# Questions tagged [contour-integration]

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

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### Contour Integral with square root

I'm a master degree theoretical physics student and while working on my thesis I've encountered the following guy: $$\int_0^{\infty}dx\frac{e^{-ax^2+ibx}}{\sqrt{x}}$$ with $a,b>0$. I wanted to ask ...
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### Contour integration of $\int_{0}^{+\infty}\frac{x^2\cos x}{\cosh x} \,{\rm d}x$

Using contour integration, find $$\int_{0}^{+\infty}\frac{x^2\cos x}{\cosh x} \,{\rm d}x$$ How to calculate it? I never worked with integrals of this type.
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### Evaluate $\int_0^{\infty}\frac{\log( x)}{x^2+a^2} \,dx$ using contour integration; Re a > 0

Evaluate $\int_0^{\infty}\frac{\log( x)}{x^2+a^2} \,dx$ using contour integration; $Re (a) > 0$ I found two questions where a > 0 but in my case I have the following condition: Re a > 0 (It ...
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### Help in understanding a clever proof of Rouché's theorem

Theorem: $f, g$ are analytic in a region $\Omega . C$ is a circle/rectangle/(simple closed curve) which along with its interior is contained in $\Omega$. Suppose that $|f|>|g|$ on $C$. Then $f$ and ...
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### Evaluation of integral with double arctan

How can we evaluate this integral? $$\int _0^{\pi }\arctan \left(\frac{\sin x}{a+\cos x}\right)\arctan \left(\frac{\sin x}{b+\cos x}\right)dx$$ I have solved this problem for $a,b\ne (-1,1)$ my result ...
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### How can I evaluate this contour complex integral?

Question: $$\oint_{c}^{} \frac{e^{{z}^{2}}}{z-2}dz$$ And the contour is this figure: Now it would have been great if they had defined the contour but no such luck. The only hint is that I can assume ...
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### How does the divergent sum $\sum_{n=1}^\infty\cos(2n\gamma)\sin(2nt)$ correctly evaluate an integral? Surely distributions don’t apply here

$\newcommand{\d}{\,\mathrm{d}}\newcommand{\res}{\operatorname{Res}}$Note: I don’t know any distribution theory myself, but I was informed by someone else and hinted to by this answer that my problem ...
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### Computing a contour integral for ranges of $r$

Compute $$\int_{|z|=r}\frac{e^{\sin(z^2)}}{(z^2+1)(z-2i)^3}\;dz$$ when $0<r<1,1<r<2$ and $r>2$. My attempt: For $r<1$, the integrand is holomorphic and by Cauchy-Goursat the ...
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