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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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Proving the closed-formed formula of a recursive expression

I was given this excercise as a practise for discrete mathematics, among some others. The instruction is to provide a closed-form formula for the recursive formula shown below. $u_n =2 + \sum_{i=1}^s ...
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Numerical Approximation Solution to Exponential Equation

I have a question about finding an approximate value $x$ for the following expression: $$\frac{(e^{x\alpha_{1}})^2 + (e^{x\alpha_{2}})^2 + \ldots + (e^{x\alpha_{n}})^2}{\displaystyle \left(\sum_{i = ...
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65 views

Closed form for $\int_0^1 e^{\frac{1}{\ln(x)}}dx$?

I want to evaluate and find a closed form for this definite integral:$$\int_0^1 e^{\frac{1}{\ln(x)}}dx.$$ I don't know where to start. I've tried taking the natural logarithm of the integral, ...
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Closed form of generalized Josephus problem

I have to solve Josephus problem where every 5th man gets killed and I have to find out the living person number and $n = 10000$ The generalized solution of Josephus problem is $J(n,k) = ((J(n-1,...
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Closed-form of $\int_{0}^{\infty} \exp {(-ax^2)}\log x \mathrm d x$ without using Laplace Transform?

I want to get the closed-form of the followng integral without using Laplace Transform but I didn't succeed with $a$ is a positive real number. $$\int_{0}^{\infty} \exp {(-ax^2)}\log x \mathrm ...
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Evaluating a Polynomic-Trigonometric-Hyperbolic Integral

Within this AoPS thread it is asked to evaluate the following integral $$\mathfrak I~=~\int_0^\infty \frac{x\sin x}{\cos x+\cosh^2 x}\mathrm dx\tag1$$ In order to be precise there is also a ...
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How to handle a differential equation whose RHS contains the derivative of the Dirac delta function?

I have a differential equation of the following form. \begin{align} \begin{split} \frac{\mathrm{d}^4\psi(\eta)}{\mathrm{d}\eta^4}-\beta^4\psi(\eta)=\psi'(\zeta)\,\delta'(\eta-\zeta) \...
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Solve the recurrence $b_1 = 2$ for $b_n = 3b_{n-1} + 5$

Solve the recurrence $b_1 = 2$ for $$b_n = 3b_{n-1} + 5$$ I've tried solving this problem using iteration, but the formula I get in the end is wrong. It is not a closed formula since there's still ...
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1answer
61 views

Simplifying $\prod\limits_{k\neq j=0}^{n-1}\frac1{\lambda_{n,k}-\lambda_{n,j}}$ for $\lambda_{n,k}=\exp\frac{i\pi(2k+1)}{n}$

I have been able to show that for $n\in\Bbb N_{\geq2}$ $$\phi(n)=\int_0^1\frac{dx}{x^n+1}=\sum_{k=0}^{n-1}\Gamma_{n,k}\log\frac{\lambda_{n,k}-1}{\lambda_{n,k}}$$ Where $$\lambda_{n,k}=\exp\frac{i\pi(...
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Floor Summation Closed Form?

Let $ a_1,a_2,a_3,...a_n $ be a set of positive integers. Does there exist any closed form for the approximation of the sum $$ \sum_{i=p}^n\operatorname{floor}\left(\frac{a_p}{i-p+1}\right) ?$$ If ...
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What is and what isn't a “closed form solution” [duplicate]

I had a friendly discussion with someone about closed form solutions. They contended that the backpropagation algorithm used in calculating the gradients of deep neural networks can't be called a ...
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Solving $\int_0^x\int_u^{u+1}\frac1{\ln t}\,dt\,du=0$ (an extension to the Ramanujan-Soldner constant)

For $u,x>0$, let $P$ be the function given by $$P(x)=\int_0^x\int_u^{u+1}\frac1{\ln t}\,dt\,du\tag1.$$ Is there a closed form for the positive root of $P(x)$, denoted by $\nu$? Can it be ...
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Interesting polynomial: $R_{k+1}(x)=x(2^{-2k-1}-1)\zeta(2k+2)+\int_0^x\int_0^t R_{k}(u)dudt$

I am trying to find the polynomial $R_k(x)$ such that $$R_k(x)=\sum_{n=1}^{\infty}\frac{(-1)^n}{n^{2k+1}}\sin(nx)$$ With the base case $$R_1(x)=\frac{x^3}{12}-\frac{\pi^2x}{12}$$ Which can be found ...
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Does $\sum\limits_{k=2}^{\infty}{\frac{|B_{k}|}{k!}(\cos(n)-1)}$ have a closed form?

I am trying to find a closed form expression of the following sum in terms of $n$ (if it exists) where $B_{k}$ is the $k$th Bernoulli number. $$\sum_{k=2}^{\infty}{\frac{|{B_{k}|}}{k!}(\cos(n)-1)}$$ ...
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Formula for this diminishing function

I'm trying to write a function for a game where points accumulated becomes less effective as they get more. Input Value: 2000 Interval: 500 For every interval reached, decrease the ...
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2answers
57 views

How to find a closed form for $\sum_{i=0}^n \binom{a+i}{b+i}i$

Wolframalpha tells me it's $$\frac{b (b + 1) \binom{a + 1}{ b + 1} - (b + n + 1) (b (n + 1) - (a + 1) n) \binom{a + n + 1}{ b + n + 1}}{(a - b + 1) (a - b + 2)}$$ but how to come up with or at least ...
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Hamiltonian paths and cycles of rook graph on $n\times2$ chessboard

According to OEIS, there are closed form for directed Hamiltonian paths (A096121) and Hamiltonian cycles (A276356) of rook graph on $n\times2$ chessboard. Are there papers which include proof of those ...
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How to prove that a recursive function with inner summation is approximately equal to some closed-form equation?

The following problem is taken from an algorithms textbook(specifically, in the context of complexity analysis of recursive algorithms.) Starting from the equation: $$nf(n) = n(n-1) + 2 \sum_{k=1}...
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Closed form for these polynomials?

I have a recurrence relation for these polynomials $p_i(x) $: \begin{align} p_0(x)&=0 \\ p_1(x)&=1 \\ p_{2i}(x)&=p_{2i-1}(x)-p_{2i-2}(x) \\ p_{2i+1}(x)&=xp_{2i}(x)-p_{2i-1}(x) \\ \end{...
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1answer
143 views

Evaluation of the sum $\sum\limits_{n=1}^{\infty}\frac1n\sin\frac1n$

I am trying to evaluate the sum $\displaystyle\sum_{n=1}^{\infty}\dfrac1n\sin\dfrac1n$. This was given in my real analysis test yesterday. I have proved that the sum exists: We know for any non-...
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4answers
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Integral $\int^{1}_{1/2}\frac{\ln(x)}{1-x}dx$

Evaluate $$\int^{1}_{1/2}\frac{\ln(x)}{1-x}dx$$ Try: Let $$ I =\int^{1}_{1/2}\frac{\ln x}{1-x}dx=\int^{1}_{1/2}\sum^{\infty}_{k=0}x^k\ln(x)dx$$ $$I =\sum^{\infty}_{k=0}\int^{1}_{1/2}x^k\ln(x)dx$$ ...
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1answer
45 views

Summation involving Binomial coefficients

$$S(m,k)=\sum_{n=m}^{k}(-1)^n\binom{n}{m}$$ Is possible to get closed form solution of this sum? We know $$T(m,k)=\sum_{n=m}^{k}\binom{n}{m}=\binom{k+1}{m+1}$$ But what if the sign alternates?
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28 views

Closed form for an integral involving an incomplete Gamma function?

I am trying to find a closed form for this integral: $$\int_{0}^{\infty}\int_{0}^{\infty}e^{-d_{p,s}^v\,x-d_{s,p}^v\, y+d_{p,p}^v\,\frac{\xi_1\,\sigma^2\,xy}{P_p\,\xi_2\,xy+\sigma^2}}\mathrm dy\...
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156 views

Evaluate $\int_{0}^{\frac{\pi}{2}}\frac{dx}{\left(\sqrt{\sin x}+\sqrt{\cos x}\right)^2}$

Evaluate $$I=\int_{0}^{\frac{\pi}{2}}\frac{dx}{\left(\sqrt{\sin x}+\sqrt{\cos x}\right)^2}$$ My try: Since $$f(x)=f\left(\frac{\pi}{2}-x\right)$$ we have: $$I=2\int_{0}^{\frac{\pi}{4}}\frac{dx}{\...
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Recurrence from closed form for $s_{a,b}(m)=(m-1)s_{a,b}(m-1)+s_{a,b}(m-2), s_{a,b}(0)=a, s_{a,b}(1)=b$

We have for $m>1$ $$s_{0,1}(m)=\sum\limits_{k=0}^{\left\lfloor{\frac{m-1}{2}}\right\rfloor}\binom{m-k-1}{k}\frac{(m-k-1)!}{k!}$$ $$s_{1,0}(m)=\sum\limits_{k=0}^{\left\lfloor{\frac{m}{2}}\right\...
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217 views

Closed form of $\int_0^\pi \ln\left(1+\sin^2(t)\right) dt$?

I attempted to evaluate this integral but seem to be getting nowhere$$I=\int_0^\pi \ln\left(1+\sin^2(t)\right) dt$$ Wolfram returns the value $I\approx 1.18266$ but was not able to provide a closed ...
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Closed form for an integral involving a generalized incomplete Gamma function?

I am trying to find a closed form for this integral: $$\int_{0}^{\infty}\int_{0}^{\infty}e^{-d_{p,s}^v\,x-d_{s,p}^v\, y+d_{p,p}^v\,\frac{\xi_1\,\sigma^2\,xy}{P_p\,\xi_2\,xy+\sigma^2}}\mathrm dy\...
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Analytical solution to the integral of ∫ x'dx?

I want to find the closed form of the integral $$\int \dot{x} dx \tag{1}$$ where $\dot{x} = \frac{dx}{dt}$. I think there should be an analytical solution. I know $$\int \dot{x} dx = \int \dot{x}^2 ...
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2answers
65 views

General closed form for $L(\phi)=\int_0^\phi \log(\sin x)\mathrm dx$ when $\phi\in(0,\pi)$?

I would like to know if there is a general closed form for $$L(\phi)=\int_0^\phi \log(\sin x)\mathrm dx,\qquad \phi \in(0,\pi)$$ Context: (below are also the extent of my search for a closed form....
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1answer
75 views

Finding the closed form for $x_n^2 = -x_{n-1}^2+6x_{n-2}^2+n$

I'm trying to find a closed form for the recurrence relation $x_n^2 = -x_{n-1}^2+6x_{n-2}^2+n$, with $x_1 = \frac{1}{4}, x_2=\frac{\sqrt{13}}{4}$ and $x_i \in \mathbb R^+$. My attempt was to let $z_n=...
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Is it true that $\int_0^1 \big(K(k^{1/2})\big)^2\,dk = \frac{7}2\zeta(3)$?

Define the complete elliptic integral of the first kind as, $$K(k) = \tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)$$ Part I. From the link above, we find some of the evaluations below, ...
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1answer
17 views

Why does ${\partial\over\partial t} w_i(tx)=\sum_{j=1}^n{\partial\over\partial x_j}w_i(tx)x_j$?

Let $w$ be a $1$-differntial form. Why does the equality holds? $${\partial\over\partial t} w_i(tx)=\sum_{j=1}^n{\partial\over\partial x_j}w_i(tx)x_j$$
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Finding $\int^{\frac{\pi}{8}}_{0}\ln(\tan x)\mathrm dx$

Finding $\displaystyle \int^{\frac{\pi}{4}}_{0}\ln(1+\cos x)\mathrm dx$ What I tried Put $\displaystyle I =\int^{\frac{\pi}{4}}_{0}\ln(1+\cos x)\mathrm dx$ \begin{align*} I&=\int^{\frac{\pi}{4}}...
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1answer
29 views

Prove Poincaré Lemma for $1$-form

Let $U\subseteq\mathbb{R}^n$ be an open set that contains $0$, and for all $t\in[0,1]$ and $ x\in U$, $tx\in\mathbb{R}^n$. Show that every closed differentiable 1-form $w$, (i.e. $dw=0$) is an exact ...
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3answers
67 views

Calculating $\int_{0}^{\pi}\text{sinh}\left(\sin\left(x\right)\right)\text{d}x$

I was wondering if it exists a beautiful exact value to $$ \int_{0}^{\pi}\text{sinh}\left(\sin\left(x\right)\right)\text{d}x$$ which has a nice graph over $\left[0,\pi\right]$, but i can't get to ...
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Integral $\int_0^\infty \frac{x^{2j}\mathrm dx}{(x^4+2ax^2+1)^{n+1}}$

I am working on an integral and I need some help. From the book Irresistible Integrals, I am given the problem of obtaining a closed form for: $$N(j,n;a)=\int_0^\infty \frac{x^{2j}\mathrm dx}{(x^4+...
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Value of a Two Variable Recursive Function

I have a function $f(x, y)$ where the base cases are $f(0, y) = y$, $f(x, 0) = 0$, and $f(x, 1) = 1$. In all other cases, $f(x, y) = max(\frac{f(i+1, y-1)*i+f(i-1, y+1)*y}{y+i})$, where $i$ ranges ...
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Integral $\int_0^1 \frac{\arctan x}{x^2-x-1}dx$

After seeing this integral I've decided to give a try to calculate:$$I=\int_0^1 \frac{\arctan x}{x^2-x-1}dx$$ That is because it's common for many integrals to have a combination of a polynomial in ...
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What is $\int_0^1 \left(\tfrac{\pi}2\,_2F_1\big(\tfrac13,\tfrac23,1,\,k^2\big)\right)^3 dk$?

As in this post, define the ff: $$K(k)=K_2(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)}$$ $$K_3(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac13,\tfrac23,1,\,k^2\right)}$$ $$K_4(k)={\...
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1answer
23 views

Closed form for an exponential recurrence

I was wondering what the closed form for a function $f(x)$ is when $f(x)=a*f(x-1)+b^\left(x-1\right)-c^\left(x+1\right)$, and $f(0) = d$ is given. Assume $a$, $b$, $c$, and $d$ are all greater than 1. ...
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2answers
40 views

Simplifying an expression with binomial coefficients and powers of $2$

$$\binom{n}{0} \cdot 2^n + \binom{n}{1} \cdot 2^{n-1} + \binom{n}{2} \cdot 2^{n-2} + \dots + \binom{n}{n} \cdot 2^{n-n}$$ Anyway to simplify this such that it can become of 'closed' form (i.e. a ...
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0answers
45 views

Closed formula for a sum

I'm trying to find out whether this sum has a closed formula: $$ \sum_{k=1}^{n}\frac{k^2a}{k^2a+b}, a,b\in\mathbb{R} $$ There isn't anything I've tried that has worked, so I'd appreciate any ideas!
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1answer
46 views

Express $\sum_{a=1}^{p-1} \lfloor{(v+qa)/p}\rfloor$ in closed form.

Here $p$ and $q$ are primes but likely not necessary for the answer. Also $p < q$ and $v\in\left\{{0,1,\dots,p*q-1}\right\}$. This problem arises from counting the number of reducible quadratics ...
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2answers
93 views

On the integral $\int_0^{\frac{\pi}{2}}x^{2n+1}\cot(x)dx$

While investigating the function $$A(z)=\int_0^\frac{\pi}{2} \frac{\sin(zx)}{\sin(x)}dx$$ I stumbled upon the integral $$\int_0^{\frac{\pi}{2}}x^{2n+1}\cot(x)dx$$ when attempting to calculate the ...
4
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2answers
72 views

How do I know whether the inverse function has a closed form?

I am interested in the function $$ y(x) := \left( x +\frac{3\pi}{2} \right) \sin(x) + \cos(x). $$ Over the range $ x \in \left[ -\frac{\pi}{2} ,\frac{\pi}{2} \right]$, this function grows ...
4
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2answers
77 views

A Probabilistic Drinking Problem

This question was asked by a fellow MSE user in chat. Motivational credits to @Quintec. Question: Bob goes to a bar and drinks a drink. On drink $n$, Bob has a $1-\left(\frac12\right)^n$ ...
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1answer
36 views

Which kinds of compositions of invertible elementary and nonelementary functions are elementary?

Let $f$ be a bijective elementary function, elementary invertible or not. Let $h$ be a bijective nonelementary function, elementary invertible or not. Which of the compositions $h(f(x))$ and $f(h(x))$ ...
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1answer
123 views

closed form for $\int_0^1\frac{\mathrm{Li}_s(x-x^2)}{x-x^2}\mathrm dx$

I am trying to evaluate $$F(s)=\sum_{n\geq1}\frac1{n^{s+1}{2n\choose n}}$$ I started off by noting that $$\frac1{n{2n\choose n}}=\frac12\int_0^1\left[x-x^2\right]^{n-1}\mathrm dx$$ So $$F(s)=\int_0^1\...
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2answers
118 views

Functions that are easy to antidifferentiate but whose inverses are hard to antidifferentiate

Some functions have antiderivatives that are harder to compute than the antiderivatives of their inverse, i.e. $\ln(x)$ is such a function. Another is $\arctan(x)$. So at the beginning of an ...
7
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1answer
139 views

closed form for $\int_{1/4}^{3/4} x^n(1-x)^n \, dx$

The integrand being a polynomial, I used the binomial formula to separate the monomials: $$\int_{\frac{1}{4}}^{\frac{3}{4}} x^n(1-x)^n \, dx = \sum_{k = 0}^{n}{ n \choose k}(-1)^{k}\int_{\frac{1}{4}}...