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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Two kind of equations involving natural log and exponentiation

I know how to solve equations using Lambert's W function like $xe^x=k$ or $e^x+x=k$ But how can I solve this two kinds of equations involving natural log ? $e^x \ln(x)=k$ and $e^x+\ln(x)=k$ I ...
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17 views

Does a better form exists for the coefficients of this product of power series?

Let $$f(a,b,t) = \sqrt{1-at}\sqrt{1-bt}$$ We take the series for $\sqrt{1-at}$ and $\sqrt{1-bt}$ around $t = 0$ and multiply them together to find $$f(a,b,t) = \sum_{n=0}^\infty ...
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1answer
46 views

How to solve equation $ x=W(a+bx^{n})+1 $?

How i can resolve the equation $x=W(a+b x^n)+1$, where $W$ is the Lambert $W$ function? thanks
2
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2answers
51 views

Evaluate $-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$ in $\gamma$.

Evaluate $\gamma$ expressed, involving Lambert function, by $$-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$$ where $\gamma<1$. I doubt that it is possible to find a value for ...
4
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398 views

No closed form for the partial sum of ${n\choose k}$ for $k \le K$?

In Concrete Mathematics, the authors state that there is no closed form for $$\sum_{k\le K}{n\choose k}.$$ This is stated shortly after the statement of (5.17) in section 5.1 (2nd edition of the ...
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3answers
53 views

Evaluate the definite integral $ \int_{\pi/6}^{\pi/2} \frac{\cos(x)}{\sin^{5/7}(x)}\, dx$

Evaluate the integral: $\displaystyle \int_{\pi/6}^{\pi/2} \frac{\cos(x)}{\sin^{5/7}(x)}\, dx$ (using substitution) Here's my attempt at solution: u = $\sin^5(x)$ $du = 5\sin^4(x) \cdot \cos(x) ...
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1answer
29 views

Closed Form Summation Example

$$ \sum_{i=1}^n (ai +b) $$ Let $n \geq 1$ be an integer, and let $a,b > 0$ be positive real numbers. Find a closed form for the following expression. In other words you are to eliminate the ...
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1answer
17 views

Summation to Closed Form conversion

I am struggling to understand basics as it related to forming a closed form expression from a summation. I understand the goal at hand, but do not understand the process for which to follow in order ...
23
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702 views

Is $\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}$ true for $m\in\mathbb N$?

Question : Is the following true for any $m\in\mathbb N$? $$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$ Motivation : I reached ...
13
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2answers
286 views

Infinite Series $\sum_{n=1}^{\infty}\frac{(m-1)^n-1}{m^n}\zeta(n+1)$

How to prove the following identity? $$\sum_{n=1}^{\infty}\frac{(m-1)^n-1}{m^n}\zeta(n+1)=\pi\cot\left(\frac{\pi}{m}\right)$$
2
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2answers
86 views

Closed form for an infinite sum over Gamma functions?

I am having quite a bit of trouble trying to find a closed form (or a really fast way to compute) for the infinite sum $$\sum_{n=1}^{\infty} a^n \dfrac{\gamma(n+1,b)}{\Gamma(n+1)\Gamma(n)}$$ where ...
3
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2answers
61 views

Find the closed-form for $\sum_{i=0}^n(-1)^i(\frac{1}{2})^i$

I start with simplifying: $$\sum_{i=0}^n(-1)^i(\frac{1}{2})^i=\sum_{i=0}^n(-\frac{1}{2})^i$$ then: $$S = 1 + (-\frac{1}{2}) + (-\frac{1}{2})^2 + ... +(-\frac{1}{2})^n$$ $$(-\frac{1}{2})S = ...
1
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1answer
20 views

Sum identity involving sin

How one can prove that $$\sum_{k=1}^n(-1)^k\sin(2k\theta)=\cos(n\pi/2+\theta+n\theta)\sec\theta\sin(n\pi/2+n\theta)?$$ It looks difficult as there is sum on the other side and product of trigonometric ...
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0answers
45 views

Is there a closed form expression for the following definite integrals?

I am looking for a closed form for these two integrals $$\int_{-\infty}^{-a}\text{d}x \frac{1}{|x|}e^{-\frac{1}{2}x^2\sigma^2}e^{i k |x|}+\int_a^{\infty}\text{d}x ...
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2answers
69 views

Richard Pavlicek's combinatorial problem

In the game of bridge, a standard deck is dealt to four players, 13 cards each. That gives a total of $\binom{52}{13,13,13,13}$ distinct deals. How many distinct deals can be dealt if all spot cards ...
5
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58 views

show that $\int_0^{\infty}\ln(2x)\frac{ax-u(1-e^{-2ax})}{\sinh^2(ax)}x^{2u-1}dx=\frac{1}{2u}(\frac{\pi}{a})^{2u}|B_{2u}|:u \in N$

$$\int_0^{\infty}\ln(2x)\frac{ax-u(1-e^{-2ax})}{\sinh^2(ax)}x^{2u-1}dx=\frac{1}{2u}(\frac{\pi}{a})^{2u}|B_{2u}|:u \in N$$ using real or complexe analysis where $B_{2u}$ is bernoulli number
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1answer
25 views

Closed-form expression of $\frac{1}{N^{n}} \left[ N^{n+1} - \sum_{k=1}^{N} (k-1)^{n} \right] $

Is there a nice closed-form expression for $$\frac{1}{N^{n}} \left[ N^{n+1} - \sum_{k=1}^{N} (k-1)^{n} \right] $$ where $n, N, k \in \mathbb{N}$. I can obtain an approximation for this for large ...
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1answer
62 views

Is there any closed form for the following series?

I am looking for any closed form expression for the series given below: $$ \sum_{m \ge 1} \frac{(xy)^m}{m(1-y^m)}. $$
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187 views

A problem of Ramanujan's interest: closed form of $1 + 2\sum_{n=1}^{\infty} \frac{\cosh(n\theta)}{\cosh(n\pi)} $

I am Brian Diaz, and I am new to the math.stackexchange community. I have been struggling with attempting to find a closed form of the following series: $$ \varphi(\theta) = 1 + 2\sum_{n=1}^{\infty} ...
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224 views

How to solve $\int_0^{\frac{\pi}{2}}\frac{x^2\cdot\log\sin x}{\sin^2 x}dx$ using a very cute way? [closed]

Few days ago my friend gave me this integral and i cant get how to solve this. The integral is:$$\int_0^{\large \frac{\pi}{2}}\frac{x^2\cdot\log{{\sin{x}}}}{\sin^2{x}}dx$$
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60 views

About the closed form for $\lim_{y\to +\infty}\left(-\frac{2}{\pi}\log(1+y)+\int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx\right)$

Recently, when facing a baby Rudin's exercise, I proved that: $$ \int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx = \frac{2}{\pi}\log(1+y)+O(1) $$ holds by integration by parts. Now I wonder if ...
4
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2answers
112 views

How to find a closed form for the derivatives of $F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,dt,$ $F(0)=\frac12$?

I have been given the function $$F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,{\rm d}t$$ for $x\ne 0,$ $F(0)=\frac12,$ and charged with finding a Taylor polynomial for $F(x)$ differing from $F$ by no ...
3
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2answers
106 views

Is there an algebraic solution to $e^{-x/a}+e^{-x/b}=1$ ($a\neq b$, $a,b$ constants)?

Is there an algebraic solution for the to find the intersection of the following two functions for values of $x\geq 0$: $$f_1(x)=1-2e^{-x/a}=f_2(x)=-1+2e^{-x/b}$$ $a$ and $b$ are positive constants. ...
4
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0answers
74 views

Help finding a closed form

I have the following function: $$\frac{2e^x}{e^{2x}+1+2x}=\sum_{n=0}^\infty \varepsilon_n\frac{x^n}{n!}$$ I would like to find a closed form for the $\varepsilon_k$. One thing that I do know is that ...
6
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3answers
138 views

Show that $\sum\limits_{i=0}^{n/2} {n-i\choose i}2^i = \frac13(2^{n+1}+(-1)^n)$

While doing a combinatorial problem, with $n$ being even, I came up with the expression $$\sum_{i=0}^{n/2} {n-i\choose i}2^i$$ for which I used wolfram to get a closed form expression of ...
0
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2answers
73 views

Generating function and its closed form

Consider the inequality $x_1 + x_2 + x_3 + x_4 ≤n$ where $x_1,x_2,x_3,x_4,n ≥ 0$ are all integers. Suppose also that $x_2 ≥ 2$, that $x_3$ is a multiple of 4, and $1 ≤ x_4 ≤ 3$. Let $c_n$ be the ...
3
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1answer
113 views

Closed form $\int_{a}^b \cfrac{1}{(1+x) \, \left[\ln(1-x)-\ln(1+x)\right]} \, \mathrm{d}x$

I am encountering an integral which involves logarithms, in particular, \begin{equation} \int_{a}^b \cfrac{1}{(1+x) \, \left[\ln(1-x)-\ln(1+x)\right]} \, \mathrm{d}x, \end{equation} where $a$ and $b$ ...
0
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1answer
32 views

Does this sum of products of binomial coefficients have a simple closed form?

Let $c,m,k$ be positive integers. Is there a simple closed form for the following sum? $$ \sum_{i=1}^{c-1} (-1)^i {c \choose i} {im \choose k} $$ Mathematica finds nothing, and Maxima's implementation ...
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53 views

Elusive closed form for card permutation problem

Does a closed form formula f(n) exist for the two rightmost columns? The two question marks are meant to be 0. The diagram is a summary of the numerical results from original question: Permutations ...
6
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2answers
185 views

Closed form of the sum $\sum\limits_{n=0}^\infty \exp(-n^3)$

I am trying to calculate the sum of the series $$\sum_{n=0}^\infty \exp(-n^3)$$ Can it be expressed in terms of known mathematical functions?
6
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3answers
118 views

Any given function $f\colon [0,1] \to \Bbb R$, what is $\int_0^1 \frac{f(x)}{f(x)+f(1-x)} \, dx$?

I have a general function $$\int_0^1\frac{f(x)}{f(x)+f(1-x)}dx.$$ How do I solve it? I have tried to split it up from $0$ to $0.5$ and from $0.5$ to $1$, but I don't know what to do next. Thanks for ...
3
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4answers
151 views

Longest chord in the intersection n disks (circle areas)

Given n disks that intersect, there is a shape in the space where they intersect. Given that, what is the longest chord, more generally longest line, that can be drawn in this space? For n=1, this is ...
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0answers
15 views

Approximate distribution of product of N normal i.i.d.?

Given $N>30$ i.i.d. $X\approx\mathcal{N}(\mu_X,\sigma_X^2)$, looking for: accurate closed form distribution approximation of $Y=\prod_{n=1}^{N}{X}$ asymptotic normal approximation of same ...
0
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1answer
49 views

Closed form for nth term - generating functions

I think I am mostly confused about what the question is asking. I read that "closed form" means that it should not be represented as as infinite sum, so I am not sure what they are asking for. Would ...
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50 views

closed-form solution for this constrained optimization

I want to find a closed-form solution for the vector $w=\left[\begin{array}{c} c\\-b \end{array}\right]$where $c$ and $b$ are column vectors, such that the following MSE is minimized: $\begin{align} ...
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1answer
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18
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318 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}) \ ...
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0answers
63 views

Closed form of generating function $r^a$

Find the closed form of the generating function of $r^a$. in this question $r$ is the variable part and $r$ assumes the values $1,2,3,4,5,6,7 \ldots$ and $a \in \mathbb R_{\geq 0}$. I would appreciate ...
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0answers
27 views

Closed form for $\sum_{k\in\mathbb{N}}\frac{k}{a\uparrow^kb}$

Let $a,b\in\Bbb{N}$. Is there a closed form for $\displaystyle\sum_{k\in\mathbb{N}}\frac{k}{a\uparrow^kb}$ ? (I use Knuth's up arrow notation) If so, how can we obtain it ? If there isn't a closed ...
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36 views

Find the closed-form of a series

Suppose that $x$ is positive number such that $x>0$. I just wonder is there existing a closed form of the series below $f(x)=\sum_{l=0}^{\infty}(2l+1)e^{-xl(l+1)}$. Is the well-known ...
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2answers
50 views

Equations involving factorial/Gamma function

Are there any known methods to formally solve equations like: 1)$x^3!+(2x^2)!-x!+3=0$ 2)$x!=e^x$ ($0$ is trivial but there must be another one) 3)$(2x!)^2+x!-1=0$ 4)$x!!+x!=7$ I don't need ...
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2answers
66 views

Solving 2nd order linear recurrence with non-constant coefficients

I am trying to find a general solution to the following definite integral: $$F_{n}{\left(a,b;z\right)}:=\int_{a}^{z}\frac{x^{n}}{\sqrt{\left(x-a\right)\left(b-x\right)}}\,\mathrm{d}x,\tag{1}$$ ...
4
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1answer
83 views

Proving that $\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$

How could we prove that $$\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$$ for $a+b>n>-\dfrac12$ ? Inspired by ...
0
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1answer
22 views

Find closed form of recursion

I know how to get the equation of the form $x^2 = Ax + B$ and then from there get $a_k = C * x_1^k + D * x_2^k$ but doesn't the original $b_k$ equation have to be of the form $7b_{k-1} - 10b_{k-2}$ ...
35
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3answers
988 views

Prove $\int_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$

Prove the following integral $$I=\int_{0}^{\frac{\pi}{2}}\dfrac{dx}{1+\sin^2{(\tan{x})}}=\dfrac{\pi}{2\sqrt{2}}\left(\dfrac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\right)$$ This integral result was ...
4
votes
1answer
56 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 ...
23
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4answers
550 views

Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$

Let $$f(a)=\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx,$$ where $\operatorname{sech}(z)=\frac2{e^z+e^{-z}}$ is the hyperbolic secant. Here are values of $f(a)$ at some ...
4
votes
2answers
161 views

Does this integral have any closed form? $\displaystyle\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$

Does this integral have any closed form? $$\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$$ I think the substitution $x=(u-1)+2\pi$ will do it, no?
0
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1answer
44 views

The $C_0-$group generated by the operator $(Af)(x)=f'(x)+a(x)f(x)$

Consider the Banach space $L^1(\mathbb{R})$ of integrable functions $f:\mathbb{R}\to \mathbb{R}$. Consider the unbounded operator $A$ defined by $$(Af)(x)=f'(x)+a(x)f(x), \ \ \ x\in \mathbb{R}$$ for ...
2
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0answers
71 views

How to find $\sum_{n \in \mathbb Z_+} \frac{2^{n-1}}{2^{2^n}}$?

I'm trying to calculte the measure of a fat Cantor set, but run into this question: How to find $$\sum_{n \in \mathbb Z_+} \frac{2^{n-1}}{2^{2^n}}$$