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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37 views

Does this tridiagonal system have a closed-form solution?

Let $$ A = \begin{pmatrix} a + c_1 & -b\\ -a & a+b+c_2 & -b\\ & -a & a+b+c_3 & -b\\ & &\ddots & \ddots & ...
16
votes
4answers
546 views

How to evaluate $I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$

Prima facie, this integral seems easy to calculate,but alas, this not's case $$I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$$ The numerical value is I=-1.122690024730644497584272... How to ...
6
votes
1answer
127 views

About the series $\sum_{n\geq 0}\frac{1}{(2n+1)^2+k}$ and the digamma function

Let we provide a closed form for $$ S_k = \sum_{n\geq 0}\frac{1}{(2n+1)^2+k} $$ for $k>0$ in terms of elementary functions. It is quite easy to check that $S_k$ can be computed in terms of the ...
0
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0answers
40 views

Euler type superdivergent

Could you explain where this came from: $$\sum _{k=0}^{\infty } (k!)^2 (-y)^k=\frac{G_{1,3}^{3,1}\left(\frac{1}{y}\mid{{0}\atop{0,0,0}}\right)+2 \left(\log \left(\frac{1}{y}\right)+\log (y)\right) ...
1
vote
2answers
85 views

Closed Form for Finite Sum: Product of two Similar Functions

I need to find a closed form expression in terms of $c$, $n$, $x$ and $y$ for $$ \sum_{j=0}^{n}\rho^{c-j}\frac{x^j}{j!}\frac{y^{c-2j}}{\left(c-2j\right)!} $$ where $c$ and $\rho$ are just constants. ...
1
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3answers
122 views

Updated: Prove completely $\int^\infty_0 \cos(x^2)dx=\frac{\sqrt{2\pi}}{4}$ using Fresnel Integrals

Prove completely $\int^\infty_0 \cos(x^2)dx=\frac{\sqrt{2\pi}}{4}$ I've tried substituting $ x^2 = t $ but could not proceed at all thereafter in integration. Any help would be appreciated. I should ...
0
votes
3answers
172 views

Exact values of the equation $\ln (x+1)=\frac{x}{4-x}$

I'm asking for a closed form (an exact value) of the equation solved for $x$ $$\ln (x+1)=\frac{x}{4-x}$$ $0$ is trivial but there is another solution (approximately 2,2...). I've tried with ...
2
votes
2answers
74 views

How do I find a closed form for this sequence?

I want to find a closed form for $$\sum_{i=0,1,...}{\left\lfloor\frac{n}{2^i}\right\rfloor}$$ e.g. ...
1
vote
0answers
62 views

Question on series being expressed in closed form

Given an integer $k$ and $0\leq \alpha \leq 1$, let $f_1(\alpha)=1/k$ and $f_{i+1}(\alpha)=\frac{(k-1)f_i(\alpha) + (f_i(\alpha)^{1/\alpha} + 1)^\alpha}{k}$. Consider the function $g(\alpha) = ...
2
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2answers
41 views

$\int _{0}^{\infty }\! \left( {\it W} \left( -{{\rm e}^{-1 -\epsilon}} \right) +1+\epsilon \right) {{\rm e}^{-\epsilon}}{d\epsilon}={\rm e} - 1$

How to prove $\int _{0}^{\infty }\! \left( {\it W} \left( -{{\rm e}^{-1 -\epsilon}} \right) +1+\epsilon \right) {{\rm e}^{-\epsilon}}{d\epsilon}={\rm e} - 1$ where W is the Lambert W function? Maple ...
1
vote
1answer
21 views

Is it possible to express this term by using “trace function”?

I wonder if we can express the following term by using "Trace function"? $$(X-\mu)^T \Sigma^{-1}(X-\mu)$$ This is the quadratic term in Multivariate Gaussian Distribution with mean of $\mu$ and ...
6
votes
1answer
1k views

Integral involving the error function of log(x)

Looking for a closed form for the integral $$\int_0^{\infty } e^{-\left(\frac{a-\log (x)}{b}\right)^2} \left(\frac{1}{2} \text{erf}\left(\frac{a-\log (x)}{b}\right)+\frac{1}{2}\right) \, ...
0
votes
0answers
31 views

closed form for $\sum_{k=0}^{n-1}\frac1{\binom{2n-1}{k}}\sum_{r=0}^{k}\binom{2n-1}{r}$?

Does there exist any closed form for the following sum? $$\sum_{k=0}^{n-1}\frac1{\binom{2n-1}{k}}\sum_{r=0}^{k}\binom{2n-1}{r}$$ Edit: Then can we find an asymptotic nice approximation as $n\to ...
28
votes
3answers
926 views

Closed form for $\int_0^\infty\arctan\Bigl(\frac{2\pi}{x-\ln\,x+\ln(\frac\pi2)}\Bigr)\frac{dx}{x+1}$

I'm trying to find a closed form for this integral: $$I=\int_0^\infty\arctan\left(\frac{2\pi}{x-\ln\,x+\ln\left(\frac\pi2\right)}\right)\frac{dx}{x+1}$$ Its approximate numeric value is ...
3
votes
5answers
110 views

Closed form for $\int \left(1-x^{2/3}\right)^{3/2}\:dx$

Find a closed-form solution to \begin{align}\int_0^1 \left(1-x^{2/3}\right)^{3/2}\:dx\tag{1},\end{align} or even more generally, is there a methodology to solving integrals of the type \begin{align} ...
1
vote
0answers
28 views

Closed form of $a_i = \lvert\{\, (e_1, \dotsc, e_r) \in [0, q)^r : e_1 + \dotsb + e_r = i \,\}\rvert $ [duplicate]

I encounter a curious sequence $a_i$ which is defined below. I wonder if it has a name and has some closed form. Let $r, q$ be positive integers. (Assume further that $q$ is a prime power if ...
10
votes
3answers
272 views

Finding the value of the infinite sum $1 -\frac{1}{4} + \frac{1}{7} - \frac{1}{10} + \frac{1}{13} - \frac{1}{16} + \frac{1}{19} + … $ [duplicate]

Can anyone help me to find what is the value of $1 -\frac{1}{4} + \frac{1}{7} - \frac{1}{10} + \frac{1}{13} - \frac{1}{16} + \frac{1}{19} + ... $ when it tends to infinity The first i wanna find the ...
0
votes
2answers
109 views

Find a formula for the nth Fibonacci Number [duplicate]

So I'm being asked to find a formula for the nth fibonacci number. I know the answer is $$x_{n}=\frac{(1+5^{1/2})^{n} -(1-5^{1/2})^n}{\sqrt{5}2^n}$$ However I don't really know how to get there. ...
4
votes
5answers
210 views

Solution of $\int \frac{1}{x^2 \sqrt{x^2+9}}dx$

I'm new of the site. I must solve this exercise: $$\int \frac{1}{x^2 \sqrt{x^2+9}}\,dx$$ I tried every substitution, but I didn't reach that I want. Can you help me, please?
0
votes
0answers
31 views

Closed form solution for DDEs?

I am solving the equation $X-A-B\exp(-Xy)-C\exp(-Xz)=0$ where $X, A, B$ and $C$ are 2x2matrices and $y$ and $z$ are scalars. What will be the closed form solution ...
1
vote
0answers
34 views

Iterations $F^n_h[f]$ of the operator $F_h[f]=D_h[f]\circ f^{-1}$

Let the $H$ be a collection of real valued invertible functions, define $f\circ g$ as composition, $f+g$ as the function $f+g(x):=f(x)+g(x)$ and define a family of functions $\{D_h\}_{h\in \Bbb ...
2
votes
1answer
44 views

Online database of formulae for series, infinite products, limits, ecc.

Around the site there are a lot of questions about closed form expressions or approximations for specific series, products, limits and whatnot. These however are hard to find to the difficulty in ...
2
votes
1answer
288 views

How can I show that $\prod_{{n\geq1,\, n\neq k}} \left(1-\frac{k^{2}}{n^{2}}\right) = \frac{\left(-1\right)^{k-1}}{2}$?

Assume $k$ positive integer. How can I show that $$ \tag 1 \prod_{{n\geq1,\, n\neq k}} \left(1-\frac{k^{2}}{n^{2}}\right) = \frac{\left(-1\right)^{k-1}}{2}? $$ I know that $$ \tag 2 ...
11
votes
1answer
397 views

Challenging identity regarding Bell polynomials

Note: [2015-03-08] A proof of the identity below was aimed to close the gap of a rather extensive elaboration of this answer of mine. The identity (1) below is part of a more complex one, which is ...
-1
votes
1answer
31 views

Pricing at the hourly rate only until this exceeds the daily rate

Pricing at the hourly rate only until this exceeds the daily rate Example: if a rent costs \$1 per hour, \$10 per day and a booking for 11 hours will be charged \$10. For two complete days it will ...
2
votes
0answers
35 views

Is there a general rule for proving that an equation has no analyticial solution

Somebody asked this here: Prove that an equation has no elementary solution But so far there is no response. The little math I know I have learnt it myself so I dont have a big picture of things. I ...
1
vote
0answers
47 views

Liouville–Hardy theorem: when is $\int f(x) \log(x) dx$ elementary?

I am currently writing a report on Liouville's theorems on integration in finite terms, and I am in the process of proving the Liouville–Hardy theorem. This is what I understand so far. Theorem ...
0
votes
1answer
39 views

How to convert this equation to closed form (regular and weighted linear regession)

I have read that the matrix form for the following summation $$ Error(w) = \sum_{i=0}^{m} w^{T}x_i - y_i $$ $w^T$ is the transpose of weights vector in linear regression $x_i$ is the ith input in ...
2
votes
1answer
108 views

Closed form for the summation $\sum_{k=1}^n\frac{1}{r^{k^2}}$

Is there any closed form for the finite sum $$\sum_{k=1}^n\dfrac{1}{r^{k^2}}$$ or infinite sum ( when $|r|<1$) $$\sum_{k=1}^\infty\dfrac{1}{r^{k^2}} ?$$ While solving this problem, I found this ...
3
votes
1answer
121 views

closed form for a double sum

How can I prove that $$\underset{k\geq1}{\sum}\left(\underset{m=-\infty}{\overset{\infty}{\sum}}\frac{\left(-1\right)^{m}}{\left(2k-1\right)^{2}+m^{2}}\right)=\frac{\pi\log\left(2\right)}{8}\,?$$I ...
-1
votes
1answer
65 views

Closed form of $\cot x=x$

I plotted the graphs of $y=\cot x$ and $y=x$. Its clear that they have infinite intersections. I tried to solve for the first root but it doesn't seem to be any known number to me. Even Wolfram Alpha ...
8
votes
2answers
85 views

show that $\int_0^{\infty}\sin(u\cosh x)\sin(u\sinh x)\frac{dx}{\sinh x}=\frac{\pi }{2}\sin u$

$$I(a)=\int_0^{\infty}\sin(u\cosh x)\sin(u\sinh x)\frac{dx}{\sinh x}:a>0$$ I started with $$\sin(a)\sin(b)=\frac{1}{2}(\cos(a-b)-\cos(a+b))$$ so $$I(a)=\frac{1}{2}\int_0^{\infty}\left ( ...
4
votes
2answers
135 views

Closed form for $ \prod_{k=1}^n (a+k^2) $

I have come across the following product: $$ \prod_{k=1}^n (a+k^2) $$ where $a$ is a positive constant. Could anyone suggest a closed form for this product? I need to approximate this for large $n$, ...
11
votes
2answers
292 views

Does there exist a closed form for $L_k$ for any $k>3$?

I defined a sequence $L_k$ as the limit of a sequence of "hyperharmonic" series in this question. I was surprised to find that $L_3=(\sqrt{13+4\sqrt2}-1)/2$, but was unable to find a representation ...
4
votes
1answer
76 views

Evaluting sum $\sum_{n=0}^\infty\frac{n^k}{n!}$

Inspired by this question,I was interested if the following sum has a closed form.Looking for $k$ integer I found the Dobinski's formula so that the sum when $k$ is natural number is $e\cdot B_k$ ...
4
votes
3answers
62 views

Hypergeometric 2F1 with negative c

I've got this hypergeometric series $_2F_1 \left[ \begin{array}{ll} a &-n \\ -a-n+1 & \end{array} ; 1\right]$ where $a,n>0$ and $a,n\in \mathbb{N}$ The problem is that $-a-n+1$ is ...
3
votes
1answer
66 views

Variation on Stokes Theorem for Manifolds (2)

Let $\omega \in \Omega^0(\mathbb{R}^{2}\setminus\{0\})$ be a $0$-form such that $d\omega=0$. Is the following statement true: For any compact, oriented, $0$-dimensional submanifold $M$ of ...
0
votes
2answers
76 views

Variation on Stokes Theorem for Manifolds

Let $n >1$ and $\omega \in \Omega^{n-1}(\mathbb{R}^{n+1}\setminus\{0\})$ such that $d\omega = 0$. Is the following statement true: For any compact, oriented, $(n-1)$-dimensional submanifold $M$ ...
4
votes
3answers
166 views

Sum: $\sum_{n=1}^\infty\prod_{k=1}^n\frac{k}{k+a}=\frac{1}{a-1}$

For the past week, I've been mulling over this Math.SE question. The question was just to prove convergence of an infinite sum, but amazingly WolframAlpha told me it had a remarkably simple closed ...
9
votes
3answers
285 views

Closed form for ${\large\int}_0^\infty\frac{x-\sin x}{\left(e^x-1\right)x^2}\,dx$

I'm interested in a closed form for this simple looking integral: $$I=\int_0^\infty\frac{x-\sin x}{\left(e^x-1\right)x^2}\,dx$$ Numerically, ...
5
votes
1answer
157 views

What is the expected number of questions answered to complete a sequence in which wrong answers send you to the start?

Given a sequence of n questions that each contain x answer choices, what is the expected number of questions answered before answering all questions correctly if answering a question incorrectly sends ...
7
votes
2answers
137 views

How to evaluate $\sum _{n=1}^{\infty } \frac{(-1)^{n+1} H_{2 n}^{(2)}}{n} = 2\zeta(3) - \frac \pi 2 G- \frac {\pi }{48}\ln 2$?

What is the best way to calculate the following sum?$$S=\sum _{n=1}^{\infty } \frac{(-1)^{n+1} H_{2 n}^{(2)}}{n} = 2\zeta(3) - \frac \pi 2 G- \frac {\pi^2}{48}\ln 2$$ I tried putting $$f(z) = ...
5
votes
1answer
138 views

How can I evaluate $\int_0^{\pi/2}\frac{x\cos{x}}{3\sin^2x+1}dx$ and $\int_0^{\pi/2}\frac{x\cos{x}}{\sin^2x+3}dx$?

I do not find the closed form of the following integrals$$\int_0^{\pi/2}\frac{x\cos{x}}{3\sin^2x+1}\mathrm dx$$ $$\int_0^{\pi/2}\frac{x\cos{x}}{\sin^2x+3}\mathrm dx$$ On the other side, I find ...
2
votes
0answers
52 views

What techniques does Mathematica use to find solutions to these sequences?

This question is related to my previous question: Need help finding a closed form for complicated sum. An answer to that question led my to try and find the general term of the following recurrence: ...
2
votes
2answers
157 views

Finding the closed form for $\sum_{n=1}^{\infty }\frac{\zeta (4n)}{\beta^{4n-1}}$ [closed]

Finding the closed-form $$\sum_{n=1}^{\infty }\frac{\zeta (4n)}{\beta^{4n-1}}$$ for $\beta\in(1,+\infty)$. I learned from this site many many important things but I till need more, so I need ...
1
vote
1answer
138 views

Need help finding a closed form for complicated sum

I'm trying to find a closed form expression for the following sequence: $$a_n=\sum_{i=1}^{n}\frac{(n-1-i+d)!}{(n-2i)!(i)!}=\sum_{i=1}^{\frac{n}{2}}\frac{(n-1-i+d)!}{(n-2i)!(i)!}$$ Where $n$ and $d$ ...
0
votes
0answers
38 views

Is there a closed form polynomial for this integral recursion?

While working on some statistical problems, I startet playing with integral recursions of the type $$p_{n+1}(x)=\int_a ^x \mathrm{d} y\; q(y)p_n(xy)$$ Here $q(y)$ and $p_0(x)$ are given polynomials, ...
6
votes
1answer
124 views

Evaluating $\int \arccos\bigl(\frac{\cos (x)}{r}\bigr)\sin^2(x){\mathrm dx}$

Following from the previous question Evaluating $\int \arccos\left(\frac{\cos(x)}{r}\right) \, \mathrm{d}x$ I now need the extra $\sin^2x$ as in the title. Of course one power of $\sin(x)$ is easy, ...
6
votes
2answers
148 views

Computing a nasty integral (probably with computer algebra system)

I'm trying to do this integral, not sure if it is possible: $$ \int_{1}^{\infty}\int_{0}^{\infty} \exp\left(\, -\,{x^{2} \over y^{2}}\,\right) \exp\left(\,-\,{y^{2} \over z^{2}}\,\right) \exp\left(\, ...
1
vote
1answer
129 views

How to calculate $ \int_0^1 \frac{(1+x)^{2r-1}}{1+x^2}dx $?

Is there a way to express $$ \int_{0}^{1}{\left(\, 1 + x\,\right)^{2r\ -\ 1} \over \!\!\!\!\!\!\! 1 + x^{2}} \,{\rm d}x $$ in a closed form with $r\in\mathbb{N}$?