# 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.

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### Residue theorem pole

How do I find poles of this function? $$f(z) = \frac{1-\cos z}{z^4 +z^3}$$ I am unable to identify zero pole order of $f(z)$ as it includes trigonometric functions and I'm new to this concept. Please ...
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i just need to solve this integral using residue theorem. $$\int_{-\infty}^{\infty}\frac{\cos(x)dx}{(x^2 +a^2)(x^2 + b^2)}$$ where $-1<a<1$. I just know i have to separate $\frac{\cos(x)dx}{(x^2 ... 0answers 19 views ### If$|f(z)|<k|z|^{-c}$, then$f(z)=\int_{-\infty}^{+\infty}\frac{f(t)}{t-z}dt$The problem is: If$f$is analytic in$\{z\in\mathbb{C}:\text{Im}(z)\geq 0\}$and there exists$k,c>0$such that$|f(z)|<k|z|^{-c}$for all$z\neq 0$, then$f$has the form $$f(z)=\dfrac{1}{2\pi ... 1answer 38 views ### How to get ILT of e^{-\alpha \sqrt{s}}/s^{3/2} by residue theorem? I posted some examples before such as$$\mathscr{L}^{-1}\left[\frac{e^{-\alpha \sqrt{s}}}{\sqrt{s}(s-A)}\right]=\frac{e^{-\alpha \sqrt{A}+At}}{\sqrt{A}} - \frac{1}{\pi}\int_{0}^{\infty}\frac{\cos{(\... 1answer 38 views ### Finding the residue at$z=z_0$of$f(z)$given… i need to resolve this problem: Given$f(z)=\frac{1}{\big(g(z)\big)^2}$, show that if$g(z_0)=0$and$g'(z_0)\neq 0$then$f$has a second order pole at$z_0$and$\text{Res}_{z=z_0}f(z)=-\frac{g''(...
Consider the following integral: $$\frac{1}{2\pi i }\oint_C f(z) dz$$ with $$f(z)= \frac{z^2+1}{z^3}$$ and $C$ is the unit circle centered at the origin. Consider two ways to do it: Enclosing the ...