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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5
votes
1answer
90 views

Solve $a^x+b^x=c$ for $x$

I need to solve an equation of the form $$a^x+b^x=c$$ with $a,b\in (0,1)$ and $c\in(0,2)$ (and I'm solving for $x\in\mathbb{R}_{>0}$). I know this admits a solution (details below), but it's such ...
1
vote
2answers
100 views

Find a closed form of the series $\sum_{n=1}^{\infty} \frac{x^n}{n^2+3n+2}$

The question I've been given is this: Using both sides of this equation: $$ \frac{x}{1-x} = \sum_{n=1}^{\infty}x^n $$ Find an expression for: $$ \sum_{n=1}^{\infty} \frac{x^n}{n^2+3n+2} $$ Any help ...
0
votes
1answer
66 views

Recurrence relation: $c_{k+1}=c_k+\frac{1}{(k+1)!}$

I have no idea how to proceed solving a recurrence relation like this. I know that the terms approach $e$ but beyond that I have no idea. The relation is$$c_{k+1}=c_k+\frac{1}{(k+1)!} \ \ ; \ c_0=1$$ ...
2
votes
1answer
69 views

Solving $x - a \log(x)=b$

Let $a>0$ and $b \in \mathbb{R}$: Assume there exists an $x >0 $ s.t. $$x - a\log(x) = b$$ holds. How can it be determined in closed-form?
4
votes
2answers
158 views

Evaluating $\sum_{n=0}^{\infty } 2^{-n} \tanh (2^{-n})$

Reading in some tables pages I found $$\sum _{n=0}^{\infty } 2^{-n} \tanh \left(2^{-n}\right)=\tanh (1) \left(1+\coth ^2(1)-\coth (1)\right)$$ I try to split in two sum using the roots of the ...
2
votes
2answers
69 views

HowTo solve this integral involving logarithm

I would like to solve integrals of the form $$I(c) := \int_0^\infty \log(1+x) x^{-c} \, dx ,$$ where $c \in (1,2)$. Mathematica says either 1) $I(c) = \frac{\pi}{1-c} \csc(\pi c)$ or 2) $I(c) = ...
1
vote
1answer
100 views

Kovacic's algorithm

Is there any reference with some example, about how to solve a "riccati" equation in this (below) form :$$y'(x)+a(x)y^2(x)+b(x)y(x)+c(x)=0$$ by Kovacic's algorithm? Or can anybody help me to ...
2
votes
1answer
44 views

Closed-Form Modular Arithmetic

Is there a way to define modulo division (or functions of modular arithmetic in general) as superposition of (elementary?) functions? For example, the multiplication is first introduced as ...
3
votes
1answer
96 views

How to prove $\int_{0}^{\infty}\frac{e^{-\left(\sqrt{x}-a/\sqrt{x}\right)^2}}{\sqrt{x}}dx=\sqrt{\pi},\,a>0$?

We know that $$\Gamma\left(\frac{1}{2}\right)=\int_{0}^{\infty}\frac{e^{-x}}{\sqrt{x}}dx=\sqrt{\pi} $$ but it seems that, for every $a>0 $ we have ...
1
vote
3answers
110 views

How do i find a closed form expression for $\sum_{k=0}^n \frac{(x-1)^k}{k+1}$?

How do I Find a closed form expression for : $$\sum_{k=0}^n \frac{(x-1)^k}{k+1}$$ Note :I have no idea how to do that, I am bad at evaluating series when we cannot use some standard series to do it. ...
2
votes
0answers
54 views

I suspect this integral has a closed form but I can't find it

$$\int_a^b \frac{\text{d}\eta}{\eta}\sin\left(A\eta^3\right)\sin\left(A(\eta-B)^3\right)$$ $$\int_a^b \frac{\text{d}\eta}{\eta}\sin\left(A\eta^3\right)\cos\left(A(\eta-B)^3\right)$$ where ...
1
vote
1answer
53 views

Better closed form for generating function $\sum \binom{n}{2k} x^k$

I have a power series $F_n(x) = \sum_k \binom{n}{2k} x^k$, which has a closed form of $F_n = \frac12 \left((1 + \sqrt{x})^n + (1 - \sqrt{x})^n\right)$. $$\begin{align} (1 + \sqrt{x})^n + (1 - ...
10
votes
3answers
327 views

Evaluating the limit of a certain definite integral

Let $\displaystyle f(x)= \lim_{\epsilon \to 0} \frac{1}{\sqrt{\epsilon}}\int_0^x ze^{-(\epsilon)^{-1}\tan^2z}dz$ for $x\in[0,\infty)$. Evaluate $f(x)$ in closed form for all $x\in[0,\infty)$ ...
3
votes
2answers
84 views

Does anyone know of a closed form solution to the following integral?

Does anyone know of a closed form solution to the following integral? $$ \DeclareMathOperator\erf{erf} \newcommand{d}{\;\mathrm{d}} \int^{+\infty}_{-\infty} \erf^{\;m}\!(x) \frac{\d^n ...
4
votes
4answers
126 views

Closed form of $\sum_{n=1}^\infty (-1)^n\frac{\sin(n\theta)}{n^3}$ for $\theta\in (-\pi,\pi)$

We have to find the closed form of the following series $$\sum_{n=1}^\infty (-1)^n\frac{\sin(n\theta)}{n^3}$$ for $\theta\in (-\pi,\pi)$. We tried to use the following form of the sine ...
0
votes
1answer
36 views

Recurrence involving square root

The recurrence equation I have is: $$ T_n = c_1 + T_{n-1} + 2\sqrt{c_2 + c_1 T_{n-1}} $$ $$ T_0 = a $$ $c_1,c_2,a$ are positive real numbers I need to somehow convert this into a linear homogeneous ...
1
vote
0answers
40 views

How to solve this kind of recurrence relation in closed form? $F(n) = aF(n-1) + bF(n-2) + cF(n-3) + dF(n-4)$

How to solve this recurrence relation in closed form? $$F(n) = aF(n-1) + bF(n-2) + cF(n-3) + dF(n-4)$$ I know how to solve recurrence relations for less than four calls by solving the ...
6
votes
3answers
212 views

Closed form for a binomial series

I am wondering if any knows how to compute a closed form for the following two series. $$\sum_{m=1}^{n}\frac{(-1)^m}{m^2}\binom{2n}{n+m}$$ $$\sum_{m=1}^{n}\frac{(-1)^m}{m^4}\binom{2n}{n+m}$$ ...
0
votes
1answer
42 views

Multiplication of 2 sums that equal another multiplication of 2 sums

I have been trying to prove a formula of mine and i come across something very interesting, well to me it is. If the formula is correct, it states that: $$ \left(\sum_{m=0}^{k-c} {k-c \choose m}{ms_1 ...
1
vote
0answers
51 views

How to Evaluate this Summation to Find a Closed Form

While taking the incomplete Bell Polynomil of $x^a$ i found out that: $$ B_{n,k}^{x^a}(x) = x^{ak-n} \sum_{m=0}^k \frac{(am)!(-1)^{k-m}}{m!(k-m)!(am-n)!} $$ Now, what i am wondering is, what is the ...
21
votes
3answers
524 views

How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$?

I am interested about some infinite product representations of $\pi$ and $e$ like this. Last week I found this formula on internet ...
1
vote
1answer
33 views

Proving the closed form for an infinite sum (related to Chebyshev polynomials)

How do I prove the following identity? For $y\not= 0$, we have $$ \sum_{n=0}^{\infty} \dfrac{1}{2y}\left( (x+y)^{n+1}-(x-y)^{n+1}\right) = \dfrac{1}{(x+y-1)(x-y-1)}. $$ I am trying to find the ...
0
votes
1answer
28 views

A closed form for the coefficients of Chebyshev polynomials

The Chebyshev polynomials are defined recursively: $T_0(x)=1$; $T_1(x)=x$; $T_n(x)=2xT_{n-1}(x)-T_{n-2}(x)$ I have been trying to find a closed form for the coefficient on the monomial $x^j$ of the ...
3
votes
2answers
242 views

Can the recurrence $C_n = 2 C_1 C_{n-1} - C_{n-2}$ be expressed in a closed expression?

I was wondering if the expression: $$ C_n = 2 C_1 C_{n-1} - C_{n-2} $$ could be expressed as a closed expression in terms of (hopefully polynomials of) $C_1$ (or $C_2$). The bases cases for this ...
6
votes
1answer
66 views

closed form for $\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{dx}{(b^2+x^2)}$

closed form for : $$\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{\mathrm{dx}}{(b^2+x^2)}$$ where $\beta$ is beta function I tried with the definition of beta and i got ...
13
votes
1answer
249 views

Evaluating $\int{ \frac{x^n}{1 + x + \frac{x^2}{2} + \cdots + \frac{x^n}{n!}}}dx$ using Pascal inversion [duplicate]

(Note: I apreciate very much who marked this as a duplicate but I would like an answer for why my proof is wrong) This is my solution, I have no clue why it failed. Let's start: define $$I_n(m) = ...
4
votes
1answer
56 views

Simplifying real part of hypergeometric function with complex parameters

I am looking for a simpler representation of the following hypergeometric function with complex parameters in terms of more basic functions and manifestly real parameters: ...
8
votes
1answer
154 views

Finding closed form for a generating function with different powers of $x$ in parameter

I'm working on a math/programming puzzle that involves an integer series defined as having a recurrence relating values $a(n)$ to $a(\lfloor\frac{n}{2}\rfloor)$ and $a(\lfloor\frac{n}{4}\rfloor)$. ...
2
votes
1answer
108 views

Closed form for $\sqrt {-1\sqrt {-2 \sqrt {-3 \sqrt {-4 \ldots}}}}$ [closed]

Does $\sqrt {-1\sqrt {-2 \sqrt {-3 \sqrt {-4 \ldots}}}}$ converge? Is there a closed form for it?
3
votes
1answer
36 views

A uncountable an closed subset of the Liouvilles Number

I am trying to "find" a closed and uncountable subset of the Liouville's numbers. $x\in L$ means that for all $n\in \mathbb{N}$ exists $p,q\in \mathbb{Z}$ with $q>1$ such that $$0<\vert ...
1
vote
0answers
96 views

A closed form for $\int x^nf(x)\mathrm{d}x$

When trying to find a closed form for the expression $$\int x^nf(x)\mathrm{d}x$$ in terms of integrals of $f(x)$ I found that $$\int xf(x)\mathrm{d}x=x\int f(x)\mathrm{d}x-\iint ...
1
vote
1answer
26 views

Eliminating a summation

I need to approach the new position $(x_t,y_t)$ at moment $t$ of a moving object at $(x_0,y_0)$ given its horizontal velocity $vx_0$, its vertical velocity $vy_0$ and some constant resistance $r$ that ...
2
votes
3answers
43 views

Prove that for any positive integer $n$ and $d$, $\sum_{k=0}^d 2^k\log_2(\frac{n}{2^k})=2^{d+1}\log_2(\frac{n}{2^{d-1}})-2-\log_2{n}$

I could prove it by induction, but I need to see how I might have discovered it for myself (cause that's what's gonna be on exam).
4
votes
1answer
106 views

Formula for $\sum\limits_{j=1}^{m-1}\frac{1}{\sin^{2p}(\frac{j\pi}{m})}$

Let $m\geq 2$ be an integer, then there is the well known formula $$\sum\limits_{j=1}^{m-1}\frac{1}{\sin^2(\frac{j\pi}{m})}=\frac{m^2-1}{3},$$ I'm interested in similar equations for the following ...
14
votes
3answers
311 views

How to compute $\sum_{n\text{ odd}}\frac{1}{n\sinh n\pi\sqrt 3}$?

I came across an old question asking to show that $$\displaystyle\sum_{n\text{ odd}}\frac{1}{n\sinh n\pi}=\frac{\ln 2}{8}.\tag{1}$$ Although I have managed to prove this formula, my proof uses ...
22
votes
4answers
299 views

How to compute $\int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx$?

How to compute the integral, $$\int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx$$ where, $\varphi = \dfrac{\sqrt{5}+1}{2}$ is the Golden Ratio?
4
votes
0answers
127 views

Is there a closed-form expression for Shapley value of glove game?

Suppose we have a coalition game with transferable utilities, with $m$ players having a right-handed glove and $n$ players having a left-handed glove. The value of a coalition is equal to the number ...
1
vote
1answer
29 views

Solution to a pde

I have a PDE system that I am trying to solve at steady state. When I make the appropriate substitutions, I get an equation of the following form: $$\frac{1+M}{M}\frac{d^2 M}{d x^2}=1$$ Is there a ...
2
votes
2answers
37 views

Help with recurrence $T(n) = T(n/2) + n$

I just need help seeing where I went wrong in this solution. $$T(n) = T\left(\frac{n}{2}\right) + n,~~~ T(1) = 0$$ By master theorem, this is $\theta(n)$. However, when I try to solve it, it ...
25
votes
4answers
547 views

Integral ${\large\int}_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}$

How to prove the following conjectured identity? $$\int_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}\stackrel{\color{#a0a0a0}?}=\frac{\sqrt[4]6}{3\sqrt\pi}\Gamma^2\big(\tfrac14\big)\tag1$$ It holds ...
5
votes
1answer
195 views

This one weird infinite product can define exponentials in terms of itself. What does it do for other constants?

What is... $$\lim_{\omega \to \infty} \prod_{N=1}^{\omega} {{1+e^{b \cdot c^{-N}}} \over 2}$$ This is similar to my other question. However, there is a constant factor rather than variable in the ...
9
votes
1answer
196 views

Closed form for ${\large\int}_0^\pi\frac{x\,\cos\frac x3}{\sqrt[3]{\sin x}}dx$

I'm trying to find a closed form for the integral below and I found the following conjecture using computer search (and some lucky guesses): $$\int_0^\pi\frac{x\,\cos\frac x3}{\sqrt[3]{\sin ...
4
votes
2answers
196 views

Integrals of Pullbacks

This is a problem from Guillemin's Differential Topology: Suppose that $f_0, f_1: X \to Y$ are homotopic maps and that the compact boundaryless manifold $X$ has dimension $k$. Prove that for all ...
0
votes
2answers
51 views

Closed form for $\int_0^\infty\frac{1}{(1+x^2)^s}\,dx$ when $s\in (0.5,\infty)\setminus\mathbb{N}$

I know that the improper integral $$ \int_0^\infty\frac{1}{(1+x^2)^s}\,dx $$ is convergent for $s>0.5$ and divergent otherwise. Furthermore, it has a closed form for $s \in \mathbb{N}$ (this can ...
2
votes
3answers
98 views

Proving $\int_0^n \left(1-\frac{t}{n}\right)^n\ln(1/t)\,dt \to \gamma$

I have to prove that $\displaystyle \lim_{n\to\infty}\int_0^n \left(1-\frac{t}{n}\right)^n\ln(1/t)\,dt= \gamma$ I tried to expand $\left(1-\frac{t}{n}\right)^n$ and swap sum and integral, which ...
0
votes
1answer
55 views

Closed Form Solution for Minimization involving Standard Normal CDF and PDF

Could someone please advice and provide detailed steps regarding any possible closed form solutions or other suggestions regarding solving a minimization problem of the type shown below? Here, $\phi$ ...
0
votes
1answer
52 views

solve logarithmic equation without numerical methods

Is there algebraic method to solve following equation for $x$: $$ a x + b \ln x + c = 0 $$ with $a , b , c$ constants without using numerical methods and ln means natural logarithm.
24
votes
2answers
589 views

Closed Form for $~\int_0^1\frac{\text{arctanh }x}{\tan\left(\frac\pi2~x\right)}~dx$

Does $~\displaystyle{\LARGE\int}_0^1\frac{\text{arctanh }x}{\tan\bigg(\dfrac\pi2~x\bigg)}~dx~\simeq~0.4883854771179872995286585433480\ldots~$ possess a closed form expression ? This recent ...
5
votes
1answer
142 views

How to use Fourier Transform with non-trivial boundary conditions such as in potential flow around a plate?

I'd specifically like to be able to solve this PDE with boundary conditions corresponding to flow around a line (plate cross-section), otherwise known as flow-tangency, with integral transforms. ...
1
vote
2answers
139 views

Closed form of partial hypergeometric sum

Can we get closed form for $$\sum_{k=0}^m \left(-\frac12\right)^k \binom{2m}{m-k}k^p,\quad p\in\mathbb{N}\,?$$ In Concrete Mathematics Knuth describes Gosper's algorithm and its Zeilberger's ...