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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13
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2answers
137 views

How to integrate $\int\limits_{0}^{\pi/2}\frac{dx}{\cos^3{x}+\sin^3{x}}$?

I have$$\int\limits_{0}^{\pi/2}\frac{\text{d}x}{\cos^3{x}+\sin^3{x}}$$ Tangent half-angle substitution gives a fourth-degree polynomial in the denominator that is difficult to factor.
1
vote
1answer
46 views

Find the value of $\int_1^4 xf''(x)dx$ when $f(1) = 2$, $f(4) = 7$, $f'(1) = 5$, $f'(4) = 3$, and $f''$ is continuous [closed]

Find the value of $\int_1^4 xf''(x)dx$ when $f(1) = 2$, $f(4) = 7$, $f'(1) = 5$, $f'(4) = 3$, and $f''$ is continuous. I'd like to see a full solution please!
0
votes
1answer
44 views

Find the closed formula for following problem. [closed]

A student borrows $\$ 3000$ on a bank credit card at a nominal rate of $18$% per year, which is actually charged at a rate of $1.5$% per month. a) If a person does not place any additional charges on ...
3
votes
0answers
46 views
1
vote
0answers
33 views

Closed form expression for $x$

We define the following relation: $$y = \frac{(1+x)^n - 1}{x}$$ Alternatively, $$y = \sum_{k = 1}^n {n \choose k} x^{k-1}$$ Where $n$ is an integer and $x$ and $y$ are positive real numbers. How ...
4
votes
3answers
188 views

Closed form for $\displaystyle \sum_{n=1}^\infty \int_0^1 \frac{x^{n-1}\ln^2(x)\ln(1-x)}{n^2} \,dx$ [closed]

I am trying to get this to equal $\displaystyle-\frac {\pi^a}{b}$ for some positive integers $a$ and $b$ . My efforts so far give: $\displaystyle \sum_{n=1}^\infty \int_0^1 ...
1
vote
0answers
33 views

Classifying functions whose inverse do not have a closed form

My initial question contained about how to classify functions whose integrals and inverses do not have a closed form. But I found this question: How can you prove that a function has no closed form ...
12
votes
3answers
179 views

Evaluate $\int_0^1\int_0^1 \left\{ \frac{e^x}{e^y} \right\}dxdy$

I want compute this integral $$\int_0^1\int_0^1 \left\{ \frac{e^x}{e^y} \right\}dxdy, $$ where $ \left\{ x \right\} $ is the fractional part function. Following PROBLEMA 171, Prueba de a), last ...
0
votes
0answers
21 views

Equation with a depended variable as argument of the tangent function: $x= K+y\tan(\sqrt{y})+y$ [duplicate]

What would you do if you need to find a relation between two variables but the one that you need to find in function of the other is the argument of a tangent (tan(y)) ? For example if I have the ...
-1
votes
1answer
14 views

Solve an equation in which a variable is an argument of a tangent: $x= K+y\tan(\sqrt{y})+y$

What would you do if you need to find a relation between two variables but the one that you need to find in function of the other is the argument of a tangent ($\tan(y)$) ? For example if I have the ...
3
votes
1answer
50 views

Find closed formula for $a_{n+1}=(n+1)a_{n}+n!$

$a_{n+1}=(n+1)a_{n}+n!$ where a0=0 and n>=0. To get the closed form, I'm trying to find an exponential generating function for the above recurrence, but it doesn't seem to be very nice. Am I going ...
3
votes
1answer
49 views

Find closed formula for the recurrence $a_{n}=na_{n-1}+n(n-1)a_{n-2}$

$a_{n}=na_{n-1}+n(n-1)a_{n-2}$ where a0 = 0, a1=1, and n >= 2. I found an exponential generating function for this recurrence, but cant seem to find the closed form because the generating function ...
2
votes
1answer
107 views

Does a closed form solution exist for $x$? [closed]

$$ \sqrt{x} + x^2 = \sqrt{2} $$ If so, how would one find it?
1
vote
0answers
23 views

Closed form of Baker Campbell Hausdorff theorem with cyclic bracket structure

I would like to know if there exists a closed form of the Baker Campbell Hausdorff theorem subject to the conditions that $[x,[x,y]] \sim x$ and $[y,[x,y]] \sim y$. The simple cases that I know a ...
8
votes
2answers
116 views

closed form for $I(n)=\int_0^1\left ( \frac{\pi}{4}-\arctan x \right )^n\frac{1+x}{1-x}\frac{dx}{1+x^2}$

$$I(n)=\int_0^1\left ( \frac{\pi}{4}-\arctan x \right )^n\frac{1+x}{1-x}\frac{dx}{1+x^2}$$ for $n=1$ I tried to use $\arctan x=u$ and by notice that $$\frac{1+\tan u}{1-\tan u}=\cot\left ( ...
1
vote
1answer
62 views

Closed form of this expression

So at my school today, we celebrated Pi Day, and we had a customary mini-olympiad. In that olympiad there was a really interesting question. There is a sequence: $\lbrace 1\rbrace, \lbrace 4, ...
1
vote
1answer
24 views

How to show $\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)}$

How to show the below equation ? $$\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)} ~~~~~(t\in \mathbb Z^+)$$
10
votes
1answer
67 views

Closed form solution for the zeros of an infinite sum

Does there exist a closed form expression for the zeros of the following equation? $$\sum\limits_{n=1}^\infty\frac{1}{n^4 - x^2} = 0 \text{ where } x \in \rm \mathbb R$$ Could you suggest a ...
3
votes
2answers
62 views

How show that $a_{n}=n$ if $ a_{n+1}+a_{n-1}=\frac{2n}{a_{n}-a_{n-1}}$

define sequence $\{a_{n}\}$ such $a_{1}=1,a_{2}=2$, and such $$ a_{n+1}+a_{n-1}=\dfrac{2n}{a_{n}-a_{n-1}},n\ge 2$$ show that:$$a_{n}=n$$ I want use without induction solve this sequence?
2
votes
1answer
89 views

Function that produces sequence 112123123412345…

I'm trying to find a function/formula for $a_n$ such that it produces the sequence $112123123412345$ and so on. I know that one possible way to do this is to find a function like $n-b_n$ where $b_n$ ...
1
vote
1answer
30 views

Supremum of a sequence: $x_n = c_1 + c_2 \sum_{i=1}^{n-1} x_i $

I am dealing with a sequence $$ x_n = c_1 + c_2 \sum_{i=1}^{n-1} x_i $$ where $x_1 > 0$ and $c_1, c_2 > 0$ are constants. I am wondering whether one can find another series, call it $y_n$, ...
5
votes
2answers
326 views

Find the ratio of integrals $\int_0^1 (1\pm x^4)^{-1/2}\,dx$

How to find this ratio $$\frac{\displaystyle \int_{0}^{1}\frac{1}{\sqrt{1+x^{4}}}\mathrm{d}x}{\displaystyle \int_{0}^{1}\frac{1}{\sqrt{1-x^{4}}}\mathrm{d}x}$$ without evaluating each integral? ...
4
votes
1answer
47 views

Recursive Sum of Previous Term and its Inverse

Can anyone help me with finding a closed form for $F_n$ where $$F_0=x_0$$ $$F_{n+1}=F_n+\frac{1}{F_n}=\frac{F_n^2+1}{F_n}$$ I could imagine this already having been done, in which case I'd appreciate ...
4
votes
0answers
41 views

Closed form solution to an ordinary differential equaiton

How to solve the following ordinary differential equation? $$y'(x)= \frac{C_1}{y(x)} +C_2 C_3 \cos\left(C_3 x\right) +C_4$$ where $C_1, C_2, C_3, C_4\in \mathbb{R}$ are all constants. It looks ...
2
votes
0answers
50 views

I am looking for a disproof of this conjecture on closed forms

Assume $I=\int_0^\infty f(x)\text{ d}x$ and $J=\sum_{n=0}^\infty f(n) ; I,J\in\Bbb{R}$ Conjecture: If $I$ has a closed form, then $J$ must carry a closed form. Can someone find a proof or ...
1
vote
1answer
23 views

Risch algorithm analogue for differential equations

I know that we can determine whether an integral has closed form, that is, is a composition of elementary functions. That problem is (more or less) solved by Risch algorithm. For differential equation ...
4
votes
2answers
148 views

Closed form 0f $I=\int _{ 0 }^{ 1 }{ \frac { \ln { x } { \left( \ln { \left( 1-{ x }^{ 2 } \right) } \right) }^{ 3 } }{ 1-x } dx }$

While solving a problem, I got stuck at an integral. The integral is as follows: Find the closed form of: $$I=\int _{ 0 }^{ 1 }{ \frac { \ln { x } { \left( \ln { \left( 1-{ x }^{ 2 } \right) } ...
8
votes
1answer
190 views

How to calculate $\int_0^\pi \ln(1+\sin x)\mathrm dx$

How to calculate this integral $$\int_0^\pi \ln(1+\sin x)\mathrm dx$$ I didn't find this question in the previous questions. With the help of Wolframalpha I got an answer $-\pi \ln 2+4\mathbf{G}$, ...
0
votes
0answers
105 views

What is close-form solution using ALS algorithm to optimize this objective function

$C \in \mathbb{R}^{m \times n}$, $X \in \mathbb{R}^{m \times n}$, $W \in \mathbb{R}^{m \times k}$, $H \in \mathbb{R}^{n \times k}$, $S \in \mathbb{R}^{m \times m}$, $P \in \mathbb{R}^{n \times n}$ ...
2
votes
2answers
85 views

How is Faulhaber's formula derived?

I have been wanting to understand how to find the sum of this series. $$1^p + 2^p + 3^p +{\dots} + n^p$$ I am familiar with Gauss' diagonalised adding trick for the sum of the first $n$ natural ...
1
vote
0answers
24 views

Does this equation has a closed form solution?

We have $K$ non-negative coefficients: $a_1,a_2,\dots,a_K,A_1,A_2,\dots,A_K$, where $A_i\geq0,\;a_i\in(0,1),\;\sum A_i<T$. The equation is: $$\sum_{i=1}^K\frac{A_i}{1-a_ix}=T,\quad x\in(0,1)$$
0
votes
0answers
28 views

Is there any closed form of the following equation?

I am looking for the closed form of the following equation: $\sum_{i=0}^\infty (a+ib)\prod_{j=0}^{i-1} e^{-\lambda(a+jb)} \lambda(a+jb)$ I tried to convert it to two smaller series as follows: ...
8
votes
3answers
120 views

General formula for the higher order derivatives of composition with exponential function

Suppose I have a function $x:\mathbb{R} \to \mathbb{R}$ and consider: $$g(t) = e^{x(t)}$$ When I start differentiating with respect to $t$ I obtain: \begin{align} g'&=e^xx'\\ ...
1
vote
0answers
53 views

Integral of the square root of a trigonmetric function

Despite my best attempts, I have been unable to evaluate the following integral: $$ \int_s^t\sqrt{9+(2+\cos3u)^2}\,du. $$ This integral showed up during an investigation of torus knots. It represents ...
6
votes
4answers
114 views

Prove $\int_0^\infty \frac{x^{k-1} + x^{-k-1}}{x^a + x^{-a}}dx = \frac{\pi}{a \cos(\frac{\pi k}{2a})}$.

I need help in proving this identity $$\int_0^\infty \frac{x^{k-1} + x^{-k-1}}{x^a + x^{-a}}dx = \frac{\pi}{a \cos(\frac{\pi k}{2a})}$$ for $0<k<a.$ It might be done using residues, but I ...
1
vote
1answer
44 views

How to find Real Part of PolyLog[3,(1-i)] in closed form

$ \Re \bigg(\text{Li}_3(1-i)\bigg)=\frac{\pi^a}{b}\ln(2)+\frac{c}{d}\zeta(e)$ has an approximate value of .8711588834109380 if $a=1 , b=-3415 , c=34 , d=39 , e=19$ are substituted into the closed ...
1
vote
2answers
42 views

Find another closed form for this sequence

I've recently come across a sequence while doing math which is: $$\{1;0;2;-1;3;-2;4;-3;...\}$$ And searching for a closed form to express the n-th therm I've came to the following: ...
2
votes
1answer
52 views

Find a closed formula for $\sum_{n=1}^\infty nx^{n-1}$ [duplicate]

Find a closed formula for $\sum_{n=1}^\infty nx^{n-1}$ I am trying to use the derivative of generalized binomial theorem, $\frac{d}{dx}[(x+1)^r=\sum_{n=0}^\infty \binom{r}{n}x^n] ...
2
votes
2answers
42 views

Why is this the closed-form solution for this series? [duplicate]

I know this is simple, but I don't know very much at all about series, and I'm wondering how it's shown that: $$ 1 + 2 + 3 + \cdots + (n - 1) = \frac{n(n - 1)}{2} $$
2
votes
1answer
127 views

Closed form or simplification of a multiple definit integral of a product of a weight averaged parameters

I am trying to obtain a closed form solution of this definite integral, or in a form at least which simplify its numerical treatment. $$\int_{x_1=0}^1...\int_{x_N=0}^1 \prod_r \left( \frac {x_r f_r} ...
4
votes
1answer
121 views

Can you get a closed-form for $\prod_{p\text{ prime}}\left(\frac{p+1}{p-1}\right)^{\frac{1}{p}}$?

When I use the Taylor expansion series for $$\log(1+x)^{1+x}+\log(1-x)^{1-x}$$ with $x=\frac{1}{p}$, $p$ prime, I believe that I can deduce $$\sum_{p\text{ ...
0
votes
0answers
23 views

Closed-form of spherical expansion of Legendre polynomial $P_k(\sin{\theta}\cos{\varphi})$

During the times of working on some problem in astro/geophysics I have come across a problem involving an expansion into spherical harmonic functions (this is the remnance of nomenclature there used ...
0
votes
1answer
56 views

Pseudo-inverse of the Cumulative Distribution Function of X

The goal of these calculations is to write a Python function that generates pseudo-random values with the distribution described below. This isn't relevant to the question (or even to this ...
8
votes
2answers
224 views

Improper Integral $\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx$

$$I=\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}\pi^3-\frac{\pi}2\log^2 2-2\pi\chi_2\left(\frac1{\sqrt 2}\right)$$ This result seems to me digitally correct? Can we prove ...
1
vote
1answer
46 views

Closed-form Solution of Log Sum

I have the series: $$\sum_{i=1}^{i=10^N} \log_5 i$$ I'm trying to figure out how to get the closed-form solution to this problem. I entered it into WolframAlpha and got that it equals: $ ...
1
vote
1answer
47 views

Definite integral of a continued fraction function

I came up with this function written as the following continued fraction (please correct me if my notation is incorrect): for $n\in\mathbb{N}$, let $$f(x;n)=x+\operatorname*{\LARGE ...
1
vote
5answers
97 views

Closed-form Solution to a Sum

I have some math questions for a programming course where it says to provide closed-form solutions for a list of sums. I've never taken an algorithms course, so I'm not quite sure what I'm doing. I ...
2
votes
1answer
108 views

Closed form for this integral $I=\int_0^{1}\frac{{\arcsin}({x^2})}{\sqrt{1-x^2}}dx$

I’m trying to find a closed form for this integral.Any help is appreciated.Thanks $$I=\int_0^{1}\frac{{ \arcsin}({x^2})}{\sqrt{1-x^2}}dx$$
1
vote
0answers
35 views

Possible closed form or approximation?

Does it have some closed form or approximation ? I tried on my own but i am not getting any idea regarding this. $$\sum_{k_1=k}^{M}\sum_{k_2=k}^{M}\frac{k_1^{-\gamma} k_2^{-\gamma} ...
2
votes
2answers
43 views

Areas where closed form solutions are of particular interest

Assuming the definition of 'Closed Form' given in the table of: Closed Form Wikipedia entry, what areas tend to have problems that are traditionally expressed in closed form? EDIT: Given the comment ...