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pantboio
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How can i compute $Res(f,z_k)$ where
$$f(z)=\frac{z-1}{1+cos\pi z}$$
and $z_k=2k+1, k\neq 0$?
$$f(z)=\frac{z-1}{1+cos\pi z}$$
and $z_k=2k+1, k\neq 0$?
pantboio said:$z_k$ are zeros of order two of denominator, but $z_0=0$ is also a zero of numerator. So $z_0=0$ is a simple pole, but all the others $z_k=2k+1$ with $k\neq 0$ are 2-poles.
so the residue is the same for all poles, maybe because of $\cos$ periodicityFernando Revilla said:Using a series expansion: $1+\cos \pi z=\ldots=\dfrac{\pi^2}{2}[z-(2k+1)]^2+\ldots$. Now,
$\dfrac{z-1}{1+\cos \pi z}=\dfrac{2k+[z-(2k+1)]}{\dfrac{\pi^2}{2}[z-(2k+1)]^2+\ldots}=\ldots +\dfrac{A_{-2}}{[z-(2k+1)]^2}+\dfrac{2/\pi^2}{z-(2k+1)}+A_0+\ldots$
So, $\mbox{Res }(f,2k+1)=\mbox{coef }\left(\dfrac{1}{z-(2k+1)}\right)=\dfrac{2}{\pi^2}$
pantboio said:so the residue is the same for all poles, maybe because of $\cos$ periodicity
Residue calculus is a branch of mathematics that deals with the computation of complex integrals. It involves finding the residues of a function at its singular points and using them to evaluate the integral.
Residue calculus has many applications in physics, engineering, and economics. It is used to solve problems in fluid mechanics, electromagnetism, and signal processing. It is also used in the analysis of financial markets and the design of control systems.
A residue is the coefficient of the term with the highest negative power in the Laurent series expansion of a function at its singular point. In residue calculus, it is used to evaluate complex integrals by using the Residue Theorem.
To calculate a residue, you first need to find the Laurent series expansion of the function at the singular point. Then, the residue can be found by taking the coefficient of the term with the highest negative power in the series. Alternatively, you can use the formula Res(f, z0) = lim(z->z0) [(z-z0)f(z)].
The Residue Theorem states that if a function has a finite number of singular points inside a closed curve, then the integral of the function around the curve is equal to the sum of the residues of the function at these points. This theorem is a powerful tool in evaluating complex integrals using residue calculus.