Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞, or as both endpoints approach limits.
1
vote
2answers
66 views
Laplace transforms please help
I really need help to find to Laplace transforms of $f(x)=x+e^{-x}$, and $g(x)=xe^x$. I'm having big troubles on the calculations. Thanks.
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 ...
4
votes
1answer
77 views
Analysis on Improper Integrals
This question is from Munkres' Analysis on Manifolds, section 15 question 1.
Let $f: \mathbb{R} \to \mathbb{R}$ be the function $f(x) = x$. Show that, given $\lambda \in \mathbb{R}$, there exists a ...
4
votes
1answer
133 views
Double integral $\int_{z=u}^{+\infty}\int_{t=u}^{+\infty}\frac{e^{-Az}}{z+B}\frac{te^{-tD}}{t-zC}\,dtdz$
I am doing research, and while calculating a closed form expression, I got a form of integration like the following:
...
2
votes
1answer
53 views
Improper Integral Question
Express $$\int_0^1x^m(1-x^n)^pdx$$ in terms of gama function and hence evaluate the integral.
I used the substitution $x^n=y$ and solving got this integral as equal to the beta-function
$${1\over ...
2
votes
1answer
58 views
Calculus - improper integrals
I have a few questions from my h.w, I hope someone can help me.
the question is:
$f:[a,\infty) \rightarrow \mathbb{R}$ is a continuous and periodic function, with period of $T>0$ .
...
1
vote
1answer
30 views
Inverse Laplace Transform. Computing the integral.
This question is related to this one, but I'm hereby taking a different approach.
Problem: Solve
$\ddot x+\delta\dot x+\omega_0^2x=\gamma\cos\omega t$.
Find the stationary points and examine their ...
0
votes
1answer
69 views
Improper integral $\int_{0}^{\infty}\frac{x^n}{x^{m+n+1}} \ dx=\frac{n! {(m-1)}!}{(m+n)!}.$
How can I prove that
$$\int_{0}^{\infty}\frac{x^n}{x^{m+n+1}} \ dx=\frac{n! {(m-1)}!}{(m+n)!}\quad ?$$
I tried to do induction on $n$ and on $m$, separately, but I could only do the base case ($n=1$ ...
0
votes
1answer
36 views
is f improperly integrable if g is not
$ f,g $ are nonnegative and locally integrable on $ [a,b) $ and
$ L := \lim_{x\to b-}\frac{f(x)}{g(x)}\ $ exists as extended real number.
If $ 0 < L \le \infty $ and $g$ is not improperly ...
0
votes
1answer
47 views
Problem on Yukawa Potential
One definition of the Yukawa potential on $R^n$ is the solution $G$ in the sense of distributions to $(-\Delta + \mu^2)G = \delta$. This 'green's function' is given by
\begin{align*}
G(x) = ...
5
votes
0answers
115 views
Evaluting $ \int_0^{\infty}\frac{v}{\sqrt{v + c}}e^{-\frac{y^2}{2(v + c)} - \frac{(u-v)^2}{u^2v}}dv$
While working on mixture (variance) of normal distribution and keep running into these two integrals
$$ \int_0^{\infty}\dfrac{v}{\sqrt{v + c}}e^{-\dfrac{y^2}{2(v + c)} - \dfrac{(u-v)^2}{u^2v}}dv,$$
...
4
votes
0answers
193 views
Computing complex principal value integral - sgn-function?
I currently face a less appealing integral which emerged computing the expectation of some random variable. It reads as (omitting all unnecessary constants except $\alpha\in(0,1)$)
$$ PV ...
3
votes
0answers
30 views
What is $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$?
I want to compute the following integral:
$\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$
with $\alpha, \beta, c$ real constants, and $\alpha>0,\beta=0$.
...
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:
$$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} ...
3
votes
0answers
163 views
2 dimensional Fourier transform integral
I'm trying to calculate the two dimensional Fourier integral
$$\iint \mathrm d\vec{R} \; (x^2 + y^2) \; e^{-2 \sqrt{ x^2 + y^2 + z^2}} \; e^{i\vec{K}\vec{R}} \;,$$
with $\vec{R}=(x,y)$. Switching to ...
2
votes
0answers
36 views
computing a difficult integral using software
I'd like to compute the following integral. I've tried SAGE but it just runs for 15 minutes then stops.. not sure what that means. If anyone wants to take a crack with mathematica or anything, please ...
2
votes
0answers
83 views
Integral form of $2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$
Being inspired by this post, I've wondered if the infinite series below may be expressed as
an intregral. I'm very curious about that.
$$2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$$
...
2
votes
0answers
117 views
Closed form of the series ,$\sum_{k=0}^{\infty}\frac{(-1)^k k!}{(k+1)^{(k+1)}}x^k$
$x,y>0$
$$f(x,y)=\int_{0}^{\infty} \frac{1}{xt+e^{y t}} dt$$
if $x=0$ then $f(0,y)=1/y$
$$f(x,y)=\int_{0}^{\infty} \frac{1}{e^{y t}(1+xte^{-y t})} dt=\int_{0}^{\infty} \frac{e^{-y ...
2
votes
0answers
95 views
Convergence of Riemann-like sums
I am trying to find suitable conditions (integrability, growth...) on a function $f:\mathbb{R}\to \mathbb{R}$ such that:
\begin{equation}
\sum_{k\in\mathbb{Z}}f(kh)h= \mathcal{O}(1),\qquad h\to 0^+.
...
1
vote
0answers
29 views
Is $f$ integrable, in the Darboux sense, on $[0,1]$?
Is
$$
f(x)=
\begin{cases}
0,\quad &x=0,\\
x\sin x,&x>0,
\end{cases}
$$
integrable, in the Darboux sense, on $[0,1]$?
I know the Darboux integral has to do with the upper sum and lower sum ...
1
vote
0answers
67 views
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 ...
1
vote
0answers
164 views
Prove this integral (about gamma function)
Prove that :
$$I\left( a\ ,\ b \right)=\int_{0}^{\infty }{\frac{{{x}^{a-\frac{3}{2}}}}{{{\left( {{x}^{2}}+\left( {{b}^{2}}-2 \right)x+1 \right)}^{a}}}\text{d}x}={{b}^{1-2a}}\frac{\Gamma \left( ...
1
vote
0answers
59 views
Analyzing the convergence of an improper integral
I have to analyze the convergence of
$$\int _{0}^{+\infty} \frac{\cos \left( x\right) -1} {x^{5 / 2}+5x^{3}}\,dx$$
I've rewritten the integral as
$$
\int _{0}^{+\infty} \frac{\cos \left( x\right) ...
1
vote
0answers
100 views
comparison test for improper integrals
Let $f$ and $g$ be continuous functions on $(a,b)$ such that $0 \le f\left( x \right) \le g(x)$ for all $x \in \left( {a,b} \right)$; $a$ can be $ - \infty $ and $b$ can be $ + \infty $.
Prove: ...
1
vote
0answers
48 views
how to calculate $\int_{[0,1]}x^{-\alpha}dx$
$0 <\alpha <1$,prove $x^{-\alpha}\in L([0,1])$,and calculate
$$\int_{[0,1]}x^{-\alpha}dx$$
Here is my method. We only need to show that $\int_{[0,1]}x^{-\alpha}dx \leq \infty$.
...
1
vote
0answers
190 views
Improper integral $\int^{\frac{\pi}{2}}_0 (\operatorname{csc} x - \frac{1}{x})\, dx$
$\def\cosec{\operatorname{csc}}$
Calculate the value of the improper integral $$\int^{\dfrac{\pi}{2}}_0\left (\cosec x - \dfrac{1}{x}\right)\, dx.$$
You may use the standard integral
$\int \cosec x \, ...
1
vote
0answers
72 views
What is suitable contour shape for $\int_0^\infty\dfrac{b^2+2ab+k}{b(b^2+ab+l)}e^{bx}~db$
$\int_0^\infty\dfrac{b^2+2ab+k}{b(b^2+ab+l)}e^{bx}~db$ . What kind of contour is suitable for this integral?
1
vote
0answers
89 views
Closed-form expression of the following double integral
How can I find closed-form expression of the following double integral
$$\int_0^{\frac{\pi}4} \int_0^\infty \frac{dr \, d\phi}{ u_1^2 + u_2^2 r + 2 u_1 u_2 \sqrt{r} \cos \phi}?$$ Please help me as ...
1
vote
0answers
75 views
Asymptotics of an integral
Consider an integral
$$
I(x) = \int\limits_{\mathbb{R}^n} e^{i\xi x } \delta(\xi^2-k^2)\chi( (\xi-k,\gamma)) \, d\xi
$$
where $x, k, \gamma \in \mathbb{R}^n$, $|\gamma| = 1$ and $(x,y)\equiv xy = ...
1
vote
0answers
140 views
Integrating the exponential and the logarithmic function together.
Ok, I want to find
$$\int\limits_0^t {{e^u}\log udu} $$ and $$\int\limits_0^t {{e^{ - u}}\log udu} $$
I'm thinking as follows
$$d\left( {{e^u}\log u} \right) = {e^u}\log udu + ...
1
vote
0answers
111 views
Improper integrals over the square and modulus square of an associated Legendre function
I am trying to evaluate integrals of the type
$$\int dz\, P^{\,\mu}_\nu(z)^2 \qquad \mathrm{and} \qquad \int dz\, \left|P^{\,\mu}_\nu(z)\right|^2$$
where $P^\mu_\nu$ are associated Legendre functions. ...
0
votes
0answers
33 views
Logarithmic accuracy
Does anyone know how the method of logarithmic accuracy works and what do I have to know about it (as far as applied Mathematics is concerned)? Any references, examples or guidelines would be ...
0
votes
0answers
98 views
Interesting integral related to floor function
Problem.
Evaluate $\displaystyle F(n, k):=\int_{0}^{1} \frac{1-\{1/x\}^n}{1-\{1/x\}^k}dx$ where $n$, $k$ are positive integers. ($\{x\}=x-\lfloor x\rfloor$)
Someone proposed this interesting problem ...
0
votes
0answers
51 views
Improper or Undefined
Let $f(x)=0 $ if $x\neq 1 $ and $f(1)=\infty $ then the Riemann integral
$\int_{0} ^1 f(x)$ $ dx $ = $ 0 $ or is it undefined?
If we take it as a legitimate function for improper Riemann ...
0
votes
0answers
60 views
a few questions about the convergence of improper integrals
Let $f(x)$ be a continuous function on $[1,\infty)$ that oscillates between positive and negative values and tends to $0$ as $x$ approaches $\infty$. Will $\int_{1}^{\infty} f(x) \ dx $ always ...
0
votes
0answers
61 views
About Laplace transform
I dont understand the following working, why the integral becomes double integral?
$$\begin{align}
& \ \ \ \int_{0}^{1}{{{\left( \frac{1}{\ln x}+\frac{1}{1-x} ...
0
votes
0answers
176 views
how integrate $\int\ln(1+\tan x)\,dx$ or $\int\ln(\sin x)\,dx$
I integrate by part I assume $dx=dv$ and $\ln(\sin x)= u$ or
I compute by Maple but its answer wasn't clearly (Maple answer: ...
0
votes
0answers
132 views
exponential times polynomial, double integration problem
Could any one help me to solve this problem?
$$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}x^my^n(x^2+y^2)^l \exp\left[\frac{-(x^2+y^2)}{4\sigma^2}\right]\sin(gx)\sin(gy) \, dx \, dy$$ where ...
0
votes
0answers
229 views
0
votes
0answers
52 views
How to estimate the integral growth rate
Let $D$ be a compact in $\mathbb{R}^n$ and $f(x,y,z)$ be a function from $D \times D \times D$ to $\mathbb{R}$ such that it has singularity if $x = y = z$ and it is continuous otherwise. How to ...
