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Questions tagged [closed-form-expressions]

For questions that specifically ask for determining a closed form of equations, integrals etc.

3
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0answers
89 views

How to extend Ritt's theorem on elementary invertible bijective elementary functions

The elementary functions of a complex variable $z$ according to Liouville and Ritt are those functions built up from the rational functions of $z$ by exponentiation, taking logarithms, and algebraic ...
-2
votes
0answers
48 views

is it possible to solve $\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$ in closed form? [duplicate]

$$\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$$ is it possible to solve this equation in closed form ? if i should use a numerical method, can you show me why i cannot get an exact solution ?
1
vote
4answers
445 views

how to solve $\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$ [closed]

I’d like to solve following equation for $x.$ If it is not possible, why I can’t? $$\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$$
0
votes
0answers
42 views

$P(\max_{0 \leq t \leq 1} \|W(t)\| \leq x)$ has no closed-form expression… right?

$P(\max_{0 \leq t \leq 1} \|W(t)\| \leq x)$ shows up in a formula for computing $p$-values for a certain statistic, where $W(t)$ is a $d$-dimensional (standard) Wiener process. My advisor says the ...
0
votes
0answers
56 views

On the necessitation of $(-1)^n$ within the series expansion of $f(x)$ concerning the usage of Ramanujan's Master Theorem

Ramanujan's well known Master Theorem states that the series expansion of the transformed function $f(x)$ has to be in form of $$f(x)~=~\sum_{n=0}^{\infty}(-1)^n\frac{\phi(n)}{n!}x^n\tag1$$ ...
4
votes
1answer
120 views

Asymptotics for sum involving Euler numbers

This first request may be easy, but the asymptotics for the next step has me scratching my head. Through an informal inductive argument I have been able to show $$ (1) \quad \sum_{j=0}^{n-1}2^{2m(n-j)...
3
votes
2answers
227 views

Infinite sum of reciprocals of squares of lengths of tangents from origin to the curve $y=\sin x$

This question is actually from MSE. I had to post it here due to the lack of response there even after placing a bounty. Here goes the question Let tangents be drawn to the curve $y=\sin x$ from ...
1
vote
0answers
89 views

Closed-form formula for Wasserstein distance between uniform discrete distribution and discrete distribution with same support

Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{...
4
votes
1answer
316 views

Approximation a sum involving log and binomial coefficient

I am wondering about the asymptotic approximation of the following expression: $$S=\sum^{N}_{i=0}\log\Bigg[\binom{\binom{N+1}{i}}{t_i}\Bigg]$$ where $$t_i=\binom{N}{i}-\binom{N-k}{i-k}+\binom{N-k}{i-...
9
votes
2answers
1k views

Difficult trigonometric integral

This question was also asked here and here. I have faced some difficulties to do the following integral: $$ I=\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta~\sin\theta\int_{0}^{\infty}dr~r^2\frac{3x^2y^...
8
votes
4answers
849 views

Find a formula for the recurrent sequence $q_{n+1}=q_n(q_n+1)+1$

Find an analytic formula for the recurrent sequence $$q_{n+1}=q_n(q_n+1)+1,\;\;q_0\in\mathbb N.$$ (The question was asked on 03.05.2018 by M. Pratsovytyi, see page 109 of Volume 1 of the Lviv ...
1
vote
1answer
227 views

On finding the critical points of $f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x$

Given some constant $a\in\mathbb{R}-\{0\}$, find $x_0$ such that $f'(x_0)=0$ where $$f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x.$$ I have managed to write $f'(x_0)=0$ as $$\...
8
votes
1answer
956 views

What is the value of this double sum in closed form?

I encountered the following double sum which requires an evaluation. Is there a closed form for this? $$\sum_{n=0}^{\infty}\frac{\sum_{k=0}^n\binom{n}k^{-1}}{(n+1)(n+2)}.$$ Incidentally, it ...
2
votes
1answer
246 views

Closed form of :$\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ for $ k$ is even integer and :$\int _{0}^{t}\exp(-x^2 \operatorname{erf}(x))dx$

This question is related to my question here such that i want to find a closed form of $\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ , for $k$ is even integer because for odd integer is $0$ as ...
1
vote
0answers
95 views

Is there any precedent in mathematics where closed-form relations between trigonometric and inverse trigonometric functions arise?

This question is connected to my current research where unexpectedly there arise connections between tronometric/hyperbolic functions and their inverses. In short, if we introduce some element $\tau$ ...
1
vote
0answers
56 views

Vibration of point load on a halfspace

The amplitude of vibration of surface of halfspace at a distance r from a point harmonic load of amplitude Q is given by $ w(r,0) = $ $ Q\over 2\pi G $ $ \int_0^\infty $ $ k^{2}\alpha pJ_0(pr)dp \...
-3
votes
1answer
101 views

How to find closed form expression for $\int^\infty_0 \frac{Ar}{1+Cr^\alpha} e^{-Br^2} dr$?

I am badly stuck in some integration here and will appreciate any help out of it. $$\int^\infty_0f(r) dr = \int^\infty_0 \frac{Ar}{1+Cr^\alpha} e^{-Br^2} dr$$ If I let $u = Br^2$, then I get $$ = \...
3
votes
1answer
119 views

Does a linear Recurrence relation with an nonlinear relation between elements can be solved by a closed formula?

I came across with this cool recurrence relation, and unfortunately i couldn't find sufficient mathematical tools to form it to a closed formula. i read several posts from math overflow saying that ...
3
votes
1answer
303 views

Solution of multivariate Geometric Brownian Motion?

It is known how to solve the SDE $dX=X\,dW$ to get a closed form expression of $X(t)=\exp(W_t-\frac{t}{2})$. The question is, is there also a way to solve \begin{equation} \begin{cases} dX=X \, dW_1+...
4
votes
0answers
311 views

Is there any closed form expression for $\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$?

Is there any closed form expression for the following serie? $$\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$$ Or at least a proof that it is an irrational number. The ...
6
votes
0answers
192 views

Closed form for 2D lattice sum

I am wondering if a closed form exists to the lattice sum $$S(a)= \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \frac 1 {(a^2+m^2+n^2)^{3/2}}$$ I am also interested in replacing $m^2+n^2$ with a ...
14
votes
2answers
859 views

When can an invertible function be inverted in closed form?

The Risch algorithm answers the question: "When can a function be integrated in closed form?", see: https://en.wikipedia.org/wiki/Symbolic_integration Is anyone aware of any work that answers the ...
4
votes
1answer
175 views

Simple integral representation for a beta function with more than two variables

The beta function $B(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}$ can be written for $\Re x, \Re y > 0$ symmetrically as $$ B(x,y) = \int_{-\frac12}^{\frac12}(\frac12-t)^{x-1}(\frac12+t)^{y-1}\,...
5
votes
1answer
311 views

Closed form for $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$

EDIT: Some additional details and corrections, I would appreciate any information about the highlighted expression. I try to solve $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$ where $I_n(x)$ is the ...
19
votes
2answers
905 views

Is there a closed form for $\int_0^\infty\frac{\tanh^3(x)}{x^2}dx$?

For $n\geqslant m>1$, the integral $$I_{n,m}:=\int\limits_0^\infty\dfrac{\tanh^n(x)}{x^m}dx$$ converges. If $m$ and $n$ are both even or both odd, we can use the residue theorem to easily evaluate ...
25
votes
2answers
971 views

Are these two new ways of representing odd zeta values as integrals known?

This is inspired by the same beautiful integral expression for $\zeta(3)$ as this question, but goes in a slightly different direction. Writing the original integral in the form $$\int_0^1\frac{x(1-x)}...
1
vote
0answers
80 views

Solving a partial difference equation with variable coefficients explicitly

I'm trying to solve the following partial difference equation with variable coefficients: $$a_{i,j,k}=-\frac{j+1}{i}a_{i-1,j+1,k-1}- \frac{k+1}{i}a_{i-1,j-1,k+1}, $$ defined on the grid $(i,j,k) \in \...
3
votes
2answers
181 views

Finding the critical points of a degree $5$ Blaschke product

The derivative of a degree $5$ polynomial $p\in\mathbb{C}[z]$ is a degree four polynomial $p'\in\mathbb{C}[z]$, and as such, the zeros of $p'$ may be found explicitly using the quartic formulae. One ...
2
votes
0answers
114 views

Reflection formula for the Hurwitz zeta function and odd zeta values

A reflection formula for the Hurwitz zeta function, which does not seem to be well known, uses half of the polynomials generated by $\frac{1}{-1+\sqrt{t-1}\cot(\sqrt{t-1}u)}$. (Look at the sections "...
1
vote
1answer
126 views

Solution of bimodal and multimodal Weibull distribution

Is there any closed form solution for $\sigma$ in a bimodal Weibull distribution function written in the following form: $$ P(\sigma) = 1- exp\Bigg(-\alpha\Big(\frac{\sigma}{\sigma_1}\Big)^{m1} -\...
15
votes
3answers
2k views

How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$?

Let $f$ be a function such that :$f:\mathbb{R}\to \mathbb{R}$ and $f^{-1}$ is a compositional inverse of $f$. I would'd like to know how do I solve this class of differential equation : $$\...
1
vote
1answer
174 views

A closed form of a summation or the taylor series expansion of some function with a closed form?

Let $Z_N = \displaystyle{\sum_{k+j\leq N}} \frac{N!N^{k+j}}{N^{N+1}}\frac{u^kv^j}{k!j!}\binom{N-j}{N-j-k}$ where $u$ and $v$ are two unknowns. My question is: Is there a closed-form for $Z_N$ or is $...
1
vote
1answer
154 views

Is there a way to get the closed form approximate result of $\int_ 0^a\frac {e^{-x - \frac {1} {x}}} {x}\, dx$ [closed]

It is known that $$\int_ 0^{\infty}\frac {e^{-x - \frac {1} {x}}} {x} dx=2 K_0(2),$$ but now I want to get the closed form approximate result of $$\int_ 0^a\frac {e^{-x - \frac {1} {x}}} {x} dx.$$ I ...
0
votes
0answers
88 views

Integral involving exponential and Marcum-Q function to a power

Do you have any suggestions to achieve a closed-form solution for the following integral: $$\int_{0}^{\infty}\exp\left[-ax\right]\times \left\{ 1-Q_{\mu}\left[\sqrt{2\kappa\mu},\sqrt{\frac{2\mu\left(\...
18
votes
2answers
720 views

Could there be a special-function counterexample to Schanuel's conjecture?

It is not too hard to show that if Schanuel's conjecture is true, then the only algebraic numbers admitting a "closed-form expression" (as defined precisely in this paper) involving $e$, $\pi$, and ...
4
votes
1answer
230 views

Does this system have a closed-form solution? $x_j^2 = \sum_{i=1}^n B_{ij} x_i$

I am interested in solving the following system of $n$ equations: $$x_j^2 = \sum_{i=1}^n B_{ij} x_i $$ for all $j\in\{1,\dots,n\}$, where $n$ is a positive integer and all the $0\leq B_{ij}\leq 1$ ...
5
votes
0answers
303 views

Determinant of a certain Vandermonde matrix

Is there a closed form expression for the determinant of the $n\times n$ Vandermonde-type matrix $$A = \left(\begin{array}{} 1&g_1 & x_1&g_1 x_1 & x_1^2&g_1 x_1^2 & \cdots &...
11
votes
2answers
389 views

How to prove that $\int _0^\infty\frac{\text{arcsinh}^nx}{x^m}dx$ is a rational combination of zeta values?

For $n\ge m\ge 2$, define $$I(n,m):= \int _0^\infty\dfrac{\text{arcsinh}^nx}{x^m}dx$$ Computer algebra systems say that the indefinite integral can be expressed in terms of polylog functions (of ...
12
votes
3answers
545 views

Is there a closed form of $\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$?

For naturals $n\ge m$, define $$I(n,m):=\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$$ with $\text{arcsinh}\ x=\ln(x+\sqrt{1+x^2} )$, so $\text{arcsinh} \frac12=\ln \frac{\sqrt{5}+1}2 $. Is it ...
7
votes
1answer
296 views

Simplifying Root of Unity Double Summation

Good afternoon. I have a particular summation, $$\zeta_{n,k}(N)=\frac{k!}{N^{n+1-k}}\sum_{j=0}^n\sum_{i=0}^{N-1}\binom{n}{j}w_N^{(j-k)i}$$ Here, the $w_N$ is the root of unity $w_N=e^\frac{2i\pi}{N}$...
1
vote
0answers
430 views

Approximation of semicontinuous functions by continuous (or smooth) functions with closed form

I'm looking for a sequence $(f_{\epsilon})_{\epsilon>0}$ of continuous (or smooth) functions approximating a semicontinuous function $f$. Here, for approximation, pointwise convergence is fine. For ...
1
vote
2answers
300 views

When is there a closed form for $\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}$?

This is a follow up on a previous question of mine. Out of curiosity, I am wondering more generally when a closed form exists for $$\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}$$ where $P$ and $Q$ are ...
7
votes
1answer
453 views

Solution to $(A+x^2)e^x=B$ with Lambert W function

Is it possible to obtain a analytical solution for $(A+x^2)e^x=B$, where we want to solve for $x$ with $A,B$ as constants?
5
votes
1answer
271 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 \, I_{k}...
4
votes
3answers
373 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
45 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[ a_1^...
1
vote
0answers
201 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 ...
3
votes
1answer
199 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
178 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
333 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 ...