All Questions
Tagged with integral or integration
1,506 questions
2
votes
2
answers
607
views
Integral calculus with Gamma function [closed]
I have to prove that for $0<\alpha<1$ and $\beta>0$,
\begin{equation}
\int_{0}^{\infty} x^{-\alpha-1}\left(e^{-\beta x}-1\right)dx=\beta^\alpha\Gamma(-\alpha),
\end{equation}
and I have ...
- 95
0
votes
1
answer
161
views
Sufficient conditions for L1 convergence of exponentials
Suppose $(X,B,m)$ is a finite measure space and $(f_n)$ is a
sequence of functions converging almost surely and in $L^2(X,m)$.
Moreover, I know that for every $n$, $\int e^{f_n(x)} m(dx)<\infty$. ...
- 650
-1
votes
1
answer
989
views
Random variable as an integral of an indicator function
This answer says that if $X$ is a random variable and $X_+ = \mathrm{max}(0, X)$, then $X_+ = \int_0^\infty I_{\{X > x\}}\mathrm{d}x$. I'd like to know how to derive this starting with $A \in \...
- 115
2
votes
1
answer
141
views
Injectivity of two sided Laplace transform
Let $\mu,\nu$ be finite Borel measures on $\mathbb R$.
Assume that there is an open interval $(a,b)$ on which the Laplace transforms exist and coincide:
$$
\int_{-\infty}^\infty e^{-tx}\,d\mu(x) = \...
- 769
0
votes
1
answer
222
views
Condition for $f^\prime$ to be absolute integrable
Suppose $f(x)$ is the probability density function of a random variable $X$, which means:
$$\int_{a}^{b} f(x) dx = 1$$
Also suppose $f$ is continuous and differentiable.
Provide a non-trivial ...
- 143
1
vote
1
answer
410
views
Takesaki lemma: existence Gelfand-Pettis integral
Consider the following fragment from Takesaki's second volume of "Theory of operator algebras" (lemma 2.4, chapter VI "Left Hilbert algebras").
In another post, it was explained ...
- 175
1
vote
0
answers
77
views
Integration over a finite-dimensional subspace of Hilbert space
Let $H$ be a separable Hilbert space with inner product $\langle,\rangle$, let $\{e_k\}_{k=1}^\infty$ be an orthonormal basis of $H$, and let $A: H\to H$ be a symmetric, positive definite and ...
- 503
1
vote
1
answer
137
views
Integral inequality implies majorization by solution of ODE
Let $f:[0, \infty)\to [0, \infty)$ be non-increasing (and not necessarily differentiable nor continuous) and satisfy $$f(t)\leq f(0)-C\int_{0}^{t}f(s)^{1/2}ds,$$ where $C>0$. How can one show that ...
- 459
7
votes
2
answers
398
views
Quantifying difficulty of integrals versus inverses
Recently, I have been discussing inverses with a tenth grade class and integrals with an eleventh/twelfth grade class, and this has led me to the following wonder:
Wonder. Is there a "reasonable&...
- 7,831
3
votes
0
answers
94
views
Question on an integral inequality
I am reading van de Vaart and Weller, Weak Convergence and Empirical Processes With Applications to Statistics. And I am stuck in the proof of Theorem 2.6.7 on page 141.
For simplicity I restae the ...
- 53
4
votes
0
answers
207
views
Integral inequality of Polya
In the book Math Problems AMM (1957), Problem 230, there is the next inequality of D. Polya:
let $a,b>0$, $0\leq x \leq a $,
$f(x)$ --- being not a linear function, and $f(0)=0$, $f(a)=b$, $f(x)\...
- 1,550
1
vote
2
answers
120
views
Integral inequality implies $f(t)\equiv 0$ for all $t\geq T$ for some finite $T>0$?
Let $f:[0, \infty)\to [0, \infty)$ be non-increasing and satisfy for all $t>t_{0}$, $$f(t)+C\int_{t_{0}}^{t}f^{\gamma}(s)ds\leq \frac{1}{t-t_{0}}\int_{t_{0}}^{t}f(s)ds,$$ where $0<\gamma<1$ ...
- 459
1
vote
0
answers
70
views
Continuity in the uniform operator topology of a map
I have a question concerning the continuity for $t>0$ in the uniform operator topology $L(X)$ of the following map: $$t\mapsto A^\alpha R(t)$$ where A is the infinitesimal generator of an analytic ...
- 39
0
votes
0
answers
124
views
Calculation of first correction to Selberg type integral
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\arcsinh{arcsinh}$Let $U \in G$, where $G$ is $\SU(N)$ matrix.
$\Tr U$ will denote the character ...
3
votes
1
answer
499
views
Does the integral $\int_0^{\infty}e^{cx^2+dx}dx/(a+bx)$ have a closed form?
The integral is
$$\DeclareMathOperator{\dm}{d\!}
\int_0^{\infty}\frac{e^{-cx^2+dx}}{a+bx}\dm x.
$$
Here I assume that $a,b,c,d$ are chosen to make this integral convergent. Rewritting the rational ...
- 375
5
votes
1
answer
322
views
Proving an integral identity
Let $ f : \left[ 0, 1 \right] \rightarrow \mathbb{R} $ be a continuous function. Knowing that: $$ \int_0^1 (2x-1)f(x)dx = 0 $$ Show that $ \exists c \in \left(0,1\right)$ such that: $$ \int_0^c (x-c)...
- 151
1
vote
0
answers
79
views
On an double integral involving Gaussian term
I want to calculate such an integral
$$
I=\int_0^{\infty}\int_0^\infty dx\,dy\ x^ny^m\exp(-ax^2-by^2-gxy+c_1x+c_2y),
$$
where $a,b,g>0$ are positive and real, $n,m$ are positive integers and $c_1,...
- 375
0
votes
1
answer
74
views
$\int_{\mathbb{R}}|p(v-r,x)-p(u-r,x)|\,dx \leq C\frac{v-u}{u-r}$
Consider $p(u,x)=(4\pi u)^{-d/2}e^{-\frac{|x|^2}{4u}},u>0,x\in \mathbb{R}^d.$
Prove that there exists $C>0$ such that for all $0<u\leq v,r\in[0,u[,$ $$\int_{\mathbb{R}^d}|p(v-r,x)-p(u-r,x)|\, ...
- 573
2
votes
1
answer
157
views
$\int_0^u\int_{[-1,1]^2}\int_{[-1,1]^2}\frac{1}{r}e^{-\alpha^2|x-y|^2/r} \, dx\,dy\,dr\leq Cu^{\epsilon}\alpha^{-2\beta}$
I am looking for a proof for the following fact: for $U>0,\beta>0,$ there exists $C>0,\epsilon>0$ such that $$\forall u\in [0,U],\alpha\in\left]0,1\right],\int_0^u\int_{[-1,1]^2}\int_{[-1,...
- 573
2
votes
2
answers
256
views
Is it possible to solve this integral?
I can't manage to solve this integral. Does it have an analytical solution?
$$\int\left(\frac{e^{x}(a-1)-1+\frac{1}{a}}{e^{x}(1-b)+1-\frac{1}{a}}\right)e^{-\frac{(x-(\mu-\frac{\sigma^{2}}{2})t)^{2}}{...
- 157
6
votes
3
answers
741
views
How do I solve the following definite integral (preferably by an asymptotic method)?
$$ \int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx $$
Note: $\mu$ here is an extremely small constant.
I have tried:
Estimating the integral by ...
- 69
1
vote
0
answers
213
views
How to compute the following integral of exponential function over an unit ball
I want to compute the value of the following integral:
$$\int_{r\in\mathbb{R}^d: \|\|r\|\|\le 1} \exp(a^Tr)dr$$.
In particular, $a$ is the coefficient and the norm $\|\|\cdot\|\|$ could be the general ...
- 21
1
vote
1
answer
190
views
Inequality and integral
Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$
Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$
...
- 573
3
votes
0
answers
283
views
Derivation of an integral containing the complete elliptic integral of the first kind
I found the following formula in "INTEGRALS AND SERIES, vol.3" by Prudnikov, Brychkov and Marichev (page 188, eq.5).
$$\int_0^{\infty} \frac{x^{\alpha-1}}{\sqrt{(a+x)^2+z^2}}K(\frac{2\sqrt{...
- 31
4
votes
1
answer
249
views
Does the analytical continuation of $\sum f(n) x^n $ always have a branch cut if $f(z)$ has a pole?
I suspect the answer to the title question is 'no', but I'm hoping to find an explicit counterexample. Also, I am requiring that $\sum f(n) x^n $ has a finite radius of convergence, otherwise, the ...
- 1,730
3
votes
1
answer
265
views
Integrating $\int_0^\infty \sqrt x e^{-4/3x^{3/2}}\left(\int_0^x \operatorname{Ai}(t)dt\int_0^x \operatorname{Bi}(t)dt\right)dx$
Show that
$$I= \int_0^\infty \sqrt x e^{\large -4/3x^{3/2}}\left(\int_0^x \operatorname{Ai}(t)dt\int_0^x \operatorname{Bi}(t)dt\right)dx$$
$$=\frac{1}{3}-\frac{\sqrt[3]{2\sqrt 3+3}+\sqrt[3]{2\sqrt 3-3}...
- 215
0
votes
1
answer
112
views
Integral and inequality
Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$
Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$
...
- 573
0
votes
1
answer
248
views
Integral with inequality
Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$
Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$
...
- 573
0
votes
2
answers
154
views
Perhaps an application of Hardy's inequality
Let $f \in H_{0}^{1}(0,1)$ and $\lambda >0$ big enough. Consider $0 <\alpha < 1$ and some $k > 0$. I would like to show the following inequality
$$
\int_{\lambda^{-k}}^{1}|f(x)|^{2}dx \leq ...
- 241
3
votes
0
answers
149
views
Monotonicity of a function defined by an integral
The question below is motivated by the related question Integral of a function changing sign and the associated answer:
Can we study the monotonicity of the following function on $(0,1)$?
$$\small f(x)...
- 395
0
votes
0
answers
88
views
Closed formula for iterated Fourier series
I'm trying to obtain a closed formula for the following integral.
\begin{align}
I_n = {} & \int_0^h \Bigr[\sum_{r_1=1}^\infty a_{1,r} \cos\left(\frac{2\pi}{h} r_1t_1\right) \\[6pt]
& {}+ b_{1,...
- 293
0
votes
0
answers
75
views
How to perform this integral to get a closed/ semi closed form
I want to get a closed/ semi-closed form of the integral given below.
$$ \int_{-\infty}^{+\infty} \exp{\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)} \text{erf}\left(\alpha \frac{x-\mu}{\sqrt{2}\sigma}\...
- 159
2
votes
0
answers
191
views
Multiple Wiener integral as Witt polynomial of Brownian motion
I know that if i have a Brownian motion $W_t$ the multiple Wiener integral
$\int_0^t \int_0^{\xi_1}...\int_0^{\xi_n} dW_{\xi_1}...dW_{\xi_n}$
can be expressed as $H_n(\int_0^t dW_s)$ where $H_n$ is ...
- 293
-2
votes
1
answer
175
views
Simple closed form for $\int \lfloor x \rfloor dx$? [closed]
Wolfram Alpha claims there is no closed form in terms of standard funcions
for $\int \lfloor x \rfloor dx$ but we believe we found
simple closed form agreeing with experimental data.
Define $i_1(x)=x -...
- 25.4k
2
votes
0
answers
252
views
Power series of the modified Bessel function of the second kind
I am looking for a power series representation of
$$ \frac{1}{K_{\nu}(x)}, $$
where $K_{\nu}$ denotes the modified Bessel function of the second kind and $\nu>-1/2$ is not an integer.
I know that ...
- 83
0
votes
0
answers
255
views
Distribution of "occupation times" of Brownian Motion
Let $B_t(\omega)$ be a standard Brownian motion and let $A\in\mathcal{B}(\mathbb R)$ be a Borel set.
I would like to find the distribution of $$Y_A(\omega):=\lambda(\{t\in[0,1]:B_t(\omega)\in A\})=\...
- 245
7
votes
1
answer
220
views
Average of polynomials over the real sphere
In quantum information, much can be done with the averaging formula
$$
\int_{\mathbb{C}P^{n-1}} (zz^*)^{\otimes t} dz = {n + t -1 \choose t}^{-1} \operatorname{\Pi}_{\mathrm{Sym}^t}$$
Here the ...
- 283
3
votes
1
answer
324
views
Integral of a function changing sign
By some numerical tests, we can see that the following function is negative on $(0,1)$:
$$\small f(x)=\int_0^\infty\frac{s^{x-1} e^{-2 s} (\pi \cos(\pi x) (s^{2 x}+(0.1)^2)-\sin(\pi x) \ln(s) (s^{2 x}-...
- 395
3
votes
4
answers
543
views
Some Log integrals related to Gamma value
Two years ago I evaluated some integrals related to $\Gamma(1/4)$.
First example:
$$(1)\hspace{.2cm}\int_{0}^{1}\frac{\sqrt{x}\log{(1+\sqrt{1+x})}}{\sqrt{1-x^2}} dx=\pi-\frac{\sqrt {2}\pi^{5/2}+4\sqrt{...
user497425
0
votes
0
answers
81
views
Fourier transform of an exponential function with radical argument divided by a radical
I have $f(t)=\dfrac{e^{-i\sqrt{(t-t_0)^2+A^2}}}{\sqrt{(t-t_0)^2+A^2}}$ where $t_0$ and $A$ are constant. I need to take the Fourier transform of $f(t)$. I made few substitutions to take it to a form ...
- 1
30
votes
4
answers
2k
views
If $X$ and $Y$ independent and identically distributed, then $E(|X-Y|)\leq E(|X+Y|)$. Are other proofs of this known?
I know a proof of the theorem that if $X$ and $Y$ independent and identically distributed, then $E(|X-Y|)\leq E(|X+Y|)$. The proof uses an integral representation of the absolute value,
$$\int_0^\...
- 409
5
votes
2
answers
2k
views
Integral of a product between two normal distributions and a monomial
The integral of the product of two normal distribution densities can be exactly solved, as shown here for example.
I'm interested in compute the following integral (for a generic $n \in \mathbb{N}$):
$...
- 305
3
votes
1
answer
146
views
Convolution between normal distribution and the maximum over $m$ Gaussian draws
$\DeclareMathOperator\erf{erf}$
Let's consider the Gaussian distribution $P_X(x)= \frac{1}{\sqrt{2 \pi \sigma^2}} e^{- \frac{x^2}{2 \sigma^2}}$. Now consider the random variable $W \equiv \max \{ X_1, ...
- 305
2
votes
1
answer
272
views
Proof of covariant convolution for a kernel function that is rotation symmetric in Fourier space
Problem Statement
Let $g:\mathbb R^{d}\to \mathbb R,d\in\{2,3\}$ be an integrable function (assumption I1). Suppose $\mathcal T$ is a rotation, and $f:\mathbb R^d\to\mathbb C$ (assumption C) is an ...
- 53
3
votes
0
answers
176
views
What is the meaning of big-O of a random variable?
I encountered this problem in a book "Pattern Recognition and Machine Learning" by Christopher M. Bishop. I excerpt it below:
screenshot of the book
In the excerpt, the big-O notation $O(\xi^...
- 31
12
votes
2
answers
1k
views
Counterexamples to differentiation under integral sign, revisited
Let $f\colon\mathbb R^2\to\mathbb R$ be a measurable function such that
\begin{equation*}
F(t):=\int_{\mathbb R}dx\,f(t,x)
\end{equation*}
exists and is finite for all real $t$. Suppose that
\...
- 128k
1
vote
1
answer
198
views
Probability of multivariant gaussian random variables in different areas
$\newcommand{\sgn}{\operatorname{sgn}}$Let $X_i$ is a gaussian random variable correlated with others. we want to find the probability of each possible case to find the expectation of following ...
- 25
9
votes
2
answers
622
views
Integral inequality: Prove $\int_0^1 f\int_0^1 1/f \leq 1$ for a certain function $f$
Let $g$ be a piecewise smooth, zero average, function over $[0,1]$ such that $\min g^2>0$. I would like to show that
$$
\int_0^1 g\sqrt{1-r/g^2}\int_0^1 \frac{1}{g\sqrt{1-r/g^2}} \leq 1
$$
for all $...
- 264
0
votes
1
answer
1k
views
The derivation of Reynolds-averaged Navier-Stokes equations
The following procedure is used to derive the Reynolds-averaged Navier-Stokes equations (Wikipedia: RANS equations)
When we talk about turbulent flows we can represent the velocity of the fluid as:
$$
...
- 1
0
votes
1
answer
257
views
Calculation of solid angle for rectangle in 6DOF [closed]
I am an undergrad trying to understand and use solid angle calculations:
I have a point source in R3 space (x_source, y_source, z_source) and a rectangle with given center (x_center, y_center, ...
- 1