1
vote
1answer
38 views

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 ...
15
votes
3answers
214 views

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.
0
votes
1answer
38 views

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 $[0,2\pi]$: $$ I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi. $$
4
votes
1answer
143 views

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 ...
4
votes
1answer
48 views

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. ...
10
votes
0answers
226 views

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 ...
0
votes
4answers
51 views

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 ...
4
votes
4answers
205 views

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 ...
14
votes
2answers
367 views

Show $\int_0^{\pi/3} \big((\sqrt{3}\cos x-\sin x)\sin x\big)^{1/2}\cos x \,dx =\frac{\pi\sqrt{3}}{8\sqrt{2}}. $

I have run a FORTRAN code and I have obtained strong evidence that $$\int_0^{\pi/3} \!\! \big((\sqrt{3}\cos\vartheta-\sin\vartheta)\sin\vartheta\big)^{\!1/2}\!\cos\vartheta \,d\vartheta ...
7
votes
2answers
176 views

How to show $\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a})$? ($a\ge0$)

$$\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a}), \ a\ge0$$ I tried to solve but came up with $\pi(2-e^{-a}) $. Could you tell me where did I do the mistake? if $x=z$ then ...
1
vote
2answers
76 views

question on integrals

Let $\displaystyle A=\int_0^1 \frac{dx}{1+x^8}$. Then which of the following are true: 1) $A\lt 1$, 2) $A\gt 1$, ...
9
votes
1answer
223 views

Calculate $\displaystyle \int_0^\infty \frac{\ln x}{1 + x^4} \mathrm{d}x$ using residue calculus

I need to evaluate this integral using calculus of residues: $$\int_0^\infty\frac{\ln(x)}{1+x^4}\mathrm{d}x$$ I know I need to consider $\displaystyle ...
4
votes
2answers
167 views

Computing $\int_{-\infty}^\infty \frac{\sin x}{x} \mathrm{d}x$ with residue calculus

This refers back to the integral of $\frac{\sin(x)}x = \frac\pi2$ already posted. How do I arrive at $\frac\pi2$ using the residue theorem? I'm at the following point: $$\int \frac{e^{iz}}{z} - \int ...
3
votes
1answer
51 views

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 ...
2
votes
2answers
225 views

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 ...
3
votes
0answers
58 views

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 ...
1
vote
2answers
124 views

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 ...
2
votes
0answers
66 views

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. ...
2
votes
2answers
246 views

$\int_0^{2\pi}e^{a \cos{\theta}}\cos({\sin{\theta}})\,d\theta$ 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 ...
4
votes
1answer
131 views

Complex Integral using Residues

This is the question: Find the integral using residue theorem. $$\int_0^{2\pi}{d\theta \over1+8\cos^2\theta} $$ I solved it like this : $$\int_0^{2\pi}{d\theta \over1+8\cos^2\theta}=\int_0^{2 \pi} ...
0
votes
1answer
78 views

A question about the residue calculus

Suppose I have a convergent definite integral of the form $$\int_{-\infty}^\infty \frac{f(x)}{x^2(e^x-1)}\text{d}x,$$ where $f(x)$ has no poles, and I want to try to evaluate it using the residue ...
12
votes
5answers
555 views

Prove $\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}$

I need to show that $$ \int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3} $$ I have already derived the result $\int_0^\infty \frac{\sin^2x}{x^2} = \frac{\pi}{2}$ using complex analysis, a result ...
23
votes
2answers
731 views

Evaluation of $\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\,\mathrm{d}x$

I plan to evaluate $$\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\, \mathrm{d}x$$ and I need a starting point for both real and complex methods. Thanks ! Sis.
4
votes
1answer
194 views

Non elementary antiderivative of $\phi(\cos x,\sin x)$ when $\phi(x,y)$ is a rational real function?

With the method of Residues, we can calculate the integral \begin{equation}\int_{0}^{2\pi}\phi(\cos x,\sin x)\, dx \end{equation} where $\phi(x,y)=\frac{p(x,y)}{q(x,y)}$, ($p,q$ are polynomials of ...
1
vote
3answers
309 views

Evalulate $\int_{-\infty}^{\infty}\frac{1}{(1+x^{2n})^2}dx$ by using residue theorem

I know the answer of the integral $$\int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx=\frac{\pi}{n\sin\left(\frac{\pi}{2n}\right)}$$where $n\in\mathbb{N}$. But how to evalulate ...
4
votes
2answers
1k views

Computing $\int_{-\infty}^{\infty} \frac{\cos x}{x^{2} + a^{2}}dx$ using residue calculus

I need to find $\displaystyle\int_{-\infty}^{\infty} \frac{\cos x}{x^{2} + a^{2}}\ dx$ where $a > 0$. To do this, I set $f(z) = \displaystyle\frac{\cos z}{z^{2} + a^{2}}$ and integrate along the ...
7
votes
1answer
1k views

Residue integral: $\int_{- \infty}^{+ \infty} \frac{e^{ax}}{1+e^x} dx$ with $0 \lt a \lt 1$.

I'm self studying complex analysis. I've encountered the following integral: $$\int_{- \infty}^{+ \infty} \frac{e^{ax}}{1+e^x} dx \text{ with } a \in \mathbb{R},\ 0 \lt a \lt 1. $$ I've done the ...