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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.

2742 questions
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Finding the residue

Finding the fourth derivative in order to get residue seems me very complicated, is there another way? $$Res\left( z=i,\frac { { e }^{ iz } }{ { \left( { z }^{ 2 }+1 \right) }^{ 5 } } \right) =\lim _{ z\rightarrow i }{ \frac { 1 }{ 4! } \frac { {…
haqnatural
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Finding the poles of a complex function in order to find the complex integral $\int_C \frac{dz}{1+z^n}$ using residues?

Could you please explain how to get the poles of the complex function $f(z) =\frac{z}{1+z^n}$. I am computing the integral $\int_0^\infty \frac{dx}{1+x^n}$ by considering the complex integral $\int_C \frac{dz}{1+z^n}$ around an arc segment enclosing…
Ryan Maguire
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Evaluate $\int_0^\infty \frac{x^2}{x^4 + 6x^2 + 13}dx$

In the context of the residue theorem, I have this integral to evaluate. The function is even, and $|\int_0^\pi\frac{R^2e^{2i\theta}iRe^{i\theta}}{R^4e^{4i\theta}+6R^2e^{2i\theta} + 13}d\theta| \leq \int_0^\pi2\frac{R^3}{R^4}d\theta \to 0$, so the…
Christian Fieldhouse
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A Problem with Residue Theorem

According to some reliable sources, the residue theorem can be stated as: the sum of residues of a function on the whole complex plane(including infinity) is zero. Now consider $$f(x) = \frac{\csc(\pi/x)}{x^2}$$ which has singularities at $+1, -1,…
Szeto
  • 11,159
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Is residue calculus applicable here?

For an integral of the form $$ I(t)\;=\;\int_{-\infty}^{+\infty}\frac{e^{i \omega t}}{1+i \omega}\,\mathrm{d} \omega $$ say, with $t>0$, a standard application of the residue calculus with a semi-circle contour integration along the upper half plane…
user431632
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Calculus of residues

How can we find the residue at $ z=0 $ of $$f(z) = \log\left(\frac{1-az}{1-bz}\right)$$ where $a, b$ are complex constants?
user437903
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How do I get the residue of the given function?

I'm reading the solution of the integral: $$\int\limits_{-\infty}^{\infty} dx\frac{e^{ax}}{1+e^x}$$ by the residue method. And I understood everything, but how to get the residue of $\frac{e^{az}}{1+e^z}$ (the book just states that the residue is…
Ivan Lerner
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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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Using the residue theorem

Is it possible to evaluate $$\int_{-\infty}^\infty \frac{x^2}{(x^2+1)^2} \, dx $$ using the residue theorem, as opposed to Calc 1 methods? How can I get started using the residue theorem? What integrand can I use in terms of $z$? I think I need to…
user295703
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Residue of 1/(z-sin(z)) at z=0

I am to find the residue of f(z)=1/(z-sin(z)) at z=0. I am confused as to which method to use. Your help will be greatly appreciated! Thanks!
mathman41
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How to calculate this integral via residues.

I get into trouble in evaluating this integral: $$ C(a)=\frac{1}{i\beta}\int_\Gamma \cot\frac{\pi z}{\beta}\frac{1}{\sin^2\frac{z}{2}}dz $$ where the contour $\Gamma$ consists of two vertical lines, (−π − i∞, −π + i∞) and (π + i∞,π − i∞).The result…
JQ Skywalker
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Residue integral solution differs from wolframalpha

Im trying to solve the following integral using the residual theorem: $$ \int\limits_0^{2\pi} \frac{\mathrm{d}\phi}{3 + 2\cos \phi} $$ Using $2\cos \phi = z - \overline z$ and $\frac{\mathrm{d}z}{\mathrm{d}\varphi} = \mathrm{i} z$ I transformed the…
user44789
  • 307
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Residue Integration

I am attempting to calculate the integral of $\frac{(1+sin(\theta))}{(3+cos(\theta))}$ from $0$ to $2\pi$. I have already changed $sin$ and $cos$ into $\frac{1}{2i(z-z^{-1})}$ and $\frac{1}{2(z+z^{-1})}$. I am really stuck now. Can anyone please…
will
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calculating the residue of a complex function in an integral

I wonder how to solve this integral: $$P(q, \omega) = \frac{-2i}{(2\pi)^4}\times\int\frac{\mathrm d\mathbf{k}~\mathrm d\omega' e^{i\omega'\eta/h}}{\left[\omega' - E(\mathbf k) + i\delta\operatorname{sgn}\left(E(\mathbf k) -…
P.A.M
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