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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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Residue for $\frac{e^z+1}{\sin z}$

I want to find the residue of $\displaystyle f=\frac{e^z+1}{\sin z}$ at $z=0$. My solution was $2$, which I got using $\displaystyle \frac{e^z+1}{\sin z}=\frac{e^z}{\sin z}+\frac{1}{\sin z}$. The residue for $\displaystyle Res_0\left( \frac{1}{\sin…
Cure
  • 4,051
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Integral of $1/(x^{2n}-a_0)^2$

I am looking for solution of $\int_{-\infty}^{\infty}1/(x^{2n}-a_0)^2 dx$ for some unknown constant $a_0$. I know that some related forms have been earlier solved with the help of residue theorem. However, I find this one particularly troublesome…
Ibrahim Bilal
  • 21
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Integrate $I=\int_e^\infty\left(\frac{\log\log y}{y(y+1)}\right)^2dy$ using residue calculus?

Can this integral be done using the residue calculus? $$I=\int_e^\infty\left(\frac{\log\log y}{y(y+1)}\right)^2dy$$ ? My (empirical) investigative attempts have been to use a keyhole contour centred at $y=e$. My working so far indicates that the…
pshmath0
  • 10,565
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Finding the residues of poles

Consider the equation $\mathcal{F}(\lambda)=0\ \ \ \forall\ \lambda = \lambda_{n},\ n \in \mathbb{N}$. I understand that the expression $\frac{d}{d\lambda}\ \ln\mathcal{F}(\lambda)=\frac{\mathcal{F'(\lambda)}}{\mathcal{F}(\lambda)}$ has poles of…
nightmarish
  • 617
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how to find residues of $\frac{e^{st}}{\cosh(a\sqrt{s})}$?

Can someone give me a hint on how to find residues of $\displaystyle\frac{e^{st}}{\cosh(a\sqrt{s})}$, where $a\in\mathbb{R}$? I am trying to solve an integral using residue method (actually inverse Laplace transform). I know the zeros of $\cosh(x)$…
Nasser
  • 2,116
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Prove on residue theorem

I have try to use the equation $$ Res(f;z_0)=\lim_{z\to z_0}\frac1{(m-1)!}\frac{d^{m-1}}{dz^{m-1}}[(z-z_0)^mf(z)] $$ But very soon I stuck, is that a good way to solve it?
user111605
  • 17
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Showing that, under additional assumptions, an entire function is not a polynomial

I'm not believing the claim (below) that is to be proven true. Show that if $f$ is entire, not zero on circles of natural radius centred at the origin and $\oint_{|z| = n}\frac{1}{f(z)}\mathrm{d}z \neq\oint_{|z| = n + 1}\frac{1}{f(z)}\mathrm{d}z$,…
johnsmith
  • 343
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Residue of $f^2$ using $f$ Laurent coefficients

I need help with the following: If $f$ is a function having $z_0$ as a second order Pole. and having the Laurent series $\sum_{k = - \infty}^{\infty} a_k (z-z_0)^k$. How could one calculate $Res_{z_0} f^2$ from the Laurent coefficients $a_k$ of…
MWSC1996
  • 29
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Using Residue theorem to evaluate integral with complex poles

I'm trying to evaluate the integral of a function $$\frac{\psi_m^T\cdot P\cdot\phi_m\cdot\exp(-\gamma\xi i)}{(\gamma_m-\gamma)\cdot B_m}$$ with respect to $\gamma$, where $\psi_m$ is a $1\times n$ row vector, P is a $n\times n$ matrix, $\phi_m$ is a…
Antigoni B
  • 11
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2 answers

Find all the singularities in the finite plane and the corresponding residues. Show the details.

I am a bit stuck on this problem: Find all the singularities in the finite plane and the corresponding residues. Show the details. $$\frac{8}{1+z^2}$$ So I know the residual is going to be the coefficient associated with the first negative…
Jwan622
  • 5,704
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Residue at a pole

In my textbook, I have this example: $$f(z) = \frac{9z+i}{z^3+z}.$$ Find the residue. So the book says it has a simple pole at $i$ but doesn't it also have a simple pole at $-i$? I can see the denominator is $z(z^2 + 1) = z(z+i)(z-i)$ so doesn't it…
Jwan622
  • 5,704
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2 answers

Looking for a curve for calculating this integral using Residue Theorem

I want to prove this equality using Residue Theorem $$\int_{0}^{\infty}\frac{x}{e^x-e^{-x}}\mathop{dx}=\frac{\pi^2}{8}$$ My Attempt: I used from variable changing $u=e^x$, However, I'm looking for a good curve which I couldn't find it yet....
Sania Sadian
  • 69
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How do I calculate $\int_{0}^{2\pi} \frac{2+4\cos(t)}{5+4\sin(t)} dt$?

How do I calculate $$\int_{0}^{2\pi} (2+4\cos(t))/(5+4\sin(t)) dt$$ I've recently started calculating integral via the residue theorem. Somehow I'm stuck with this certain integral. I've substituted t with e^it and received two polynoms but somehow…
Jorge
  • 13
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Finding the residue of an infinite series

I want to find the residue of $$ \sum_{n=1}^{\infty} \frac{z^n}{n!(1-z^n)} $$ at $z=1$. I've tried $$ \sum_{n=1}^{\infty} \frac{z^n}{n!(1-z^n)} = \sum_{n=1}^{\infty} \left( \frac{1}{n!(1-z^n)} - \frac{1}{n!} \right) = 1-e +\sum_{n=1}^{\infty}…
Rowan
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How to apply the method of steepest descent to solve this integral?

I want to reproduce Eq. (11) of this paper. It is the result of solving the integral $$ \int_{0}^{\Lambda} \text{d}q \frac{q (e^{i q r} - e^{-iqr})}{q^2 - x} $$ (where $\Lambda = \pi$ and $x = \omega/\sqrt{\Delta k}$, with $\omega$ real and $\Delta…
Kappie001
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