# Tagged Questions

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### Solve $\mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ]$ using contour integration

I wish to evaluate $y(t) = \mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ]$, where $\mathscr{F}$ represents the Fourier transform, and U(t) represents the ...
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### Application of Residues

So in applying the residue theorem to solve improper real integrals, we agree to take our semicircles to be as large or as small as necessary such that all the poles we wish to work with lie inside ...
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### Convergence of improper real integrals using the residue theorem

Let $f:\mathbb{R} \to \mathbb{R}$ be a funktion that satisfies $$|xf(x)| \to 0 \qquad (|x| \to \infty)$$ If $f$ can be extended to a holomorphic function with finitely many singularities $S$ and the ...
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### Need help with $\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx$

I need you help with this integral: $$\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx.$$ Mathematica says it does not converge, which is apparently false.
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### Integration by Residue Theorem - Is This Integral equal to Zero?

$$\int_{-\infty}^{+\infty}\frac{\sin(x)}{2x^{5}-3jx^{3}+2x}dx=0$$ ...
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### Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$

I want to evaluate the following integral via complex analysis $$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a >0$$ Which function/ contour should I consider ?
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### Calculating $\int_0^\infty \frac {\sin^2x}{x^2}dx$ using the Residue Theorem.

I am trying to compute the following integral using the Residue Theorem but am quite stuck: $$\int_0^\infty \frac{\sin^2x}{x^2}dx$$ I have tried applying Jordan's lemma, having written $\sin(x)$ as ...
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### Integral $\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$

I want to solve the integral $$\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$$ Which function and contour should I consider ?
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### perturbative series expansion of integral via complex integration

Define for real $x>0$ and $\epsilon>0,$ the function $$f(x,\epsilon):= \int_{\epsilon}^\infty \frac{\mathrm{d}s}{s} \frac{1}{\sinh^2 s/2} e^{-sx}.$$ Question: is it possible to compute ...
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### Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Evaluate the integral $$\int_{-\infty}^{\infty} \frac{1}{(1 + x^2)^{n+1}} dx.$$ I know that it equals $2\pi i$(the sum of the residues; at $z_k$) where $z_k$ are the poles of the function. I ...
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### Integrating $\int_0^{\infty} \frac{dx}{1+x^3}$ using residues.

I want to calculate the integral: $$I \equiv \int_0^{\infty} \frac{dx}{1+x^3}$$ using residue calculus. I'm having trouble coming up with a suitable contour. I tried to take a contour in the shape ...
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### Evaluating a Real Improper Integral by Residues

I am having trouble evaluating this improper integral due to its integrand and the singularities that are present. The question reads as Show that ...
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### Improper integrals are “not totally Improper”

Question is to evaluate $$\int _{-\infty}^{\infty} \frac{dx}{(x^2+a^2)^2}\text {for } a>0$$ Idea is to calculate this using complex analysis/residue theory/contour integration. Approach is ...
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### Evaluating $\int_{-\infty}^\infty \frac{dx}{\cosh(x-a)\cos(2x)}$

I have been asked to evaluate $$\int_{-\infty}^\infty \frac{dx}{\cosh(x-a)\cos(2x)}$$. I'm deliberating on whether this indefinite integral exists or not. The integrand diverges when ...
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### How to calculate $\int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4}$?

I would like to calculate $$\int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4}$$ by means of the Residue Theorem. This is what I tried so far: We can define a path $\alpha$ that consists of half a ...
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### Finding a generalization for $\int_{0}^{\infty}e^{- 3\pi x^{2} }\frac{\sinh(\pi x)}{\sinh(3\pi x)}dx$

$\;\;\;\;$I was reading the introduction of Paul J. Nain's book "Dr. Euler's fabulous formula" where he talks about the sense of beauty in mathematics and quotes the G.N.Watson as saying that a ...
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### Cauchy principal value of $\int_{- \infty}^{\infty}e^{-ax^2}\cos(2abx) \,dx$

How do I find out the Cauchy Principal value of $\int_{-\infty}^{\infty}e^{-ax^2}\cos(2abx) \,dx\,\,\,\,\,\,\,\,a,b>0$ using complex integration? The answer is $\sqrt{\frac{\pi}{a}}e^{-ab^2}$, and ...
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### Improper integral equal to -pi with square root and Cauchy principal value

I'd like to know if the following proof for the value of $I$ is correct, and if there is a simpler solution to it. Also, I will probably encounter more improper integrals like this in the future, and ...
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### Integral $\int_0^\infty \exp(ia/x^2+ibx^2)dx$

Compute the integral: $$\int_0^\infty \exp\left(\frac{ia}{x^2}+ibx^2\right)\,dx$$ for $a$, $b$ real and positive. I tried complex variables, but don't really know how to ...
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### evaluate $\int_0^\infty \dfrac{dx}{1+x^4}$ using $\int_0^\infty \dfrac{u^{p-1}}{1+u} du$

evaluate $\int_0^\infty \dfrac{dx}{1+x^4}$using $\int_0^\infty \dfrac{u^{p-1}}{1+u} du = \dfrac{\pi}{\sin( \pi p)}$. I am having trouble finding what is $p$. I set $u = x^4$, I figure $du = 4x^3 dx$, ...
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### A generalized integral need help

I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$ What I do is use a Reciprocal subsitution, easy to show that: ...
I came across the following integral in a book (Kato's Perturbation Theory for Linear Operators, $\S$3.5): $\int_{-\infty}^\infty (a^2+x^2)^{-n/2}\,dx$ where $n$ is a non-negative integer and $a$ is ...