# Questions tagged [closed-form-expressions]

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

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### 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 ...

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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 ?

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**4**answers

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?
$$\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$$

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**0**answers

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 ...

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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**

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**1**answer

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)...

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**2**answers

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 ...

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**0**answers

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**

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**1**answer

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-...

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**2**answers

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**

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**4**answers

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 ...

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**1**answer

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 $$\...

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**1**answer

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

**1**answer

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 ...

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**0**answers

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 $\tau$ ...

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**0**answers

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 \...

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**1**answer

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
$$ = \...

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**1**answer

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 ...

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**1**answer

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+...

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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 ...

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**0**answers

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 ...

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**2**answers

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 ...

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**1**answer

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}\,...

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**1**answer

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 ...

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**2**answers

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 ...

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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)}...

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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 \...

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**2**answers

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 ...

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**0**answers

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 "...

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**1**answer

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} -\...

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**3**answers

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 : $$\...

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**1**answer

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 $...

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**1**answer

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 ...

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**0**answers

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(\...

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**2**answers

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**

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**1**answer

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 $0\leq B_{ij}\leq 1$ ...

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**0**answers

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 &...

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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 ...

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**3**answers

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 ...

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**1**answer

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 $w_N=e^\frac{2i\pi}{N}$...

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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 ...

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**2**answers

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 ...

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votes

**1**answer

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**

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**1**answer

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}...

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**3**answers

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 ...

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**0**answers

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^...

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**0**answers

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

**1**answer

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 ...

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**0**answers

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 ...

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**2**answers

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 ...