# Tagged Questions

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 compute this integral on contour

How to compute this following integral? $$\int \limits_{C}^{}\frac{e^{az^2}\,dz}{z^4+1}$$ Given $a>0$ and $C:=\{z: |z+1| = 1\}$ is positively oriented. Where should I start? What would $f(z)$...
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### Show that the polynomial $p(z)=z^5+7z-1$ (in $\Bbb{C}[z]$) has one real root with absolute value smaller than one.

Show that the polynomial $p(z)=z^5+7z-1$ (in $\Bbb{C}[z]$) has one real root with absolute value smaller than one and that the rest of the roots are in $\{1<|z|<2\}$. Ok, so this exercise is ...
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### finding residue for complex analysis

I am having a tough time finding the residue for a function, suppose my test function is $$\frac{z^2}{{(z^2+a^2)}^2}$$ while I could determine the poles to be $+-ai$ and I know the formula to find ...
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### integral vs. residue at infinity

I have an issue with residues at infinity. I am computing the integral $\displaystyle{\int_{C_3^+(0)} \dfrac{e^{3z}}{z^2(z^2+2z+2)} dz}$ Since all three poles ($0$ of order 2, $1\pm i$ of order 1) ...
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### contour integral branch cut

I need some help to solve the following integral by contour integration. $$\int_{0}^{1} x^a (1-x)^{1-a}\,\mathrm{d}x$$ I attached my ideas and a picture of the paths to fix the labels. Kind regards,...
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### Evaluating the integral $\int_{-1}^{1} \frac{1}{(1+x^{2})(1-x^{2})^{1/4}}dx$

I've been trying to find a way to integrate $\int_{-1}^{1}\frac{1}{(1+x^{2})(1-x^{2})^{1/4}}dx$ using contour integration, but I'm having a hard time coming up with a contour to use. Since I have a ...
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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 ...
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### how to evaluate this definite integral $\int_0^\infty\frac{\sin^2(x)}{x^2}dx$? [duplicate]

For $\int_{0}^{\infty}\frac{\sin^2(x)}{x^2}dx$. I considered using residue theorem. But since the function inside is holomorphic except for a removable singularity at the origin. So whatever contour I ...
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An old complex analysis exam question: Evaluate $$\large I(a) = \int_{-\infty}^{\infty} e^{-\frac{1}{2}x^2+iax}dx$$ So far, I have completed the square in exponent, and now I have the integral $$... 3answers 57 views ### Residues of poles Find Res_{f}\left ( z_{0} \right ), where, f\left ( z \right )=\frac{1}{z^{4}+4}, for z_{0}=1+i Now, the definition for$$Res_{f}\left ( 1+i \right ) =\lim_{z \to z_{0}} \left\{\left ( z-\left ...
Title says it all. I'm noticing a trend of failure on Mathematica/WolframAlpha's parts when trying to compute either the Laurent expansions or the residues of functions like $\sin\dfrac{1}{z}$ about \$...