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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17
votes
3answers
533 views

Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.

What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
1
vote
0answers
30 views

Closed-form expressions of $\sum_{n=1}^\infty \frac{\sin^2(an) e^{-bn^2}}{n^2}$

Does anybody know if there's a closed-form expression of this series? $$\sum_{n=1}^\infty \frac{\sin^2(an) e^{-bn^2}}{n^2}$$ where $a$ and $b$ are strictly positive. It's easy to see that it's ...
1
vote
2answers
56 views

Find a closed form for the equations $1^3 = 1$, $2^3 = 3 + 5$, $3^3 = 7 + 9 + 11$

This is the assignment I have: Find a closed form for the equations $1^3 = 1$ $2^3 = 3+5$ $3^3 = 7+9+11$ $4^3 = 13+15+17+19$ $5^3 = 21+23+25+27+29$ $...$ Hints. ...
5
votes
1answer
123 views

General Solution of $y'(x)+p(x)e^{r(x) y(x)}=q(x)$

I solved the case for the non-homogenous constant coefficients case and I wondered if there is a way to find a general solution to a non-constant coefficient case. I don't know how to approach this at ...
8
votes
1answer
270 views

Closed-form of integral $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

I'm looking for a closed form of this definite iterated integral. $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$ From Vladimir ...
2
votes
1answer
21 views

Closed Form Solution for Recurrence Relation

Is it possible to calculate the closed form solution for the following recurrence relation? $$ T(n) = T\left(\frac{n}{2}\right) + T\left(\frac{n}{2} + 1\right) + \frac{n}{2} $$ I am trying to teach ...
16
votes
6answers
566 views

Closed form for $\int_{0}^{1/2}\left(2x - 1\right)^{6}\ \log^{2}\left(2\sin\left(\pi x\right)\right)\,{\rm d}x$

How can I find a closed form for the following integral $$ \int_0^{1/2}\left(2x - 1\right)^{6}\ \log^{2}\left(2\sin\left(\pi x\right)\right) \,{\rm d}x $$
7
votes
1answer
212 views

How to find this integral $\int_{0}^{1}\frac{x}{1-x^4}\arctan{\frac{x-x^5}{1+x^6}}\,dx$

Find the integral value $$ I=\int_{0}^{1} {x \over 1 - x^{4}}\,\arctan\left(x - x^{5} \over 1 + x^{6}\right)\,{\rm d}x $$ My good friends gave me this problem, and I can't solve it. Using computer I ...
23
votes
3answers
553 views
1
vote
1answer
34 views

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

$$\int_{-\infty}^\infty \!\!\text{d} r\dfrac{1}{r}e^{\frac{-(r-r_0)^2}{\delta^2}}\sin(k r)$$ Where $\delta>0$, $r_0\in \mathbb{R}$. Can anyone help me with this? it seems to me there has to be a ...
0
votes
2answers
49 views

Fixing the closed form of $\sum_{k=1}^nk\sin^2(kx).$

I've been working on finding the closed form of this:$$\sum_{k=1}^nk\sin^2(kx).$$ Using the fact that:$$\sum_{k=1}^nku^k={u\over (1-u)^2}\bigg[nu^{n+1}-(n+1)u^n+1\bigg]\forall u\ge 1\quad (1)$$ I ...
1
vote
0answers
13 views

Simplified expression of $ _2F_1((K-1)a,K,Ka,x) $

Is there any simplified expression of this Hypergeometric function $ _2F_1((K-1)a,K,Ka,x) $ Thanks!
1
vote
2answers
59 views

Trying to find the closed form for the nth term of $\frac{1}{1-x^4}$

I know that $\frac{1}{1-x^4}$ is the generating function for the sequence (1, 0, 0, 0, 1, 0, 0, 0, 1, ...) I don't know how to find the closed form for the nth term though. Itried messing around with ...
-2
votes
1answer
34 views

Evaluating the series with arctangents: $\sum_{r=1}^\infty \tan^{-1}\frac{2r}{2+r^2+r^4}$

If $$S=\sum\limits_{r=1}^\infty\tan^{-1}\left(\frac{2r}{2+r^2+r^4}\right)$$ then what is cot S? Options: A) 1; B) 3; C) 1/3; D) 2 Does it converge? I don't really know how to find the ...
4
votes
2answers
82 views

How solve $\int \frac{dx}{(x^2-x)^x}$ [on hold]

I want solve $$\int \frac{dx}{(x^2-x)^x}$$. thanks for help
0
votes
2answers
61 views

How to find the generating function and the closed form for the generating form

I'm trying to find the generating function and the closed form for the generating form for this sequence: $0,1,-2,4,-8,16,-32,64...$ I've tried the following: I think it's an index shift so that's ...
1
vote
1answer
18 views

Is a finite continued fraction a closed-form expression?

We had a discussion regarding this answer on Electrical Engineering. The answer in question discussed a finite continued fraction. We're wondering whether it's a closed-form expression or not. ...
2
votes
1answer
25 views

Closed form for $x+2^ax^2+3^ax^3+…+n^ax^n$

I was wondering if there was a closed form for $$f(x)=x+2^ax^2+3^ax^3+...+n^ax^n+...$$ I have tried to find one but I had no luck. If you divide by $x$ and then integrate you get ...
3
votes
1answer
14 views

How does the recursion relation work in the solution to this differential equation (using series)?

Sorry for the vague title but it would not let me post the first step and last step of this equation (too many characters!). How does $$\dfrac{a_0}{3n(3n-1)(3n-3)(3n-4)\cdots 9 \cdot 8 \cdot 6 \cdot ...
1
vote
1answer
15 views

Find a closed form equation of the following sequence: ${0,0,-2,0,4,0,-6,…}$

Find a closed form equation of the following sequence: ${{0,0,-2,0,4,0,-6,...}}$ I know $1+-1^n$ = 0 if n is odd and 1 if n is even. However finding alternating signs when plugging in only even ...
1
vote
1answer
25 views

Is there a closed form to $a_{n+2}=\frac{(n+1)(n-2)a_{n+1} + (4n+3)a_n - a_{n-1}}{(n+2)(n+1)}$ in terms of $a_0$ and $a_1$?

Is there a closed form solution to $$a_{n+2}=\dfrac{(n+1)(n-2)a_{n+1} + (4n+3)a_n - a_{n-1}}{(n+2)(n+1)}$$ that can be written in terms of $a_0$ and $a_1$ given the fact that that $$a_2 = \dfrac{2a_1 ...
2
votes
1answer
30 views

Closed-form of prime zeta values

The prime zeta function is defined as $$P(s)=\sum_{p\,\in\mathrm{\mathcal P}} \frac{1}{p^s},$$ where $\mathcal P$ is the set of prime numbers. It converges for all $\Re(s)>1$. There is a related ...
7
votes
1answer
120 views
+100

Integral/infinite sum related to Bessels which pop up in optical coherence theory

In propagating partially coherent optical fields, the following integral pops up: $$I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta,$$ where $a$ and $b$ are real numbers. If we ...
2
votes
0answers
39 views

Closed-form of $\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$

Is there a possibility to find a closed-form for $$\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$$ We have $$I=\int_0^1\frac{Li_2^3(-x)+x^4Li_2^3(-\frac{1}{x})}{x^3}\,dx$$ After repeatedly ...
12
votes
4answers
496 views

Closed form of $I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx$

Does the integral below have a closed-form: $$I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx,$$ where $\tan^{-1} (\cdot)$ is inverse tangent function. ...
8
votes
2answers
84 views

Closed- form of $\int_0^1 \frac{{\text{Li}}_3^2(-x)}{x^2}\,dx$

Is there a possibility to find a closed-form for $$\int_0^1 \frac{{\text{Li}}_3^2(-x)}{x^2}\,dx$$
0
votes
0answers
18 views

General question: Algorithm or procedures to show existence of closed forms of any infinite series?

Having seen many questions regarding finding closed form of integrals or infinite series, and some users providing either the final answer or detailed solution, and also reading how one finds a closed ...
3
votes
2answers
143 views

Solving a 2 independent variables (2nd degree) recurrence relation

Changes to the recurrences and definition are changed! See here: $f(n, 1) = 2n^2 $ and $f (n, k) = 0$ for $k \geq 2n$ and for $k < 0$ and $f(n, 2n-1) = 1$ for all $n$. Question: Is it possible ...
5
votes
1answer
28 views

Conditional iterations constant.

Let $f(0)=2.$ Define for positive integers $n$ : $f(n+1) = \frac{3}{2} f(n)$ if $f(n)$ is even. $f(n+1) = \frac{3}{2}(f(n)+1)$ if $f(n)$ is odd. We now have $\lim_{n->\infty} \dfrac{4* (3/2)^{n} ...
4
votes
2answers
94 views

How to solve $\int_{0}^{2\pi} \frac{\cos(50x)}{5+4\cos(x)} dx\,?$

I encountered this integral and tried to solve it. As you can expect I could not solve this and thought I will ask it here. The integral is: $$\int_{0}^{2\pi} \frac{\cos(50x)}{5+4\cos(x)}\, dx$$ I ...
7
votes
4answers
218 views

How to evaluate $\int_0^\infty \frac{1}{x^n+1} dx$

Noticed that the integral $$\int_0^\infty \frac{1}{x^n+1} dx$$ is often approached with partial fraction decomposition, but this gets increasingly ugly as $n$ gets bigger. Is there a neat trick to do ...
2
votes
0answers
16 views

Closed-form expectation of CES function of a random variable?

I am faced with the following function, called CES (constant elasticity of substitution), of the continuously-distributed random variable $\epsilon$: $f(\epsilon) = (a^\sigma + ...
5
votes
1answer
197 views

Closed-form formula for the $n^{\rm th}$ term of ${1,1,1,1,\ldots, 1}, {2,2,2,2,\ldots, 2},\ldots, {k-1, k-1}, k.$

Let $k$ be a positive integer. Consider a finite sequence $L_k(n)$ given by $$\underbrace{1,1,1,1,\ldots, 1}_{k\text{ terms}}, \underbrace{2,2,2,2,\ldots, 2}_{k-1\text{ terms}},\ldots, ...
4
votes
1answer
30 views

$(1-t^2)\frac{\mathrm{d}^2y}{\mathrm{d}t^2}-t\frac{\mathrm{d}y}{\mathrm{d}t}+(a+2q (1- 2t^2))y=0$

So I have to solve $$(1-t^2)\frac{\mathrm{d}^2y}{\mathrm{d}t^2} -t\frac{\mathrm{d}y}{\mathrm{d}t}+(a+2q (1-2t^2))y=0$$ All substitutions seem to fail, some trigonometric ones fail less than the rest, ...
6
votes
3answers
95 views

what's the summation of this finite sequence?

$a$ and $b$ are positive integers. The summation is $$\sum\limits_{x = 1}^a {x\left( {\begin{array}{*{20}{c}} {a + b - x}\\ b \end{array}} \right)} .$$ Any closed-form expression? I thought it ...
8
votes
1answer
85 views

Sum of Harmonic numbers $\sum\limits_{n=1}^{\infty} \frac{H_n^{(2)}}{2^nn^2}$

Finding the closed form of: $$\sum\limits_{n=1}^{\infty} \frac{H_n^{(2)}}{2^nn^2}$$ where, $\displaystyle H_n^{(2)} = \sum\limits_{k=1}^{n}\frac{1}{k^2}$ It appears when we try to determine the ...
1
vote
2answers
42 views

How find the following integral?

I want find a closed Form for below integral$$\int \frac{1}{-1-aX+\frac{1}{2}bX^2} dX$$. thanks for help
0
votes
2answers
32 views

Find a closed form of $\sum_{i=0}^{n}\frac{x^i}{\left(1-x^2\right)^i}$.

Let $\displaystyle f(x) = \sum_{i=0}^{n}\dfrac{x^i}{\left(1-x^2\right)^i}$ While solving a problem I came up with this function which requires me to solve this function into a closed form. How do I ...
10
votes
1answer
116 views

An integral $\int^\infty_0\frac{\tanh{x}}{x(1-2\cosh{2x})^2}{\rm d}x$

I would like to enquire about the possible methods of computing the following integral $$\color{blue}{\int^\infty_0\frac{\tanh{x}}{x(1-2\cosh{2x})^2}{\rm d}x=\ ?}$$ A possible way I see of doing this ...
15
votes
5answers
204 views

Evaluating $\sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_{2n}}{n}$

I would appreciate to understand the main steps giving the evaluation of this series: $$ S=\sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_{2n}}{n}$$ where $H_n$ is the harmonic number. I've tried with no ...
6
votes
3answers
246 views

The Integral of Multiple Tangent Functions

I need help to find the numerical values to the precision at least $50$ digits (the closed forms if possible) for the following integrals \begin{equation} ...
1
vote
1answer
248 views

Closed form for general recursive function

Does a closed form exists for general recursive functions? my guess is not, but what types can be solved or what are the constraints on a recursive function so it has a closed form, what are some ...
8
votes
6answers
184 views

Easiest way to find $\Re\int_{0}^{\pi/2} e^{e^{i\theta}} d\theta$

How do we find $$\Re\left[\int_0^{\large\frac{\pi}{2}} e^{\Large e^{i\theta}}d\theta\right]$$ In the shortest and easiest possible manner? I cannot think of anything good.
5
votes
0answers
97 views

Value of a Sine-Like Infinite Product

Does the following infinite product have a "nice" closed form? $$ P = \prod_{k=2}^{\infty} \left(\left(1 - \frac{1}{k^2}\right)^\dfrac{(-1)^k}{k}\right) $$ I know that without the power one could ...
5
votes
1answer
131 views

Dilogarithm in closed form

Is there a closed form expression for \begin{align} e^{\Large\frac{i\pi}3} \text{Li}_{2}\left( \frac{e^{\Large\frac{i\pi}3} }{2}\right) + e^{-\Large\frac{i\pi}3} \text{Li}_{2}\left( ...
1
vote
1answer
51 views

Help in simplifying this nasty expression obtained after binomial expnasion

I have arrived to the following expression and was wondering if anyone can help me further simplify to something nicer, $$F= 1- [1-\text{exp} (- \alpha(N) ) ]^N= 1- \sum_{k=0}^{N} \binom{N}{k} ...
16
votes
2answers
238 views

An integral by O. Furdui $\int_0^1 \log^2(\sqrt{1+x}-\sqrt{1-x}) \ dx$

The following integral was proposed in a paper by O. Furdui, namely $$\int_0^1 \log^2(\sqrt{1+x}-\sqrt{1-x}) \ dx$$ and then the generalization $$\int_0^1 \log^2(\sqrt[k]{1+x}-\sqrt[k]{1-x}) \ ...
1
vote
0answers
40 views

Zeros of $f$ in a disk

If $f$ holomorphic in a domain $U$ and $f(z)\neq 0$ for all $z\in U$ then every zero of $f$ is such that $f(q)=0$ and $\det(Df_{p})>0$. Using that I have to prove that if $f$ keeps that conditions ...
23
votes
3answers
749 views

Closed form of $\displaystyle\mathscr{R}=\int_0^{\frac{\pi}{2}}\sin^2x\,\ln\big(\sin^2(\tan x)\big)\,\,dx$

Inspired by Mr. Olivier Oloa in this question. Does the following integral admit a closed form? \begin{align} \mathscr{R}=\int_0^{\Large\frac{\pi}{2}}\sin^2x\,\ln\big(\sin^2(\tan x)\big)\,\,dx ...
0
votes
0answers
28 views

List of functions $\chi_{s,a}(n)$ defined on a Group such that $\chi_{s,a}(n)\in{s,a}$ and depending on the parity

Question Let $(G,\cdot,e)$ be a non-commutative group and $s,a \in G$ .I'm looking for interesting functions $\chi_{s,a}:\Bbb N \rightarrow G$ witht this property $$\chi_{s,a}(n)= \begin{cases} s, ...