# 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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### How to solve the following integral $\int_0^1 \dfrac{x^4}{\sqrt{x(1-x)}}dx$ (residue theorem)

How to solve the following integral $\int_0^1 \dfrac{x^4}{\sqrt{x(1-x)}}dx$ (although we may use normal real integral to solve, I wonder if contour analysis can also help?) The question offers a hint ...
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### Finding the residue of a mod of a function…

What is the residue of $$f(z)=\frac{1}{|z+c||z-c|}$$ at $z=c$ and $z=-c$. I know to find the residue without mod in the denominator, but I have no idea of finding the residue with a mod in the ...
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### Integral Value through Complex Integration (residue theorem)

I'd like to know how to evaluate the integral $$I=\int_0^\infty\frac{e^{-s^2}\sin(s)}{s}\,ds=\frac{\pi}{2}\text{erf}(1/2)$$ through the residue theorem. My first steps were to expand $\sin$ as ...
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### How to realize ILT by residue theorem?

$$F(s)=\frac{e^{-x \sqrt{s}/\sqrt{D}} (e^{2C \sqrt{s}/\sqrt{D}} + e^{2x \sqrt{s}/\sqrt{D}})}{s^{3/2} (e^{2c \sqrt{s}/\sqrt{D}} -1)}$$ By the residue theorem. I tried, but didn't realize. The ...
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### Contour integral of fractional function

So, I have to solve the following integral $$\int_{0}^{\infty} \frac{\sin^{2}{x}}{x^{2}} dx$$ I'm aware that this has been asked about before on this site, but I want comments on my attempted ...
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### Evaluate : $\displaystyle\int\limits_{\gamma }\frac{\log (1+z)}{1+z}dz$

Computer $$\displaystyle\int\limits_{\gamma }\frac{\log (1+z)}{1+z}dz$$ Where : $$\gamma =\{ |z|=1~ ; ~\Re z≥0,\Im z≥0 \}$$ I try : $z=e^{it}$ then $dz=ie^{it}dt$ And $t\in [0,\frac{π}{2}$ then ...
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### Complex Integration (Residue Theorem)

How do I integrate $\oint_{C:\left | z \right |= R}^{}\frac{e^{^{\frac{1}{z}}}}{z^{2}+1}dz$ , with $0< R< 1$ ? I am supposed to use the residue theorem but there's no Laurent series around z=...
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### Determining a meromorphic function by its poles.

Given a series of complex numbers $\{a_j\}_{j\in\mathbb{Z}}$, there exists a meromorphic function having simple poles only at all $\mathbb{Z}$ with the residues equal to $\{a_j\}_{j\in\mathbb{Z}}$ ...
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### Calculate:$\int_{0}^{\infty}\frac{\ln x}{(x+1)^{3}}\mathrm{d}x$ with contour integration

Calculate: $$\int_{0}^{\infty}\frac{\ln x}{(x+1)^{3}}\mathrm{d}x$$ My try: Keyhole integration: $\displaystyle \frac{\pi i\ln R\cdot e}{(Re^{\theta i}+1)^{3}}\rightarrow 0$ (we take $r$ as large as we ...
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### why this claim is wrong ? (as it leads to $\int_{0}^{\infty}\frac{\sin x}{x}dx=0$)
let's define $\begin{array}{c} f=\begin{cases} \frac{\sin x}{x} & x\neq0\\ 1 & x=0 \end{cases}\end{array}$ f is holomorphic on $\mathbb{C}$ as it equals to it's taylor series. therefore, for ...