# Questions tagged [integration]

Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...

1,454
questions

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### Prove that the regularized incomplete beta function monotone with each of its parameter

Consider the regularized incomplete beta function $I_x(a, b)$ with $x \in [0,1]$ and $a, b > 0$. I am hypothesizing that the function is monotone decreasing with respect to $a$ and monotone ...

2
votes

1
answer

71
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### Proof of the monotonicity of a regularized incomplete beta function

I want to prove the monotonicity of $I_r(nr, 2+(1-r)n)$ on $n$ but has no clues. The $I$ is the regularized incomplete beta function, defined as follows:
$$I_r(nr, 2+(1-r)n)=\frac{\int_0^r x^{nr-1}(1-...

0
votes

1
answer

106
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### How to prove that $ \sum_{k=1}^\infty \frac{\sin kx}{k^z} = \frac{1}{\Gamma(z)} \int_0^\infty \frac{t^{z-1}e^t\sin x}{1–2e^t\cos x+e^{2t}}dt? $ [closed]

I need it to show that $\displaystyle\sum_{k=1}^\infty \frac{\sin k}{k^3} = \frac{2\pi^2-3\pi+1}{12}$

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### Solve a definite integral that represents the variation of a capacitance with a given angle [migrated]

I am trying to find an explicit form of the following definite integral. I have tried Mathematica but it failed to give me an answer. I know this returns something related to a trigonometric function. ...

0
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1
answer

73
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### Orthogonal space of polynomials

Let $f \colon [0,+\infty) \to \mathbb R$ be a continuous function. Assume that for any non-negative integer $n$, the function $f(t) t^n$ in integrable in $(0,+\infty)$ and
$$
\int_0^{+\infty} f(t) t^n ...

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1
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60
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### PDF of the difference of two Beta Prime distribution

I am struggling to find the PDF of the difference of two Beta Prime distribution.
Definition
A random variable is said to have a Beta Prime distribution $\text{B}'(\alpha, \beta)$ with $\alpha, \beta&...

-1
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0
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87
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### Sufficient conditions for $1/f\in L^p$

This is a very simple question that I have not found a satisfactory answer for. When is the reciprocal of a function $f:[-1,1]\to\mathbb{R}$ in $L^p([-1,1])$? In other words, when is $\int1/f^p\,dx<...

1
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2
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161
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### Definite integral of Gaussian divided by hyperbolic cosine

Question: is there a nice formula for $\int_{-\infty}^{+\infty} \frac{e^{-rx^2}}{e^x + e^{-x}} dx$ ? For a real parameter $r > 0$.
Maybe useful:
consider the bilateral Laplace transform
$$J(z) = \...

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1
answer

150
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### Averaged measure in integrations

Consider
\begin{align}
& F(n,x)\equiv \int_0^x \cdots g (x_5)\int_0^{x_5} ~\int_0^{x_4} g (x_3)~~\int_0^{x_3} ~\int_0^{x_2} g (x_1)~~A(x_1)\,dx_1\cdots dx_n
\end{align}
where $g(x)$ is a measure. ...

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votes

1
answer

197
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### Cauchy reduction formula with measure (a variation)

The Cauchy reduction formula conveniently compresses $n$ integrations of a function $F(x)$ into a single integral. Here I am interested in reducing the following "curved-space" ...

3
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2
answers

403
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### Closed form for $ \int_{0}^{1} \dotsi \int_{0}^{1} \frac{x_1^q + \dotsb + x_n^q}{x_1^p + \dotsb + x_n^p} \, \mathrm{d}x_1 \dotsm \mathrm{d}x_n $

I asked this question on MSE, but received no answer.
Recently, reading this problem, I found out that
$$ \lim_{n\to \infty} \int_{0}^{1} \dotsi \int_{0}^{1} \frac{x_1^q + \dotsb + x_n^q}{x_1^p + \...

1
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0
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### Generalization of the triangle inequality to complex exponents: What is $P\left(\left| x^{a+bi} + y^{a+bi} \right| \ge \left|z^{a+bi}\right|\right)$?

Let $x \le y \le z$ be the length of the sides of a triangle whose vertices are uniformly random on the circumference of a circle. In this question, it was proved that if $a \ge 1$, then the ...

13
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1
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718
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### If $(a,b,c)$ are the sides of a triangle, then the probability $P(ax + by \ge c) = \frac{4}{\pi^2}\chi_2(x) + \frac{4}{\pi^2}\chi_2(y)$

Posting this question in MO since it is unanswered in MSE
Let $(a,b,c)$ be the side of a triangle. In its most general linear form, the triangle inequality can be expressed as: Does $ax + by \ge c$ ...

1
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1
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326
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### Cauchy reduction formula with measure

The Cauchy reduction formula is
$$\int_{0}^{x}{\cdots{\int_{0}^{x_{3}}{\int_{0}^{x_{2}}{F(x_1)\,dx_1\cdots dx_n}}}}
=\frac{1}{(n-1)!}{\int_{0}^{x}{(x-t)^{n-1}F(t)\,dt}}$$
Consider a generalisation
$$\...

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votes

0
answers

91
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### Indefinite integral of li(x) dependant expression

I have the following $f(x)$ function:
$f(x)=\frac{4 (n-1)^x n^{1-x} \operatorname{li}\left(1-\left(\frac{n-1}{n}\right)^x\right)}{2 \text{li}\left(1-\left(\frac{n-1}{n}\right)^x\right)-3}$
When I try ...

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0
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### Integral of gradient of a function times a vector fields, null whatever the function, implies null divergence and tangential limits conditions

I'm reposting this question from math.stackexchange, as I haven't got answers so far.
At the beginning of "Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de
...

10
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1
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666
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### Find the area of the region enclosed by $\sin^p x+\sin^p y=\sin^p(x+y)$, the $x$-axis and the $y$-axis (comes from a probability question)

Consider the graph of $\sin^p x+\sin^p y=\sin^p(x+y)$, where $x$ and $y$ are acute, and $p>1$.
Here are examples with, from left to right, $p=1.05,\space 1.25,\space 2,\space 4,\space 100$.
Find ...

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0
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75
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### An integral over the sphere in $\mathbb{R}^d$

Let $S^{d-1}$ be the unit sphere in $\mathbb{R}^d$.
Let $|x-y|$ denote the euclidean distance between to points $x$ and $y$ in $\mathbb{R}^d$.
Is there a nice expression for the following (maybe ...

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0
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30
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### Error on spline interpolation for integration

I am interested in studying the integral
$$
I \equiv \int_m^M dx \, f(x) d(x)
$$
where $f(x)$ is an analitcally known function (mildly oscillating),
while $d(x)$ is sampled at a finite, discrete set ...

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0
answers

67
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### Integrating $\int_{-\infty}^{\infty}\exp(-x^2/(1-ix))(1-ix)^{-1/2}{\rm d}x$

I want to calculate the integral
$$
I=\int_{-\infty}^{\infty}\exp\left(\frac{-x^2}{1-ix}\right)(1-ix)^{-1/2}\ {\rm d}x.
$$
I got two ways to deal this integral. The first, noting that
$$
\int_{-\infty}...

3
votes

0
answers

100
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### Maximal-type inequality for a Borel probability measure supported on a subset of $L^2(\mathbb{R}^d)$

Let $\mu$ be a Borel probability measure on $L^2(\mathbb{R}^d)$ for $d\ge 1$ which is moreover supported on the unit sphere
$$S=\{\phi\in L^2(\mathbb{R}^d): \| \phi\|_{ L^2(\mathbb{R}^d)}=1\}.$$
Let ...

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5
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### Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\sin y}{\sin(x+y)}$ and the $x$-axis (comes from a probability question)

This question resisted attacks at MSE, so I am posting it here.
Here is the graph of $\dfrac{\sin x}{\sin y}=\dfrac{\sin x+\sin y}{\sin(x+y)}$.
Find the area of the region enclosed by the curve and ...

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0
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78
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### Possible closed form of the integral contains delta function?

I want to evaluate an $N$-dimensional integral defined by
$$
P(s,k)=\int_{-\infty}^{\infty}{\rm d}x_1\dotsi\int_{-\infty}^{\infty}{\rm d}x_N\ \delta\left(\Biggl|\frac{\sum_{m\ne n}^N\exp(i(m+n)k)(x_m-...

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0
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129
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### Help me find the antiderivative of $W(W(x))$ where $W$ denotes the Lambert W Function

Let $W$ denote the Lambert W Function. I must know the antiderivative of $W^2 = W(W(x))$.
I'm already convinced this function is not elementary. This does nothing to settle up my curiosity, as I ...

3
votes

1
answer

231
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### Expectation on a Polish space

I was wondering, if given a Polish space $X$, and given some probability measure $p$ on $X$, can the expectation of an $X$-valued function be taken? In particular, would the integral
$\int_X x dp$ ...

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0
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### Discretization of oscillating integral

Suppose I am interested in computing
$$
I \equiv \int_0^B dx \, g(x) f(x)
$$
where $B$ is a known upper bound for the integral,
$g(x)$ is a known oscillating function and
$f(x)$ is a smooth function ...

0
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1
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95
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### Numerical integration with integrable singularity

Suppose I have a numerical estimation of discrete samples of a smooth function $C(t)$ at $t = a, \dots, T = Na$ and I want to (numerically) compute the integral of $f(t) = \frac{C(t)}{\sqrt{t}}$. In ...

4
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1
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253
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### Approximating a finite sum with an integral

Consider the following sum (with $a$ being a real number and $N$ an even integer)
$$S(a, N) = \sum_{m=1}^{N/2} \frac{4}{N+1} \sin^2\left( \frac{2\pi m}{N+1} \right)\sin \left( 2 a \cos \left( \frac{2\...

2
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1
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282
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### A strange functional inequality

Let $f,g \in C([-2,2],\mathbb R_+^*)$ even and concave real functions.
Is it true that
$$
\int_0^1 f\big(\cos(x^{-1})+\sin(x^{-1})\big) \cdot g\big(\cos(x^{-1})-\sin(x^{-1})\big) \mathrm{d}x\\ \leq f(...

0
votes

0
answers

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### Schauder basis of L^p on general domains

Let $\Omega$ be a non-empty regular compact subset of $\mathbb{R}^n$; i.e. the closure of its interior is itself; for some $n\in \mathbb{Z}^+$. Let $1\le p<\infty$. When does the space $L^p(\...

1
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1
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103
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### An inequality for the Darboux integral

As we know, if $f(x)$ are Riemann integrable, we have
\begin{gather}
\left|\int_a^b f(x)~\mathrm{d}x\right|\leq \int_a^b |f(x)|~\mathrm{d}x.
\end{gather}
So, for Darboux integrals, such as the upper ...

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1
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### Densities, pseudoforms, absolute differential forms and measures, differential forms, etc

Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here.
Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...

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0
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130
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### Can this integral be solved analytically

I have an integral of the form
$$\int_{t_1}^{t_2} \frac{\sum_{i=1}^n a_i e^{b_i t}}{\sum_{i=1}^n c_i e^{d_i t}} dt$$
Where $a_i,b_i,c_i,d_i$ are $4n$ real constants, and $t_1,t_2$ are positives. Is ...

1
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0
answers

63
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### Derivative with respect to initial condition for the solution of an SDE

Suppose we have an SDE (assuming the Lipschitz continuous conditions required for the existence of the solution):
\begin{align}
dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t
\end{align}
and define its ...

1
vote

1
answer

74
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### Calculating an integral involving Haar measure on orthogonal projections

Let $\sigma_{n, m}$ denote the uniform measure over matrices $U \in \mathbb{R}^{n \times m}$ satisfying $U^T U = I_m$. Let $1_k \in \mathbb{R}^k$ denote the vector with all entries equal to $1$.
I am ...

0
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1
answer

76
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### Uniform convergence of differential quotients in $L^1$

I know that the question arose already in other contexts. However, I think this question might be different. If I have $f\in W^{1,1}$ then it is obvious that $\frac{f(x+t)-f(x)}{t}$ converges ...

5
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### How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?

In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...

3
votes

1
answer

215
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### How to find the coefficient of $x^k$ in the expression $\prod_{p=2}^n (1+xp) $

I got this general formula for $ n\in N$ (I showed it here)
$$\int_0^1 \left(\frac{x}{1-x} \ln x \right)^n dx=n \sum_{p=0}^{n-1}a(n,p+1) (-1)^{n-p} \zeta(p+2)+n! $$
where $a(n,k)$ is the coefficient ...

0
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0
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### Is there a closed form of $\int\frac{x\ln x}{\arctan x}dx$?

Consider the integral$$\int\dfrac{x\ln x}{\arctan x}dx\label1\tag1$$While I have made some progress on computing the antiderivative, I would like to know whether or not there is a closed form of it.
...

1
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0
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90
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### Conjectured closed form of $\int_0^1\frac{\ln^3(1+x)\,\ln^3x}x\mathrm{d}x$

I posted this question on Math Stack Exchange, but there were no helpful comments or answers
https://math.stackexchange.com/q/4874446/1298448
How to integrate $${\displaystyle \int_0^1\frac{\ln^3(1+x)\...

4
votes

2
answers

671
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### Conjectured closed form of $\int\limits_0^1 \frac{\ln y \operatorname{Li}_2 (-y)}{1-y^2} \, dy$

Let's state with $\psi^{(1)}$ the trigamma. Calculate the order:
$$\mathcal{S} = \sum_{n=1}^\infty (-1)^{n-1} (\psi^{(1)}(n))^2$$
(Cornel Ioan Valean)
I uploaded this question here https://math....

7
votes

1
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334
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### Exponential trigonometric integral

I want to compute the normalization constant of some probability density on SO(3). After some simplification, I arrive at the following double integral:
$$ \tag{1}\label{eq:1}
\int_0^{2 \pi} \int_0^{\...

2
votes

0
answers

156
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### Taylor coefficients of the integral of the ordered exponential

Let $A$ be a continuous $2\times 2$ matrix-valued function on $[0,1]$. Define $X_A$ as the solution of
$$
X_A'(t) = A(t) X_A(t), \qquad X(0) = I.
$$
In other words $X_A$ is the ordered exponential of $...

5
votes

1
answer

341
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### Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$

After reading some meta posts, I've decided to post this question on MathOverflow since I didn't receive any comments or answers on MSE
Certainly, I apologize for any oversight. Here's a more refined ...

1
vote

0
answers

70
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### Integral inequality related to the (mixed?) moments of two functions

For $k\in \mathbb{Z}_+$ and $t\in [0,1]$ we set
$$
S_{k}(t) = \{(t_1,\ldots, t_k)\colon 1\ge t\ge t_1\ge\ldots\ge t_k\ge 0\}.
$$
Let $a$ and $b$ be two continuous functions on $[0,1]$. Introduce
$$
...

0
votes

1
answer

98
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### Demonstrating bound on integral

I found this question on another forum and it got me interested because of how tight the bound is: prove that
$$\int_0^\infty \frac{\arctan(x)}{x^2 + 4}\,dx > \frac{\pi}{4}.$$
The difference ...

1
vote

0
answers

116
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### Integration in polynomial time

The work of Friedman and Ko and
Müller guarantee the polynomial time computability of the integrals of analytic functions inside the circle of convergence. But do algorithms have practical value? Is ...

7
votes

1
answer

467
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### How to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?

Here in my answer I got the real part for the polylogarithm function at $1+i$ for natural $n$
$$ \Re\left(\text{Li}_n(1+i)\right)=\left(\frac{-1}{4}\right)^{n+1}A_n-B_n $$
where
$$ B_n=\sum_{k=0}^{\...

1
vote

1
answer

74
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### Characterization of an integral operator with a Bessel kernel

I am considering the following integral operator: $$K(\sigma)(\theta)=\int_0^{2\pi} \sigma(\theta') J_0(|e^{i\theta}-e^{i\theta'}|)\,d\theta',$$ where $J_0$ is the Bessel function of order $0.$
I am ...

2
votes

0
answers

90
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### Techniques for computing integrals on $G/K$

Edit: I originally tried writing a simpler question, but to make more clear what I want I wrote the general question below
Let $G$ be a compact group with compact subgroup $K\subseteq G$ (everything ...