Tagged Questions
4
votes
1answer
82 views
Integrate: $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$
Can this integral be solved with contour integral or by some application of Residue theorem?
$$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catlan constant}$$
It has two ...
0
votes
3answers
77 views
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 ...
3
votes
3answers
137 views
Integral $\int_0^\infty \exp(ia/x^2+ibx^2)dx$
Compute the integral:
\begin{equation}
\int_0^\infty \exp\left(\frac{ia}{x^2}+ibx^2\right)\,dx
\end{equation}
for $a$, $b$ real and positive. I tried complex variables, but don't really know how to ...
1
vote
2answers
111 views
Evaluate $\int\limits_0^\infty \frac{\cos(ax)}{\cos(bx)}\frac{1}{1+x^2}dx$
I would like to show that
$$\text{PV}\int_0^\infty \frac{\cos(ax)}{\cos(bx)}\frac{1}{1+x^2}dx = \frac{\pi}{2}\mathrm{sech}(b)$$
using complex analysis. $a$ and $b$ are real numbers and $a \neq b$.
...
0
votes
2answers
66 views
Improper integral of a rational function whose denominator is of degree at least two greater than that of the numerator
There's a technique in complex analysis (involving residue calculus) to solve the improper integral (from $-\infty$ to $\infty$) of a rational function whose denominator is of degree at least $2$ ...
1
vote
1answer
90 views
Existence of Riemann-Liouville Integral
The Riemann Liouville integral is defined as:
$\frac{1}{\Gamma\left(\nu\right)}\int\limits _{h}^{t}\left(t-\xi\right)^{\nu-1}f\left(\xi\right)d\xi$
It is supposed it does exist for all $\nu>0$ and ...
3
votes
0answers
115 views
Is there a closed form expression for this integral?
I've been trying to find a closed form expression/series expansion for the following integral without success:
$$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} ...
3
votes
1answer
115 views
Integral $\int_{-\infty}^{\infty}\frac{e^{r \arctan(ax)}+e^{-r \arctan(ax)}}{1+x^2}\cos \left( \frac{r}{2}\log(1+a^2x^2)\right)dx$
How can I show that
$$\int_{-\infty}^{\infty}\frac{e^{r \arctan(ax)}+e^{-r \arctan(ax)}}{1+x^2}\cos \left( \frac{r}{2}\log(1+a^2x^2)\right)dx = 2\pi \cos \left( r\log(1+a)\right)?$$
$a \in ...
1
vote
1answer
164 views
Integral $\int_{-\infty}^{\infty}\frac{\cos(s \arctan(ax))}{(1+x^2)(1+a^2x^2)^{s/2}}dx$
Prove that:
$$\int_{-\infty}^{\infty}\frac{\cos(s \arctan(ax))}{(1+x^2)(1+a^2x^2)^{s/2}}dx = \frac{\pi}{(1+a)^s}$$
where $a,s \in \mathbb{R}^{+}$.
This looks difficult. What would be a good start? ...
2
votes
0answers
83 views
Integral form of $2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$
Being inspired by this post, I've wondered if the infinite series below may be expressed as
an intregral. I'm very curious about that.
$$2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$$
...
1
vote
2answers
124 views
Calculating $\int_{-\infty}^{\infty}\frac{\sin(ax)}{x}\, dx$ using complex analysis
I am going over my complex analysis lecture notes and there is an
example about calculating $$\int_{-\infty}^{\infty}\frac{\sin(ax)}{x}\, dx$$
that I don't understand.
The solution in the notes ...
4
votes
1answer
59 views
evaluate $\int_0^\infty \dfrac{dx}{1+x^4}$ using $\int_0^\infty \dfrac{u^{p-1}}{1+u} du$
evaluate $\int_0^\infty \dfrac{dx}{1+x^4}$using $\int_0^\infty \dfrac{u^{p-1}}{1+u} du = \dfrac{\pi}{\sin( \pi p)}$. I am having trouble finding what is $p$. I set $u = x^4$, I figure $du = 4x^3 dx$, ...
1
vote
1answer
113 views
How does this calculation of $\int_0^\infty(\sin^2x)/x^2\ dx$ work?
I'm having trouble with two steps in a calculation of
$$\int_0^\infty\left(\frac{\sin x}{x}\right)^2\ dx$$
in a book.
They take the contours $C_R$ composed of upper half-circles ...
4
votes
3answers
192 views
$\int_0^\infty\frac{\log x dx}{x^2-1}$ with a hint.
I have to calculate $$\int_0^\infty\frac{\log x dx}{x^2-1},$$
and the hint is to integrate $\frac{\log z}{z^2-1}$ over the boundary of the domain $$\{z\,:\,r<|z|<R,\,\Re (z)>0,\,\Im ...
6
votes
1answer
113 views
Is this a correct way to calculate $\int_{-\infty}^\infty\frac{x\sin(\pi x)}{x^2+2x+5}dx?$
I have this integral to calculate: $$I=\int_{-\infty}^\infty\frac{x\sin(\pi x)}{x^2+2x+5}dx.$$
I think I have done it, but I would like to make sure my solution is correct.
I take the function ...
8
votes
1answer
170 views
Proof of Cauchy's Beta Integral $\int_{-\infty}^\infty \frac{dt}{(1+it)^x(1-it)^y}$
The Cauchy's Beta Integral is given by
$$\int_{-\infty}^\infty \frac{dt}{(1+it)^x(1-it)^y}=\frac{\pi 2^{2-x-y}\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}$$
I would like to know how it is proved.
2
votes
1answer
59 views
Residue Calculus Integral computation
I ran into this problem when I was doing some residue computations.
For real $a\neq0$, compute,
$$I=\int_{-\infty}^{+\infty} \frac{e^{iax}}{(x+i)^3} $$
Be sure to treat both cases when $a<0, ...
1
vote
1answer
88 views
integration by parts/ improper integral question
This is from an old qualifying examination question.
Let $a>1$ be fixed. Show that
$$ \displaystyle A_N=\pi i a \int_1^N t^{a-\frac{3}{2}}e^{\pi i t^a} dt $$
converges to some complex number as ...
0
votes
1answer
85 views
Improper integration [duplicate]
Possible Duplicate:
How to evaluate these integrals by hand
I am trying to evaluate the following: $$\int_{-\infty}^\infty \frac{\cos x}{e^x+e^{-x}}\, dx$$ using the residue theorem but I ...
3
votes
3answers
162 views
Improper integration involving complex analytic arguments
I am trying to evaluate the following:
$\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a>1$ and $a \in \mathbb{R}$/
Any help will be much appreciated.
2
votes
1answer
104 views
Definite integral involving hyperbolic cosine
I have had no experience so far with hyperbolic functions so any help will be appreciated. This is on the chapter of complex integration but I would especially appreciate it if you could turn this ...
3
votes
2answers
112 views
Computation of a certain integral
I would like to compute the following integral. This is for a complex analysis course but I managed to around some other integrals using real analysis methodologies. Hopefully one might be able to do ...
1
vote
3answers
203 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 ...
9
votes
4answers
265 views
How to evaluate this integral $\int_{-\infty}^{+\infty}\frac{x^2e^x}{(1+e^x)^2}dx$?
I need to evaluate $$\int_{-\infty}^{+\infty}\frac{x^2e^x}{(1+e^x)^2}dx$$
I think the answer is $\frac{\pi^2}{3}$, but I'm not able to calculate it.
13
votes
5answers
803 views
Evaluating $\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$
I've been trying to evaluate the following integral from the 2011 Harvard PhD Qualifying Exam. For all $n\in\mathbb{N}^+$ in general:
$$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$
...
3
votes
2answers
106 views
Evaluating $\int_0^{\pi} \frac{\rho ie^{i\theta}d\theta}{1+{\rho^2 e^{2i\theta}}}$ as $\rho$ tends to $\infty$
I'm reading Boas' book and there's a part where she assumes that the following integral goes to $0$ as $\rho$ tends to $\infty$. The argument is that the numerator contains $\rho$ while the ...
1
vote
3answers
133 views
Vanishing of a certain improper complex integral
Suppose $f,g \in \mathbb{C}[z]$ are polynomials with $\text{deg } g - \text{deg } f \geq 2$. Is it true that $\int_{-\infty}^{\infty} e^{iz}f(z)/g(z) dx = 0$, where $z = x + iy$ and $y > ...
0
votes
3answers
93 views
Improper integration using complex methods
Sorry for my English if there is any mistake. The exercice for which I need help is the following:
Compute using complex methods:
$I=\int_1 ^\infty \frac{\mathrm{d}x}{x^2+1}$
i) Choose the complex ...
3
votes
2answers
341 views
Are Complex Substitutions Legal in Integration?
This question has been irritating me for awhile so I thought I'd ask here.
Are complex substitutions in integration okay? Can the following substitution used to evaluate the Fresnel integrals:
...
2
votes
2answers
905 views
Evaluate improper integral $(\cos(2x)-1)/x^2$
Consider the following improper integral:
\begin{equation}
\int_0^\infty \frac{\cos{2x}-1}{x^2}\;dx
\end{equation}
I would like to evaluate it via contour integration (the path is a semicircle in ...
0
votes
1answer
411 views
How to calculate the principal part of improper integral?
How to calculate the principal part of this improper integral via contour integration?
\begin{equation}
P\int_{0}^{+\infty}\frac{dx}{x^2+x-2}
\end{equation}
I have seen some examples where you ...
6
votes
2answers
353 views
Taking the derivative under a principal value integral
I'm interested in showing that:
$$
\frac{d}{dt}P \; \int_{-\infty}^{\infty} \frac{\phi(x)}{x-t}dt = P \int_{-\infty}^{\infty}\frac{\phi(x)-\phi(t)}{(x-t)^2}dt
$$
where $\phi(x)$ is a test function ...
4
votes
0answers
193 views
Computing complex principal value integral - sgn-function?
I currently face a less appealing integral which emerged computing the expectation of some random variable. It reads as (omitting all unnecessary constants except $\alpha\in(0,1)$)
$$ PV ...
