Questions tagged [closed-form]

A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".

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$I(x) = -\int_0^1 \frac{1}{z}\ln\left(\frac{1-x z + \sqrt{1-2 x z+ z^2}}{2}\right)\,dz$

Is there a closed form integral for $$I(x) = -\int_0^1 \frac{1}{z}\ln\left(\frac{1-x z + \sqrt{1-2 x z+ z^2}}{2}\right)\,dz$$ for $-1 < x < 1$? This integral is related to Legendre polynomials ...
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Determine closed forms for generating functions

I'm studying generating functions for my study of number theory and I'm trying to solve some exercises. I need to find closed forms for the generating functions of the following sequences: a) $a_n=n^3$...
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A Regularized Beta function limit: $\lim_{a\to0}\frac{1-\text I_\frac zb(a,b)}a$

The goal is to “generalize” the Exponential Integral $\text{Ei}(x)$ using the Regularized Beta function $\text I_z(a,b)$: $$f(b,z)=\lim_{a\to0}\frac{1-\text I_\frac zb(a,b)}a$$ Some clues include: $$\...
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Finding a better approximation for $f(L) = \left\lfloor{\frac{1}{4}\sum_{n=1}^{L-1}\left\lfloor n+300\times2^{n/7}\right\rfloor}\right\rfloor$

I have the following equation: $$f(L) = \left\lfloor{\frac{1}{4}\sum_{n=1}^{L-1}\left\lfloor n+300\times2^{n/7}\right\rfloor}\right\rfloor$$ And im looking for either a closed-form solution or an ...
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How can we prove a closed form for $\frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right)+\text{Li}_2\left(2+\sqrt{3}\right)$?

I have been working on a problem in number theory that I have reduced to the problem of showing that the two-term linear combination $$ \frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right) + \...
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1 vote
1 answer
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Sum in closed form

Let $n$ be a nonnegative integer. I am interested in finding a closed form for the sum $$S(x)=\sum_{j=n}^\infty(n+j)(n+j-1)\cdots(j+1)x^j.$$ If I'm not mistaken, we get $$\sum_{j=0}^\infty(n+j)(n+j-1)\...
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3 votes
3 answers
143 views

Closed-form solution of recurrence relation

I have the first-order non-homogeneous recurrence (defined for $n \geq 1$): $$f(n) = f(n-1) \; \frac{n-1}{n} + 1$$ with base case $f(1) = 1$. Looking at the values of the sequence ‒ $1, 1.5, 2, 2.5$ ‒ ...
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Is there a closed-form expression for the first k terms in a binomial series?

This is kind of two questions in one. Firstly, does the following expression have a closed form? $$\sum_{i=0}^k \binom{n}{i}x^i$$ $$\text{(first $k$ terms in binomial series)}$$ where $k$ is some ...
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Closed form for a recursive sequence: reference request

Playing around with this WolframAlpha widget, I found that the sequence satisfying $$f(n)=4f(n-1)+2f(n-1)^2$$ and $$f(1)=4$$ is given by $$f(n)=\cos(2^{n-1}\cos^{-1}(5))-1.$$ Could you tell me where I ...
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2 answers
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Is there a closed form solution to $x^a + x^b - x^{a+b} = 0$ for $x > 0$, given $a,b > 0$? [closed]

The title basically explains everything. Unfortunately I was stuck right from the beginning, so I cannot provide any more info. However, I provided a plot of the function for some values of $a$ and $b$...
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What does the formal definition of "closed-form" say about finite sums exactly?

I have looked through the online literature and there seems to be conflicting answers to this question. Consider the finite sum $$\sum_{i=0}^{\lfloor n/2 \rfloor} {n-1\choose i}$$ Is this expression ...
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Closed form for $\rm{Li }_2\left( -{\frac {i\sqrt {3}}{3}} \right)$

In my personal research with Maple i find this closed form : $$\operatorname{Li }_2\left( -{\frac {i\sqrt {3}}{3}} \right)={\frac {{\pi}^{2}}{24}}+{\frac {\ln \left( 2 \right) \ln \left( 3 \right) }...
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4 votes
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formulas for binary expansion of irrational number between $0$ and $1$

One can write any irrational number between $0$ and $1$ composed as closed expression of popular known numbers, such as, for example, the expression $$\frac{1}{\sqrt{2}}$$ in binary by successively ...
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6 votes
1 answer
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Closed form for definite integrals invovling Jacobi elliptic functions

In a 1879 work, Glaisher proves the following closed forms $$\int_{0}^{K\left(k\right)}\log\left(\text{sn}\left(z;k\right)\right)dz=-\frac{1}{4}\pi K^{\prime}\left(k\right)-\frac{1}{2}K\left(k\right)\...
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1 answer
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Is there a closed-form formula for this function from positive integers to positive integers?

Suppose we are given $n$ letters, $n$ a positive integer. We are also given a binary operation $+$. I want to know how many ways we can group these letters with parentheses. For example, if $n=2$, ...
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2 votes
1 answer
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Closed form formula instead of recursive sequence [closed]

I'm trying to create a computational model for a neuroscience project, but the computation times are too long for it to be useful. In particular, there is an iterative recursive step that is too slow (...
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Finding closed form Linear algebra solutions to equations using Signum operator.

I am a bit uncertain as to how one would solve equations that uses the sign function. In this case the sign function is applied component wise. Here is an example of the type of equations I am looking ...
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6 votes
0 answers
122 views

Multipole-like integral on the unit disc

I am interested in computing the following integrals \begin{align} I (\mathbf{x}_{1}) {} & = \!\! \int_{\mathbf{D}} \!\! \mathrm{d} \mathbf{x} \, \frac{|\mathbf{x}_{1} \!-\! \mathbf{x}|}{\sqrt{1 \!...
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1 vote
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Trig sum and residues

I am interested in the sum $$ \sum_{n=1}^k 2\biggl[\sin\biggl(\frac{n\pi}{2k+2}\biggr)\biggr]^{-2g} $$ where $k,g$ are integers. It is not too hard to show that this can also be expressed as $$ -1-2\...
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How to get closed form when $a(n)\alpha+b(n)\beta_0+c(n)\beta_1=3^m\alpha+\cdots$?

Let $n=(1b_{m-1}b_{m-2}\cdots b_1b_0)_2$, and we know $$\begin{aligned}g(n)&=a(n)\alpha+b(n)\beta_0+c(n)\beta_1\\ &=(\alpha\beta_{b_{m-1}}\beta_{b_{m-2}}\cdots\beta_{b_1}\beta_{b_0})_3\\ &=...
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20 votes
2 answers
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How to prove $\int_0^1 \frac{\arctan^2(x)\ln\left(\frac{x}{(1-x)^2}\right)}x \, \mathrm{d}x=G^2$?

A while back I made a post asking for examples of integrals which evaluated to famous irrational constants (or constants that were very likely irrational but yet unproven to be). The top answer in ...
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  • 4,990
2 votes
1 answer
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Finding the value of $\sum^{\infty}_{n=3,5,7,9....}\frac{2n^2 \exp(-\pi n/2)}{\exp(\pi n)+1}$

I want to evalutate: $\displaystyle \tag*{} \sum \limits ^{\infty}_{n=3,5,7,9....}\dfrac{2n^2 \exp \left(-\pi n/2\right)}{\exp(\pi n)+1}$ This question is inspired from my previous question which ...
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1 vote
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Finding the value of $\sum \limits ^{\infty}_{n=3,5,7,9....}\dfrac{2n \exp \left(-\pi n/2\right)}{\exp(\pi n)+1}$

From my previous question, I changed the sum into: $\displaystyle \tag*{} \sum \limits ^{\infty}_{n=3,5,7,9....}\dfrac{2n \exp \left(-\pi n/2\right)}{\exp(\pi n)+1}$ Since my previous question, I ...
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  • 552
2 votes
1 answer
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Conjecture about equality of integrals with specific 2nd and 3rd order polynomial multiplying a function of $\sin^2(\pi t)$

Is it true that, for a general class of functions $f,$ $$2\int_0^1(y^3 -\frac{1}{4})f(\sin^2{(\pi \, y)})\,dy \, \overset{?}{=} \, \int_0^1(1-y)(1-3y)f(\sin^2{(\pi \, y)})\,dy$$ The above conjecture ...
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1 vote
1 answer
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Calculation of a specific integral involving the Levy distribution

It is well-known that for a standard Brownian motion $(B_t)_{t\geq 0}$ the first hitting time of the level $1$ $$T_1 := \inf\{t> 0 : B_t = 1 \}$$ has standard Levy distribution, this means that $...
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  • 3,754
3 votes
1 answer
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Behavior of self convolution of 1/n

I wish to find a closed form, or a good upper bound for $\sum_{i=1}^{i=n-1} \frac{1}{i \times (n-i)}$. I can specify a lower bound of $(n-1)/n^2$, which looks like $1/n$ because we have $n-1$ terms ...
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0 votes
0 answers
30 views

Closed form of equation with summation to infinity: brownian motion in a confined 1D box

I would like to implement this equation, which represent the brownian motion in one-dimension (1D) of a particle in a confined 1D box, which initial position is $x_0$ at time $t_0$ and final position ...
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4 votes
2 answers
170 views

Closed form of $\sum_{i=0}^{\infty} x^i (i!(i+1)!)^{-1/2}$

I am looking for a closed form of this series $$ \sum\limits_{i=0}^{\infty} \frac{x^i}{\sqrt{i!(i+1)!}},\:x \geq 0. $$ The main problem is that this series contains roots of factorials. But many ...
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1 vote
0 answers
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Can this recurrence relation be solved in explicit form?

Problem We have the recurrence relation $$a(j, k, n) = \begin{cases} 1 & : k = 0 ∧ j = n − 1 \\ 0 & : k = 0 ∧ j < n − 1 \\ a(j, k − 1, n) + (−1)^{k + n − 1 − j} × \frac{n!} {k! (n − k)!} × \...
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9 votes
1 answer
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Is there a closed form of the Laplace Limit Constant: $x$ such that $\frac{xe^{\sqrt{x^2+1}}}{\sqrt{x^2+1}+1}=1$ using library functions?

The Laplace Limit Constant $\lambda$ is well know constant which is the $y$ value of the global extrema of: $$x\,\text{sech}(x):$$ Therefore: $$x=\max(x\,\text{sech}(x))=-\min(x\,\text{sech}(x))=1....
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0 votes
0 answers
37 views

Closed Form solution to $\min_{x} ||Ax-b||_{1}$

Given a system of linear equations over $\mathbb{R}^n$ $$Ax=b$$, where $A \in \mathbb{R}^{m \times n}, x \in \mathbb{R}^n, b \in \mathbb{R}^m, m >n$, I want to minimize the following objective, ...
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2 votes
0 answers
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Which explicit expressions for $\Bbb E [ f(Z\sqrt{t} +x)]=\int_{\Bbb R} f(u \sqrt{t} + x) e^{-u^2 /2} du$ are known, where $Z \sim \mathcal{N}(0,1)$?

I would like to collect a list of known explicit expression of the form $$h(x,t) = \Bbb E [ f(Z\sqrt{t} +x)],$$ where $f:\Bbb R \to \Bbb [0,\infty )$ is a measurable, non-negative function and $Z \sim ...
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  • 3,754
6 votes
2 answers
324 views

When is it necessary to define a new function?

For example: Lambert $W$ is a non-elementary function that can be defined as a solution for $x$ to $x\cdot e^x$, but $\int{\frac{1}{\ln(t)}dt}$ is also supposed to be nonelementary. How do we know ...
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  • 121
3 votes
1 answer
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How to find a closed form formula for such integral using special functions

I am trying to find a closed form formula for this integral $$\int_{}^{} \left(1-x^2\right)^n \frac{\mathrm{d^i} }{\mathrm{d} x^i} \left(1-x^2\right)^n dx $$ where $i=0,1,2,...\in \mathbb{N}$ and $n\...
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11 votes
0 answers
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Why Dottie$=2\sqrt{I^{-1}_\frac12(\frac 12,\frac 32)-I^{-1}_\frac12(\frac 12,\frac 32)^2} = \sin^{-1}\big(1-2I^{-1}_\frac12(\frac 12,\frac 32)\big)$?

Introduction: For some background information on the Dottie Number D, see the great posts at: What is the solution of $\cos(x)=x$? Some definitions: The “solution” to Kepler’s equation is Kepler E: ...
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Convert this recurrence relation to its closed-form

I struggle with the following relation that I need (hope) to convert to a closed-form. I write in detail only the point at which I am stuck, as I can take it from there to finish the task. The ...
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1 vote
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What is the difference between closed-form expression and closed-form function?

What is the difference, if any, between a closed-form expression and a closed-form function? I have seen those terms used interchangeably, but is there any difference between them, and if so, what is ...
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2 votes
2 answers
128 views

Trigonometric Integrals and Hypergeometric function

I'm dealing with these two integrals: $$I_1=\int_{-\pi}^{\pi} \frac{\cos(x)\cos(nx)}{(1+e \cos(x))^3} \mathrm{d}x, \quad I_2=\int_{-\pi}^{\pi} \frac{\sin(x)\sin(nx)}{(1+e \cos(x))^3}\mathrm{d}x$$ is ...
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4 votes
3 answers
416 views

Can you solve any mathematical function?

For any finite mathemathical function (consisting of addition, subtraction, division, multiplication, exponentiation, trigonometry) can you find $x$ in $f(x) = y$ where $y$ is a number you want? Is it ...
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7 votes
1 answer
132 views

Generalized integrals for Bessel Moments $\int_{0}^{\infty} x^4K_0(x)K_1(x)^3 \ln(xK_1(x))^2\text{d}x=\frac{1}{32}$

Let $I_\nu(x)$ be the modified Bessel functions of first kind with order$\text{ }\nu$, $K_\nu(x)$ be the modified Bessel functions of second kind with order$\text{ }\nu$. Prerequisite Information: ...
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3 votes
3 answers
234 views

How to solve $e^x \ln(x) = a$?

I was wondering if it was possible to solve equations of the form $e^x \ln(x) = a, \;a > 0$ in terms of the Lambert $W$ function $W(x)$? I understand that fixed point iteration or the Newton-...
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1 vote
2 answers
79 views

Is it Possible that some Non-Analytically Integrable Functions might actually have Analytical Integrals?

Is it Possible that some Non-Analytically Integrable Functions might actually have Analytical Integrals (that have not yet been discovered)? When reading posts like these (e.g. List of functions not ...
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1 vote
1 answer
39 views

Closed form for KL divergence equation

For a fixed $p\in(0,1)$ and $y>0$ is it possible to find a closed form for $x\in(0,1)$ such that $$x\log\left(\frac{x}p{}\right) + (1-x)\log\left(\frac{1-x}{1-p}\right) = y ?$$ I've tried a bunch ...
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0 answers
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Does there exist a closed-form analytical solution for the European barrier option?

Right now I am studying barrier options. I have been able to calculate the price of a European barrier option with Monte Carlo. However, the calculations are quite time-consuming. Therefore, I was ...
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  • 1
0 votes
0 answers
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Zeros of a closed-form solution for a recurrence relation

I'm trying to solve the constant-recursive sequence $D_n= \left(b + (b-2)x^2\right) D_{n-1} - \left((1-b)x\right)^2 D_{n-2}$, where $x = 1/2$, and with initial conditions $D_0 = 1$ and $D_1 = b$. The ...
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  • 101
7 votes
0 answers
174 views

Evaluate $\int_{0}^{\infty }\!{\frac {\ln \left( x \right) \arctan \left( x \right) }{{{\rm e}^{\pi\,x}}-1}}\,{\rm d}x$

I have the idea of this integral when I see $$\int_{0}^{\infty }\!{\frac {\arctan \left( x \right) }{{{\rm e}^{\pi\,x}}-1}}\,{\rm d}x$$ and so I know that the closed form is $${\frac{1}{2}}-{\frac {\...
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  • 193
2 votes
0 answers
46 views

Closed form for $\prod_{j=1}^n\prod_{k=1}^{m_{j}-1} (x_j-kN^{-1})$

Let $n,N\in\mathbb{N}$ and $m_1,x_1,\ldots,m_n,x_n\in\mathbb{N}$. I'm trying to rewrite following expression $$\prod_{j=1}^n\prod_{k=0}^{m_{j}-1} (x_j-kN^{-1})$$ I am only interested in the summands ...
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  • 1,712
5 votes
2 answers
110 views

Closed form expression for series involving Legendre polynomials

Given $-1 \leq x \leq 1$ and $0 \leq \eta \leq 1$, I am interested in computing $$ E(x,\eta) = \sum_{\ell = 0}^{+ \infty} |P_{\ell} (0)|^{2} \, P_{\ell} (x) \, \eta^{\ell} , $$ with $P_{\ell}$ the ...
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6 votes
0 answers
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A generalized "Rare" integral involving $\operatorname{Li}_3$

In my previous post, it can be shown that $$\int_{0}^{1} \frac{\operatorname{Li}_2(-x)- \operatorname{Li}_2(1-x)+\ln(x)\ln(1+x)+\pi x\ln(1+x) -\pi x\ln(x)}{1+x^2}\frac{\text{d}x}{\sqrt{1-x^2} } =\...
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-1 votes
1 answer
88 views

What is the closed form solution for this infinite sum? [closed]

We are working on a problem related to order statistics. This requires the computation of the following infinite sum. Let i be a positive integer. Let real numbers $$\alpha >0$$ and $$1>\beta>...
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