frac{1}{\sqrt{2\pi}}\int_{-\infty }^{\infty }\frac{k}{2}\sqrt{\frac{\pi}{2}}e^{-2|k-\pi|}dk$

Hi someone can help me with this integral, I can imagine that it is solvable with method of residues but in this case I do not know where to start. $$\frac{1}{\sqrt{2\pi}}\int_{-\infty }^{\infty…
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Evaluate the integral $\int_{0}^{2\pi}\frac{1}{2-\cos^2(x)} dx$ using complex analysis

Evaluate $$\int_{0}^{2\pi}\frac{1}{2-\cos^2(x)} dx$$ For this problem I tried using residue theorem to find the value of the integral but the computation seems rather messy. Is there a better way to find the integral? Thanks.
user112358
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Improper integrals with Jordan's lemma, choice of contour?

$$ f(z) = \frac{e^{-iz}}{z^2+9}, \:\:\: z \in \mathbb{C} $$ I am trying to compute $$ I = \int_\limits{-\infty}^{\infty} f(z)\:dz $$ The zeros of the denominator give that $f$ is analytic in $\mathbb{C}\backslash\left\{-3i,3i\right\}$. The residues…
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Curvilinear integral of a derivate

I have $ g = e^{iz} / z^{1/3} $ and I have to calculate the integral of g' over a circumference of center (0,0) and radius $ \pi $ , oriented counterclockwise. I really have no idea how to do this job. I tries using the residues theorem…
PeppeDAlterio
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Computing residue of $f(z)=\frac{z-\pi}{\sin^2z}$ at $z = \pi$

I know how to find residuum simple function but now I have function $$f(z)=\frac{z-\pi}{\sin^2z}$$ and I have to calculate residuum in $\pi$ (that is, $ \operatorname{Res}_{z=\pi}f(z)$). When I calculate the limit in $\pi$ it's infinity. So in $\pi$…
zxc
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Computing the residue of a complex function

I have the following contour integral (with C the positively oriented unit circle centered at the origin): $$ \frac{-i}{4}\int_{C}\frac{\left(z^2+1\right)^2}{z\left(-z^4+3z^2-1\right)}dz $$ It has isolated singularities inside $C$ at $z = 0,…
user405561
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Strange roots expansion and guessed estimation of the magnitude of a root

What expansion happened to the roots? Looks like a series but it can't be? It is a root not a function, indeed $a$ should be a parameter? Furthermore, even if it was considered as a function, what type of expansion is it? Taylor? How do we know…
Euler_Salter
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How do we know that there are residues?

How does someone know a priori (I mean without doing calculations, as here!) that $A,B,C$ are the residues? I mean of course if we use the standard formula for residues on the RHS we get $$\lim_{z\to…
Euler_Salter
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Calculation of Partial Fraction Decomposition Constants by Residues

If $p(x)$ and $q(x)$ are polynomials such that $\deg q(x) > \deg p(x)$ then the rational function $p(x)/q(x)$ has a partial fraction expansion $$ \frac{p(x)}{q(x)} = \frac{a_1}{(x-q_1)^{m_1}}+\frac{a_2}{(x-q_2)^{m_2}}+\cdots…
JMJ
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Residue of $\frac{1}{\cos z}$ at $z=\pi/2$

What is the residue of $$\frac{1}{\cos z}$$ at $z=\pi/2$? Given that $$\cos z=\prod_{k\in\mathbb{Z}}(z-z_k)=(z-\pi/2)(z+\pi/2)(z^2-(3\pi/2)^2)(z^2-(5\pi/2)^2)\cdots,$$ the point $z=\pi/2$ is a simple pole (order $n=1$) and so usning the definition…
yngabl
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Difficult residue $Res(z+2)^2{cos{z\over{1+z}}}$ at $z=-1$

$Res( (z+2)^2{cos{z\over{1+z}}})$ at $z=-1$ Usually calculating residues is easier. In this case the point is essential singularity,so I tried to expand cos into series, and expand $ z\over{1+z} $ and combine the series to find $a_{-1}$, but it gets…
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