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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Does this series have a closed form?

A friend of mine asked me if I could find a closed form for the series: $$ S = \sum_{n=-\infty}^{\infty} (n-h)^{\alpha} e^{-\beta(n-h)^2}, $$ with $\alpha,\beta > 0$. I don't even know how to ...
2
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0answers
25 views

Closed form expression for a sum

I want to calculate a sum of the form $$\sum_{k=0}^m \frac{\Gamma[m+1+\alpha-k]^2}{\Gamma[m+1-k]^2}\frac{\Gamma[x+k]}{\Gamma[x]k!}$$ where $m>0$ and belongs to integers and $\alpha$ takes half ...
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1answer
31 views

Sequence closed expression or others

What are closed expression or any other expression (involving integrals, specials functions...) for $\sum_{k=0}^{n}(n-2k)^t\frac{n!}{k!(n-k)!}$ where $t>0$ integer Thank you
5
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0answers
67 views

Closed-form of $\int_0^{\pi/2} \arctan(x)\cot(x)\,dx$

I'm looking for a closed-form of the following integral problem. $$I = \int_0^{\pi/2} \arctan(x)\cot(x)\,dx.$$ The numerical approximation of $I$ is $$I \approx ...
0
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1answer
19 views

Closed form expression for $\sigma$

A student I'm tutoring came to me with a problem in which he needs to find a closed-form expression in $\sigma$, $E(|Y|)$. $Y$ has a normal distribution with mean $0$ and standard deviation $\sigma$. ...
0
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1answer
37 views

Closed form for this 2 variable recurrence?

I'm trying to find a closed form for this two variable recurrence, but Wolfram Alpha does not seem to understand the input. $$ \begin{cases} a_{0,1} = 1 \\ a_{0,i} = 0 \quad \forall i\neq1 \\ ...
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0answers
31 views

Manipulation of summations

this question branches off another question that can be seen here Now we begin be taking a look at the following expressions: $$ \sum_{k=1}^{n-l} \sum_{j-0}^m \frac{\ln(g)^{m-j}}{g^k} ...
1
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1answer
23 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 ...
2
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0answers
104 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 = ...
6
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1answer
103 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
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1answer
25 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 ...
7
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0answers
203 views
+50

Closed 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 ...
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0answers
20 views

Transcendental quadratic equation involving Lambert's $W $function

Trying to solve an equation between a logarithm and a rational polynomial I rearranged the therms to looks like this. $3W^2(x)+(1-x)W(x)-e^x=0$ Are there any ways in which i can solve this without ...
1
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2answers
20 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 ...
4
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1answer
121 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 ...
2
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1answer
67 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 ...
4
votes
3answers
74 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 ...
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0answers
30 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
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4answers
349 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 ...
3
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1answer
80 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
29 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
52 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
99 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
1answer
92 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
58 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. ...
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0answers
49 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
31 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 ...
0
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0answers
4 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 ...
20
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3answers
659 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
98 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} ...
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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 ...
11
votes
3answers
257 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
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2answers
70 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
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5answers
128 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
24 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
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0answers
29 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
35 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
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1answer
115 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 ...
2
votes
0answers
76 views

Solve $\gcd(a+b+c,a^2+b^2+c^2)=1$ [closed]

is it possible to solve this for closed forms expression? It seems simple but I been trying it for a long time, still nothing. Thnx
10
votes
1answer
297 views

Challenging identity regarding Bell polynomials

Note: A proof of the identity below will 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 stated in Part 3, ...
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1answer
19 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
22 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
21 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
30 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
votes
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
89 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
82 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 ...
0
votes
1answer
55 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 ...