Questions tagged [improper-integrals]
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 $\infty$ or $-\infty$, or as both endpoints approach limits.
7,906
questions
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Estimation of a gamma function-like integral
A random variable $X$ has a pdf: $$f(x) = \frac{1}{k!} \cdot x^k \cdot e^{-x}$$
Prove that $$P(0<X<2\cdot(k+1)) > \frac{k}{k+1}$$
There are no conditions about $k$, so it can be non-integer.
...
- 339
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0
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For which values of the parameter $S$ is the integral convergent?
I feel like I should take lower limit and I did but I couldn't proceed. I need help. Should I use some comparison test 1
$$\int \limits^{2}_{1} \frac{(1- \cos \pi x)(x-1)^S}{x^3 -1} dx$$
1
vote
2
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Laplace transform of $\sin(\omega t)$
I am learning about the Laplace transform and I know I got the answer to this example question wrong, but I'm trying to figure out if I just made a calculus or algebra type error, or if I'm ...
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2
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Can we convert the following integral equation to a differential equation:$h(r)= \int_0^\infty\frac{f(x)}{e^{r x} + 1} dx$?
Can we convert the following integral equation to a differential equation:$$h(r) =\int_0^\infty\frac{f(x)}{e^{r x} + 1 } dx?$$ Here, $f(x)$ is a non-trivial 'nice' function( whatsoever condition is ...
- 373
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Is the Riemann Liouville fractional integral compact operator? [closed]
I am about to figured out that is the Riemann Liouville fractional integral compact operator or not? where f is continuous function in [0, b].
0
votes
2
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53
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Integral of a Generalized Laguerre Polynomial [closed]
I am looking for the solutions to the following integral:
$$
I_{n} =
\int_{0}^{\infty}x^{4}
\operatorname{L}_{n}^{3}\left(x\right)
{\rm e}^{-\left(n + 3\right)x/2}\,{\rm d}x,\qquad n \in\mathbb{N}_{0}
...
- 19
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votes
0
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Finding a general expression for the improper integral $\int_0^\infty K_1( ( k^2+\alpha^2)^{1/2}r)\sin(kz)\,\mathrm{d}k$
$\newcommand{\on}[1]{\operatorname{#1}}$
In solving a classical fluid mechanics problem involving flow in porous media, I encountered a delicate infinite integral ...
1
vote
0
answers
57
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How do I evaluate $\int_0^1 \int_0^\infty \frac{\ln t}{1 + \alpha t^2} \mathrm{d}t \, \mathrm{d}\alpha$? [closed]
How would I evaluate the integral
$$
\int_0^1 \int_0^\infty \frac{\ln t}{1 + \alpha t^2}\, \mathrm{d}t \, \mathrm{d}\alpha?
$$
- 49
1
vote
0
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An integral identity involving a generalized hypergeometric function.
Let $\theta \ge 0$, $S\ge 0$, $\zeta \ge 0$ and $x \ge 0$ be real numbers and let $q \ge 1$ be an integer. Then the following identity below holds true:
\begin{equation}
G_\zeta(x) := \int\limits_0^\...
- 11.5k
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2
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Could we approximate $\int_0^1\frac{1}{x^4}dx$ using a Riemann sum?
We know that in one dimension, the integral $\int_{0}^{1}\left(1/x^{4}\right){\rm d}x$ is not finite. But could we approximate this integral using a Riemann Sum ?.
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1
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Prove the closed form of $\int_0^T \exp\left(\frac{ia}{T-\tau}+\frac{ib}{\tau}\right)\frac{d\tau}{\left[\sqrt{(T-\tau)\tau}\right]^3}$.
While working through the Dover book "Quantum Mechanics and Path Integrals", I stumbled across a problem requiring me to use the following identity given in the book's appendix.
$$\int_0^T \...
- 78
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votes
1
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How do I find an equivalent for this integral?
I am trying to derive concentration bounds for the spectral norm of some rank-one matrices under Gaussian measure. My objective is to obtain a bound with respect to both the number of samples $N$ and ...
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3
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how to solve improper integral
$$\int^{a}_{l}\frac{a^3 - x^3}{(a^2 - x^2)^{3/2}} {\rm d}x$$
I have obviously separated the integrals into
$$\int^{a}_{l}\frac{a^3}{(a^2 - x^2)^{3/2}} {\rm d}x$$
$$\int^{a}_{l}\frac{- x^3}{(a^2 - x^2)^...
- 31
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1
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Sufficient condition for integrability
Let $f : (a,b)\to\mathbb R$ be a continuous function. Is is integrable, in general? What if the domain was $[a,b]$? If it's only generally integrable in the second case, is it because of uniform ...
- 610
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Integral divergence implies summation divergence
Assume $f:(0,1)\times (0,1) \to \mathbb{R}$ is a nonnegative continiuos bivariate function such that
$$\int_0^1 \int_0^1 f^2(x,y) dx dy = \infty,\quad \int_0^1 \int_0^1 f(x,y) dx dy = 1.$$
Can we ...
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