# Tagged Questions

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### Complex contour integral with sign function:$-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dp$

I am trying to evaluate the integral: $-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dx$ with sgn$(x)$ the sign function and $a$ positive real. Naively applying 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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### Computing $\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t$

I'd like to calculate the following integral on the interval $[0,2\pi]$: $$I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi.$$
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### Integral Using Harmonic Functions

Evaluate the integral: $$\int^{2 \pi}_0 \dfrac{\cos^2 \theta}{|2e^{i\theta}-z|^2} \, d \theta \qquad \mbox {when} \, |z| \neq 2.$$ Now, I thought about trying to change this to look like a Poisson ...
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### Quarternionic Analysis

What is/are the current understanding/opinions about Quarternionic Analysis as a generalization of Complex Analysis with respect to a "Quarternionic Residue Calculus" (if such a thing exists)? i.e. ...
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### Integrating $\int_0^\infty\frac{\log (1+z^2)}{e^z-1}dz$ using residue calculus.

I've been looking at how to integrate the following definite integral using the residue calculus, but can't seem to get my thoughts together. I know the $\log$ term is a multivalued function and the ...
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### Integrating $t^{2r-1} / t^{2k} (1+t^2)^{r+1}$

Let $k$ and $r$ be natural numbers such that $1 \leq k \leq r$. I want to calculate $$\int_0^\infty \frac{t^{2r-1}}{t^{2k}(1+t^2)^{r+1}} dt.$$ Since the integrand is an odd function the standard ...
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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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### Since $A(\alpha)=\int_0^{2\pi}\,d\theta\,\,\frac{a-i\cdot{b}\cos(\pi+\theta+\alpha)}{c-i\cdot{d}\cos(\pi+\theta+\alpha)}$, is $A(0)=A(\pi/5)$?

I would like to understand if the result of following integral $$A(\alpha)=\int_0^{2\pi}\,d\theta\,\,\frac{a-i\cdot{b}\cos(\pi+\theta+\alpha)}{c-i\cdot{d}\cos(\pi+\theta+\alpha)}$$ is or not ...
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### Integrating $\int_{0}^{\infty} \frac{(\log x)^2}{x^2+x+1}$ using residue theorem [duplicate]

Just out of curiosity, how does one integrate something like this using residue theory? $$\int_{0}^{\infty}\frac{(\log x)^2}{x^2+x+1} dx$$ According to Wolfram Alpha, the answer is ...
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### Hint to compute the following integral

Can someone give a hint on how to solve the following integral? $$\int_0^{2N\pi}\frac{(-R\cos t)(\xi t-r)+\xi R\sin t}{(R^2+(\xi t-r)^2)^{3/2}}dt$$ I've tried some substitutions. First I've splitted ...
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### Determine the integral $\int_0^\infty \frac{\mathrm{d}x}{(x^2+1)^2}$ using residues.

Determine the integral $$\int_0^\infty \frac{\mathrm{d}x}{(x^2+1)^2}$$ using residues. This is from Section 79, Brown and Churchill's Complex Variables and Applications. In order to do this. We ...
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### Evaluate $\int_0^\tau \frac{t\sin(t z)}{z\cos(t z)-\sin(tz)}\text{d}t$

I'm trying to evaluate the following definite integral. Mathematica gives me a complicated expression which I think I can simplify, but I was wondering if there was a "nice" way to evaluate it. ...
### $\int_0^{2\pi}e^{a \cos{\theta}}\cos({\sin{\theta}})\,d\theta$ using residues
How do I find the following integral by converting it into a complex integral and then using residue theorem? $$\int_0^{2\pi}e^{a \cos{\theta}}\cos({\sin{\theta}})\,d\theta$$ My approach is as ...