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3
votes
1answer
137 views

On the search for an explicit form of a particular integral

Let $f$ be integrable over the interval $(0, 1)$, and $$I_n = \int_0^{1} x^n f(x) \, \mathrm{d}x.$$ Suppose $f(x) = f(1-x)$; we can then show that $$I_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \, ...
3
votes
3answers
215 views

Convergence of expansion for fractional iteration

I was reading about tetration here. The site mentions that the convergence of the expansion for fractional iteration is unproven. However, I was interested in reading more literature about convergence ...
1
vote
0answers
22 views

PDF of points at the intersection of a sphere and hyperboloid in n dimensions

I'm studying a statistical mechanics problem and I have two conserved quantities: $$ E = \sum_{k=0}^{M} \left[ a_1^2(k) + a_2^2(k) + b_1^2(k) + b_2^2(k)\right] $$ $$ H = \sum_{k=0}^{M} 2 k \left[ ...
0
votes
0answers
47 views

Can these integrals be represented in closed form?

Prevuously asked at Mathematics but received no answer. This paper in the formula F.3.6 (page 271) gives the following formula for the derivative of Hurwitz Zeta function: $$\frac ...
0
votes
0answers
53 views

Partial Vandermonde Circulant Determinant Expression

Consider following partial Vandermonde type, circulant matrix $\begin{bmatrix} x_1 & x_2 & 0 & \dots & 0 & x_n\\ x_1^2 & x_2^2 & x_3^2 & \dots & 0 & 0\\ \vdots ...
0
votes
0answers
83 views

Closed form for convolution of two-dimensional Gaussian with characteristic function of a disk

Is there a closed form expression for the convolution of a two-dimensional (elliptical) Gaussian function with the characteristic function of the interior of an ellipse? The motivation is that I have ...
3
votes
1answer
114 views

Is it possible to get an equation with two exponentials and a bessel function in closed form?

Is it possible to get the equation below into closed form? I have tried using integration tables but I haven't found anything that matches. Are there any other methods to achieve a closed form ...
1
vote
0answers
76 views

The hypertriangular function of $n$

I'm looking for papers or recent results on the hypertriangular function of $n$: $$H_t(n)= \displaystyle\sum\limits_{k=1}^{n} k^k$$ This is A001923 in the OEIS. I don't have much experience with ...
5
votes
2answers
318 views

Does this equation has a closed-form solution for $t$? ($(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i)$)

We are given $n\in \mathbb N^+$ and $p\in[\frac{1}{2},\frac{n+1}{n+2}]$. Our goal is to find $t\in[0,1]$ such that $$(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i$$ Is there a closed-form ...
2
votes
2answers
300 views

What summations of elementary trig functions are known to have (elementary) closed forms?

I've been trying to find a closed form of $\displaystyle \sum_k{\tan{(k)}}$ that contains only elementary functions, and I think I may be onto something. But rather than reinvent the wheel, I want to ...
54
votes
2answers
3k views

Is it possible to express $\int\sqrt{x+\sqrt{x+\sqrt{x+1}}}dx$ in elementary functions?

I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here: Is it possible to express ...
2
votes
1answer
362 views

Conjectured closed form for definite integral

Let $K(x)$ be the complete elliptic integral of the first kind (the argument is the parameter $m = k^2$). Let $$ A = \int_0^1 \arcsin(K(x)) dx$$ With precision $1000$ decimal digits $\Re A = ...
3
votes
2answers
391 views

Sum of series $a^{i^2}$

Is there any closed form known for the expression $\sum_{i=1}^\infty a^{i^2}$ where $|a|<1$? Thanks!
5
votes
2answers
586 views

Random walk by simplex vertices

I apologize if this question is well-known, but I was unable to find it mentioned anywhere. There exists a bug which moves around in $r$-space. The bug begins at the origin of this $r$-space. If the ...
0
votes
1answer
253 views

Permutations of letters under some conditions

Let $W(p,q,r,s)$ be the number of permutations of the letters which satisfy the following conditions : Condition 1 : The letters are consist of $P,Q,R,S$. Condition 2 : The number of letter ...
3
votes
1answer
163 views

Infinite series - analytical solution

Analytical Solution is required for: $$\sum_{n=0}^\infty (2n+1)\exp(-n(n+1)x),$$ $$\sum_{n=0}^\infty (2n+1)^2\exp(-n(n+1)x),$$ $$\sum_{n=0}^\infty n(n+1)(2n+1)\exp(-n(n+1)x),$$ $$\sum_{n=0}^\infty ...
-1
votes
1answer
164 views

Generic way to solve f(x+1) - f(x) = g(x) when g(x) is given [closed]

All I have been looking around for a general way to solve the problem of $f(x+1) - f(x) = g(x)$, where $g(x)$ is given. Has this problem been studied before? If there does not exist such a general ...
4
votes
1answer
419 views

sum of integral part of n/k

Is there any direct formula or algorithm better than the brute force (O(n) algorithm by iterating from 1 to n) way to calculate the sum \begin{equation} S = \sum\limits_{i=1}^n [{\frac{n}{i}}] ...
0
votes
2answers
186 views

looking for f(x) in f(x) = a.exp((x-f(x))/b) [closed]

not a math expert, but this problem is really bugging me. whichever way i turn it, i cant find an expression for f(x) which satisfies f(x) = exp(( x - f(x)) /b ). I can also express the problem as ...
6
votes
1answer
331 views

Convergence and Closed Form of an Integral Involving Bell Numbers

1. Does the following integral converge ? $$\int_0^\infty \frac{b(x)}{B(x)} dx$$ where $$b(x) = \sum_{n=1}^\infty \frac{n^x}{n^n} \qquad and \qquad B(x) = \sum_{n=1}^\infty \frac{n^x}{n!}$$ 2. ...
0
votes
0answers
125 views

multivariate integral calculation in closed form

I am looking for a closed form for the below integral but since I don't have the necessary backgrounds I am not able to solve it: i know the final solution is in the form of modified Bessel functions ...
4
votes
0answers
131 views

Integrate the exponential of sum of circular differences?

Given positive integer $N$ and parameters $T>0$, $a$, $b$, what is $\int_{t_1=0}^T \cdots \int_{t_N=0}^T e^{a(t_1+\cdots+t_N)+b(|t_1-t_2|+\cdots+|t_{i-1}-t_i|+|t_N-t_1|)} dt_1 \cdots dt_N$ ? Any ...
11
votes
1answer
234 views

An elementary expression for $_3F_2(1,1,9/4;2,2;-1)$

Consider the following series: $$S=\sum_{n=1}^\infty\frac{(-1)^n\ \Gamma\left(\frac{5}{4}+n\right)}{n^2\ \Gamma(n)}.$$ It can be expressed in terms of a hypergeometric function: ...
1
vote
1answer
106 views

Estimate the scale of the power series with Poisson pdf/pmf-like terms

I would like to have an estimate for the series $$P(t) = \sum\limits_{k = 0}^\infty (e^{-t}\frac{t^k}{k!})^m,$$ where $e$ is the base of natural logarithm, $k!$ is the factorial of the integer $k$, ...
1
vote
2answers
273 views

Any closed form for series like $F(x)=\Sigma_{i=p}^{\infty}x^p$,p is prime$?

Any closed form for series like $$F(x)=\Sigma_{i=2}^{\infty}x^p,\text{p is prime}$$ or $$F(x)=\Sigma_{i=0}^{\infty}x^{i!}$$? More generally,we can obtain a power series from decimal expansion of a ...
45
votes
7answers
2k views

How closed-form conjectures are made?

Recently I posted a conjecture at Math.SE: $$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$ where $J_\mu(x)$ and $Y_\mu(x)$ ...
19
votes
3answers
656 views

Is this combination of generalized polygamma and dilogarithm actually zero? $\Im\;\psi^{(-2)}(1+i)+\frac1{4\pi}\text{Li}_2(e^{-2\pi})-\log\sqrt{2\pi}+\frac{5\pi}{24}+\frac12$

I encountered this quantity in my calculations and tried to simplify it. Approximate numeric calculations suggested it could be zero (more precisely, it is certainly less than $10^{-4\times10^3}$ in ...
0
votes
2answers
97 views

Fitting algebraic expression to a number [algorithm]

I know that it may turn out useless, but this is precisely the reason why I'm asking. Does any one know of an existing piece of code that would find me the best approximation to a given irrational ...
4
votes
1answer
474 views

Is there a closed form expression/series expansion for $\int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{az+b^2z^2}\Gamma(z)\Gamma(1-z)dz$ ?

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} ...
8
votes
5answers
794 views

$\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$

I'm trying to solve the integral $\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$, where $s$, $r$ and $m$>1 are positive integers. My question is whether a closed form ...
7
votes
1answer
337 views

Experimental mathematics: how are floating point equations discovered/converted to exact equations?

the 2005 AMS article/survey on experimental mathematics[1] by Bailey/Borwein mentions many remarkable successes in the field including new formulas for $\pi$ that were discovered via the PSLQ ...
1
vote
1answer
191 views

Expectation under a t-distribution

Given parameters $\lambda, \nu>0$, a covariance matrix $R$, a mean vector $\mu \in R^p$, the Arellano-Valle and Bolfarine's generalized $t$ distribution is given by (see, for example, the book by ...
1
vote
1answer
193 views

Growth of a particular sequence

Not sure if this is research level? While analyzing a particular algorithm, I came across the following series: $\sum_{k=1}^{m-1}\frac{3^{k}}{\prod_{i=1}^{k-1}2^{2^{i}}}$ Is there a closed form ...
3
votes
2answers
262 views

an equation in fractions

I have an equation of the form $\sum_{i=1}^{m}{\frac{1}{a_{i}-x}}=\sum_{j=1}^{n}{\frac{1}{b_{j}-x}}$ and would like to express $x$ as a an approximate explicit function of the $a_{i},b_{j},m,n$. Have ...
0
votes
2answers
198 views

A certain sum with q by the power of binomial (n 2)

Is there a closed form to the following sum: $\sum_{n=0}^{\infty}a^nq^{n(n-1)/2}$ for all $a>0$ and $0\lt q\lt 1$ ?
0
votes
3answers
815 views

Simplifying finite sum over 1/(ax+b)

Can I simplify: \begin{equation} \sum_{x=x_0}^{x_1} \frac{1}{ax+b} \end{equation}
7
votes
2answers
398 views

New results on Chow's notion of closed-form numbers?

In an interesting article (available here), Timothy Chow proposes that a closed-form number be defined as an element of the smallest subfield of $\mathbb{C}$ that is closed under $\exp$ and a chosen ...
7
votes
3answers
760 views

Expectation of a simple function of multivariate gaussians iid rvs

I would like to compute analytically the following expected value: $$ E\left( \frac{X_i^2}{\sum_j \lambda_j^2 X_j^2}\right) $$ where the $X_i \approx N(0,1)$ are iid. It seems to be an elementary ...
7
votes
1answer
608 views

Sum of subset of geometric series: a^2^n

The formula for 1 + a + a^2 + .... where 0 < a < 1 is $\frac{1}{1-a}$, but what if you wanted to sum only those where the exponent is a power of 2? That is, $S = a + a^2 + a^4 + a^8 + \cdots$ ...