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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1answer
37 views

Multi-index power series

What is closed-form expression for the summation $$ S(n,m)=\sum_{|\alpha|=m} p^{\alpha} = \sum_{\alpha_1 + \cdots + \alpha_n = m} \prod_{i=1}^n p_i^{\alpha_i} $$ as a function of $n$ and $m$? Here ...
8
votes
1answer
219 views

Other integral related to Ahmed's integral

I have a doubt regarding the evaluation of the following integral : $$ \int_0^\frac{1}{\sqrt{5}} \frac{\tan^{-1}\left({\sqrt{(1 + x^2)/2}}\right)} {(1 + 3x^2)\sqrt{1 + x^2}}\,du = ...
8
votes
1answer
123 views

How to integrate $\frac{x^{2}\log {\sin x}}{1+x^{6}}$

I recently stumbled upon a question $$\int_0^{\infty}\frac{x^{m-1}\log^{a}x}{1+x^n}dx$$ I was able to evaluate it,but I am curious if there exists a closed form for, ...
2
votes
1answer
30 views

Is there a closed form for a sequence invariant under “Cauchy square”?

For two sequences $a=(a_n), b=(b_n),$ define the Cauchy product as $a*b=(c_n),$ where $c_n=\sum_{k=0}^{n}a_kb_{n-k}.$ Then is there a closed form expression for a sequence $(a_n)$ whose product with ...
10
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0answers
379 views

Analytic form of: $ \int \frac{\bigl[\cos^{-1}(x)\sqrt{1-x^2}\bigr]^{-1}}{\ln\bigl( 1+\sin(2x\sqrt{1-x^2})/\pi\bigr)} dx $

Background: On my quest to solve difficult integrals, I chanced upon this site: http://www.durofy.com/5-most-beautiful-questions-from-integral-calculus/ Good problems for me, (novice), although I ...
3
votes
2answers
43 views

Explicit formula for IFS fractal dimesnion

Is there an explicit formula for finding the box counting dimension of an arbitrary IFS fractal, such as the IFS fern or any other random IFS fractal? If not, is there at least a summation, or ...
5
votes
1answer
153 views

Closed form of $\int_{0}^{\pi/2}x\cot\left(x\right)\cos\left(x\right)\log\left(\sin\left(x\right)\right)dx$

I would like to know if there exists a closed form for this integral $$\int_{0}^{\pi/2}x\cot\left(x\right)\cos\left(x\right)\log\left(\sin\left(x\right)\right)dx.$$ I tried the relation ...
3
votes
1answer
75 views

closed form for $\int_{0}^{1/2}\frac{x\cos\left(x\pi\right)^{2}\cos\left(2\pi kx\right)}{\sin\left(x\pi\right)}dx,k\in \mathbb{N}$

I would know if exists a closed form for $$\int_{0}^{1/2}\frac{x\cos^{2}\left(\pi x\right)\cos\left(2\pi kx\right)}{\sin\left(\pi x\right)}dx,k\in\mathbb{N}.$$I tried integration by parts without ...
5
votes
3answers
85 views

Closed form for the partial sum $\sum\limits_{k = 1}^n \frac{\ln k}k$

I'd like to find a closed form for this partial sum: $$\sum\limits_{k = 1}^n \frac{\ln k}k$$ Using the properties of the logarithms, I converted the above into $$\ln\left(\prod_{k = 1}^n ...
0
votes
0answers
34 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 & ...
14
votes
4answers
446 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 ...
4
votes
1answer
92 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
30 views

Euler type superdivergent

Could you explain where it come from $$\sum _{k=0}^{\infty } (k!)^2 (-y)^k=\frac{G_{1,3}^{3,1}\left(\frac{1}{y}| \begin{array}{c} 0 \\ 0,0,0 \\ \end{array} \right)+2 \left(\log ...
1
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2answers
71 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
104 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
162 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
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2answers
62 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
60 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
37 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
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1answer
18 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 ...
5
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
21 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 ...
25
votes
3answers
798 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 ...
2
votes
5answers
100 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
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0answers
22 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
263 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
89 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
140 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
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0answers
30 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
32 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
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1answer
41 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
219 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
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1answer
368 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 ...
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1answer
21 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
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0answers
28 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
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0answers
28 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
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1answer
35 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 ...
0
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1answer
35 views

Closed Form of n(mod7) [closed]

For an integer n,what is the closed form as a function of n, if it exists, of n(mod7)={0,1,2,3,4,5,6,0,1,2,3,4,5,6,0,1,2,3,4,5,6,0,...,n(mod7)}? The closed form of n(mod8) uses trigonometric ...
2
votes
1answer
94 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
100 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
63 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
81 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
125 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
288 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
69 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
54 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
56 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
67 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
148 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 ...
8
votes
3answers
240 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, ...