Tagged Questions

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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3
votes
2answers
65 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
36 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
13 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
14 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 ...
0
votes
1answer
22 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
0answers
29 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 ...
2
votes
1answer
29 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 ...
8
votes
2answers
77 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
16 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 ...
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 + ...
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, ...
5
votes
1answer
27 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} ...
8
votes
1answer
81 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 ...
4
votes
2answers
92 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 ...
9
votes
1answer
109 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 ...
6
votes
3answers
93 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 ...
5
votes
1answer
168 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, ...
5
votes
0answers
100 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 ...
1
vote
2answers
47 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} ...
3
votes
2answers
140 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
110 views

How to find closed-form of $\int_{0}^{+\infty} \operatorname{sech}^2 (x^2)\,dx$

How to find this integral closed form: $$I=\int_{0}^{+\infty}\operatorname{sech}^2{(x^2)}\,dx$$ where $\operatorname{sech}{(x)}$ is defined as secant of hyperbolic function. This problem ...
1
vote
0answers
38 views

New identity for sums of Bessel functions?

I've come across the following proposed identity: $$ ...
2
votes
3answers
40 views

Closed form for certain trigonometric integral

\begin{align}&\mbox{Is there a closed form for} \\[2mm]&\int_0^{\pi/2} \sin^{2}\left(\, nx\,\right)\sin\left(\, mx\,\right)\cot\left(\, x\,\right) \,{\rm d}x\ \quad\mbox{where}\quad m, n\ ...
0
votes
0answers
27 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, ...
2
votes
0answers
36 views

Closed form of a “harmonic” alternating dilogarithm sum

Does the following sum $$ S = \sum_{n\geq 2}(-1)^n \mathrm{Li}_2(2/n) = 1.14434\ 42096\ 91982\ 23727\ 39852\ 45805\ldots $$ have a closed form in terms of known constants? Neither the inverse ...
5
votes
2answers
58 views

An infinite exponential sum

I was trying to create a problem for a test I'm writing, and I ended up attempting to evaluate $$\sum_{n=1}^{\infty} \dfrac{1}{e^n-1}.$$ This definitely converges, but I have no idea how to go ...
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 ...
0
votes
1answer
47 views

How to solve integrals using series?

Many places I have seen when solving integrals you change a lot of it into sums. Finding $\int_{0}^{\pi/2} \dfrac{\tan x}{1+m^2\tan^2{x}} \mathrm{d}x$ Is just an example. So in general, how do you ...
0
votes
1answer
75 views

$\displaystyle\int_0^{\infty} \dfrac{\mathrm{d}x}{(x^4+ax^3+bx^2+cx+d)^m}$

It's a generalization for $$F(a,m)=\int_0^{\infty}\dfrac{\mathrm{d}x}{(x^4+2ax^2+1)^{m+1}}$$ that's being evaluated at Irresistible integrals, George Boros and Victor H. Moll 2004. I wonder if there ...
21
votes
3answers
719 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 ...
6
votes
3answers
240 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} ...
2
votes
1answer
98 views

summation of a finite sequence?

What is the summation of the finite sequence: $$\sum\limits_{i = 1}^n {\frac{1}{i}\left( {\begin{array}{*{20}{c}} {2i - 2}\\ {i - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {2n + 2 - 2i}\\ ...
18
votes
0answers
357 views

Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $

I need the method which can find this integral (the closed-form if possible). $$ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}\,dx $$ I used the relationship between $\tan x$ and $\tanh x$ but it didn't ...
5
votes
1answer
117 views

what is the summation of such a finite sequence?

The summation is: $$\sum_{i=0}^n \binom{2i}i \binom{2n-2i}{n-i}$$ The answer is $4^n$. How to prove it, and how to think out it?
8
votes
2answers
110 views

Closed form of $\displaystyle\int_{0}^{\pi/4}\int_{\pi/2}^{\pi}\frac{(\cos x-\sin x)^{y-2}}{(\cos x+\sin x)^{y+2}}\, dy\, dx$

Can the following double integral be evaluated analytically \begin{equation} I=\int_{0}^{\Large\frac{\pi}{4}}\int_{\Large\frac{\pi}{2}}^{\large\pi}\frac{(\cos x-\sin x)^{y-2}}{(\cos x+\sin ...
16
votes
2answers
233 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}) \ ...
5
votes
0answers
110 views

Integral of a product of five Bessel functions of order $0$

Does the following integral have a closed form? $$ \mathcal{J}(2,3,5,7,11) = \int_0^\infty x J_0(x\sqrt{2})J_0(x\sqrt{3})J_0(x\sqrt{5})J_0(x\sqrt{7})J_0(x\sqrt{11})\,dx. $$ I know that some similar ...
9
votes
2answers
91 views

How to prove $\int_{0}^{-1} \frac{\operatorname{Li}_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2} $

$\def\Li{\operatorname{Li}}$ I wonder how to prove: $$ \int_{0}^{-1} \frac{\Li_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2} $$ I'm not used to polylogarithm, so I don't know how to tackle it. ...
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 ...
13
votes
1answer
161 views

Evaluating by real methods $\int_0^{\pi/2} \frac{x^5}{2-\cos^2(x)}\ dx$

$\def\Li{{\rm{Li}}}$I'm sure you guys can briefly get the result by some methods of complex analysis, but now I'm only interested in real analysis methods of proving the result. What would you propose ...
5
votes
1answer
60 views

using complex or real analysis solve $\int_{0}^{\pi/2}\frac{x^m}{\sin x}dx$

closed form for $$\int_{0}^{\frac{\pi}{2}}\frac{x^m}{\sin x}\ dx$$ I slove it for some m but in general i failed. I tried by part , by substitution,by using $\sin x =\frac{e^{ix}-e^{-ix}}{2i}$ . I ...
3
votes
2answers
81 views

Evaluation of $\int_0^\pi \! \ln\left(1-2\alpha\cos x+\alpha^2\right) \mathrm{d}x$

I have got a trouble with integral $$\int_0^\pi \! \ln\left(1-2\alpha\cos x+\alpha^2\right) \, \mathrm{d}x,\quad |\alpha|<1.$$ My teacher said there are two ways of solving such ones, if there is ...
1
vote
3answers
31 views

Closed form of a sum

I am trying to derive the closed form of the sum $\sum\limits_{i=2}^n \frac{1}{i(i-1)}$ which Mathematica tells me is $\frac{n-1}{n}$. I am completely baffled on how to arrive at this result. The ...
9
votes
0answers
50 views

Is there a simple expression for $\sum_{n=0}^{\infty} \left[ (4x)^n \frac{(n!)^2}{(2n+1)!} \right]^2?$

Is there a simple expression for the power series $$\sum_{n=0}^{\infty} \left[ (4x)^n \frac{(n!)^2}{(2n+1)!} \right]^2?$$ This question came up in a quantum mechanics problem. Mathematica only returns ...
4
votes
0answers
35 views

Write $\sum_{k=1}^nk\sin(kx)^2$ in closed form

$\underline{Given:}$ Write in closed form $$\sum_{k=1}^nk\sin(kx)^2$$ using the fact that $$\sum_{k=1}^nku^k=\frac u{(1-u)^2}[(n)u^{n+1}(n+1)u^n+1]$$ $\underline{My\ Work:}$ I substituted ...
6
votes
1answer
136 views

Prove this closed-form of sum of ${_4F_3}$ hypergeometric functions

I think the following identity is true. How could we prove it? $${_4F_3}\left(\begin{array}c 1,1,1,1 \\\tfrac54,2,2\end{array}\middle|\,1\right) + ...