# Tagged Questions

Questions on the evaluation of integrals along a locus in the complex plane.

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### Evaluating $\int_c\frac{1}{\sin\frac{1}{z}}\text{d}z$ over $C= \{z\big\vert|z|=\frac{1}{5}\}$

Evaluate $$\int\limits_{|z|=\frac{1}{5}} \frac{1}{\sin\frac{1}{z}}\text{d}z$$ My attempt: I know that this function has non isolated singularity at $0$, and simple poles at $\frac{1}{n \pi}$. ...
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### Prove that a complex-valued entire function is identically zero.

Suppose $f$ is entire and $$\iint_\mathbb{C}|f(z)|^2dxdy < \infty$$Prove that $f\equiv 0.$ So far I have: Suppose $f$ is bounded. Then $f$ is constant by virtue of Liouville and so the ...
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### Contour Integral finding Poles

I have an integral to solve using appropriate contour integrals; The question is like this. $$\int\limits_0^{2\pi}\frac{\cos(n\theta)d\theta}{1+2p\cos(\theta)+p^2}$$ $$-1<p<1$$ So I thought ...
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### Contour Integral of $1/(zcos(z))$ around the circle $\lvert z \rvert = n\pi$

This is part of a past exam question from a second year undergraduate complex variable theory course. I am attempting to show that the integral in the title tends to zero as n goes to infinity. The ...
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### Complex analysis contour integration calculation check if it is right

Calculate $\int{ \frac{e^z}{z^2(z^2+3)}}dz$ over the rectangle $x=2,x=-2 ,y=2,y=-2$. What i did is find the roots of $z^2+3$ break the $\frac{1}{z^2+3}$ into $\frac{-i/6}{z-3i} +\frac{i/6}{z+3i}$ ...
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### Evaluating $\int_{0}^\infty \frac{\log x \, dx}{\sqrt x(x^2+a^2)^2}$ using contour integration

I need your help with this integral: $$\int_{0}^\infty \frac{\log x \, dx}{\sqrt x(x^2+a^2)^2}$$ where $a>0$. I have tried some complex integration methods, but none seems adequate for this ...
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### How to find $L = \int_0^1 \frac{dx}{1+{x^8}}$

Let $L = \displaystyle \int_0^1 \frac{dx}{1+{x^8}}$ . Then $L < 1$ $L > 1$ $L < \frac{\pi}{4}$ $L > \frac{\pi}{4}$ I got some idea from this video link. But got stuck while evaluating ...
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### Evaluating the real integral $\int_{0}^{2\pi}\frac{1}{2+\sin\theta}d\theta$ using complex analysis

I thought it's value would be zero, since the complex integrand: $$\Im\left(\int_{C}\frac{1}{2+e^{i\theta}}d\theta\right)$$ Where $C$ is the unit disc, is nonsingular. Also $e^{iz}\ne -2$ for any $z$...
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### Contour Integration along a line segment

This may sound like a silly question, but was just wondering if someone can clear this up for me Consider the line segment joining the points a,b. Therefore we have f(t)=a+t(b-a) where t is ...
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### Calculating Infinite Real Integrals Using Residues

I want to calculate the following real integral using residues and I am unsure how to proceed. $$\int_{-\infty}^{+\infty}\frac{1- x^2}{1+ x^4} dx$$ I know I must change this to a contour integral so ...
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### What is the method to use the generalised Cauchy Integral Formula

Past Paper Question: a) State the generalized form of Cauchy’s integral theorem b)Evaluate $$\displaystyle f(z)=\int_{\gamma}\frac{z^2}{\biggr(z-\dfrac{\pi}{4}\biggl)^3} dz$$ where $\gamma$ is ...
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### $\int \frac{\cos z}{z(z+2)}\mathop{\mathrm{d}z}$

$$\int \frac{\cos z}{z(z+2)}\mathop{\mathrm{d}z}$$ traversing the unit circle counterclockwise. So the singularities are $z=0$ and $z=-2$ but the second is outside the unit circle so it isn't ...
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### $\int \frac{2+\sin(z)}{z} dz$

Please bear in mind that I am trying to teach myself complex integration having never taken a course in complex analysis, so assume I know very little. $\int \frac{2+\sin(z)}{z} dz$ traversing the ...
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### Splitting a contour integral into separate integrals?

Suppose you had a complex contour integral of the form $$\int_{\alpha + \beta} f(z) \; dz$$ Is this equivalent to $$\int_{\alpha} f(z) \; dz + \int_{\beta} f(z) \; dz$$ Thanks
The question come from a Summation like this $${ \sum _{ { z=i\omega }_{ n } } { \frac { -\alpha E\pi }{ 4{ z }^{ 3 }\sqrt { -\alpha -z } } } }$$ I can use Cauchy theorem to transform it to a ...