Questions tagged [closed-form]
A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".
3,693
questions
1
vote
0
answers
188
views
Closed form for $A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$
Consider the double sums :
$$A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$$
$$A_4 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^4 + b^4}$$
Is there a closed form for $A_3$ ...
- 16.4k
2
votes
1
answer
115
views
How to integrate $\int_{0}^{\infty} \frac{1}{x^2 (\tan^2 x + \cot^2 x)} \,dx$
How to integrate $$\int_{0}^{\infty} \frac{1}{x^2 (\tan^2 x + \cot^2 x)} \,dx$$
Let $z = e^{ix}$. We write the Fourier series
$$
\frac1{\tan^2x+\cot^2x} = \frac1{\left(\frac{z-z^{-1}}{i(z+z^{-1})}\...
6
votes
2
answers
165
views
Integrating $\int_{0}^{1} \left(\frac{\arctan(x) - x}{x^2}\right)^2 \,dx$
how to integrate $$\int_{0}^{1} \left(\frac{\arctan(x) - x}{x^2}\right)^2 \,dx$$
Attempt
$$=\int_{0}^{1} \left(\frac{\arctan(x) - x}{x^2}\right)^2 \,dx = \int_{0}^{1} \frac{1}{x^4} \cdot (\arctan(x) -...
3
votes
2
answers
228
views
How to calculate this sum $\sum_{n=1}^{\infty} \frac{H_n \cdot H_{n+1}}{(n+1)(n+2)}$
How to calculate this sum $$\sum_{n=1}^{\infty} \frac{H_n \cdot H_{n+1}}{(n+1)(n+2)}$$
Attempt
The series telescopes. We have
$$=\frac{H_n \cdot H_{n+1}}{(n+1)(n+2)} = \frac{H_n \cdot H_{n+1}}{n+1} - ...
2
votes
1
answer
99
views
Integrating $\int_{0}^{\infty} \frac{\tanh \left(\frac{x}{2}\right)+\tanh (2 x)}{x}\left(e^{\frac{3 x}{2}}-1\right)^{2} e^{-4 x} \mathrm{~d}x$
Show that
$$\int_{0}^{\infty} \frac{\tanh \left(\frac{x}{2}\right)+\tanh (2 x)}{x}\left(e^{\frac{3 x}{2}}-1\right)^{2} e^{-4 x} \mathrm{~d}x=\boxed{2\log\left(\frac{768\sqrt2\pi^4}{25\Gamma^3\left(\...
1
vote
0
answers
50
views
Is there a Closed-Form Solution for L2 Regularization Raised to a Power?
Recently, I came across a modified L2 regularization term as stated in the equation below, where $\gamma$ is a positive number.
$$
\lambda'(w^Tw)^\gamma
$$
I'm curious if a closed-form solution ...
- 11
4
votes
2
answers
211
views
How to calculate this integral $\int_{0}^{\infty} \left(\frac{x - 1}{\ln(x)}\right)^2 \cdot \frac{1}{1 + x^n} \,dx$
How to calculate this integral
$$ \int_{0}^{\infty} \left(\frac{x - 1}{\ln(x)}\right)^2 \cdot \frac{1}{1 + x^n} \,dx$$
This is how I start
$$f(a)=\int_{0}^{\infty} \frac{(x^a-1)^2}{\ln^2(x)}\frac{1}{...
0
votes
1
answer
34
views
"Nice" form/bounds for $\ln\prod_{i=1}^n \left(1 + \frac{x_i^2}{n}\right)$
Does the expression
$$\ln\prod_{i = 1}^n\left(1 + \frac{x_i^2}{n}\right)$$
have either
a "nicer" closed-form, or
quantitative upper/lower bounds on such a form?
For me, "nice" ...
- 7,058
5
votes
3
answers
461
views
Integration of $ \int_{0}^{\frac{\pi}{2}} x \log(1-\cos x) \,dx $ [closed]
Question: What is the closed form of this following integral? $
\int_{0}^{\frac{\pi}{2}} x \log(1-\cos x) \,dx.$
Here is my solution
we know that
$$\displaystyle{\sum\limits_{n = 1}^\infty {\frac{{\...
0
votes
0
answers
30
views
Calculate probability of Gaussian random variable exceeding threshold when sampled from different Gaussian distribution
I am wondering if there is a closed-form expression for the probability that a Gaussian random vector $\boldsymbol{X}$ falls in-between some bounds as specified by a different Gaussian random variable ...
3
votes
0
answers
96
views
Probability that two random integers have at most $n$ prime factors in common
I want to calculate the probability that "picking at random" two integers, they have at most $n$ prime factors in common.
Reading this nice answer, I found out the probability to have ...
- 923
3
votes
1
answer
172
views
Show $\int_{0}^{\infty}\frac{\sin x^{2}}{\pi-x^{2}}dx=\frac{\sqrt{\pi}}{2}\left[C(\sqrt{2})+S(\sqrt{2})\right]$
Show that:
$$\int_{0}^{\infty}\frac{\sin x^{2}}{\pi-x^{2}}dx=\frac{\sqrt{\pi}}{2}\left[C(\sqrt{2})+S(\sqrt{2})\right]$$
where $C$ and $S$are the Fresnel integrals defined as:
$$\displaystyle C(u)=\...
0
votes
0
answers
49
views
Sum over exponential function lying within the argument of another exponential function
I have faced the following sum in my calculation:
$$\sum_{l_1=0}^{N_1-1}\sum_{l_2=0}^{N_2-1}\sum_{l_3=0}^{N_3-1}\exp{[-ia(q_xl_1+q_yl_2+q_zl_3)]}\times\exp{-i[q\beta\times\text{e}^{ia[k_xl_1+k_yl_2+...
- 344
2
votes
2
answers
227
views
Evaluate the infinite product $ \prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right )$
Question statement
Evaluate the infinite product
$$\displaystyle{\prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right ) }$$
My try
Because of the square of $\displaystyle{x}$ , we can consider $...
3
votes
2
answers
139
views
Orthogonal trajectories to family of curves $\left\Vert x \right\Vert_p=1$ where $x\in\mathbb{R}^2$
I have been trying to find the orthogonal trajectories of the family of $p$-norm curves $\left\Vert x \right\Vert_p=1$, where $x\in\mathbb{R}^2$ and $p>0$. I eventually reached a step where I must ...
- 375
5
votes
5
answers
376
views
How to evaluate $\int_{-\infty}^{+\infty}\frac{\cos x}{\left(1+x+x^2\right)^2+1}\mathrm{~d}x$
Question
$$\int_{-\infty}^{+\infty}\frac{\cos x}{\left(1+x+x^2\right)^2+1}\mathrm{~d}x$$
Wolfram alpha says it is
$$\int_{-\infty}^{\infty} \frac{\cos(x)}{\left(1 + x + x^2\right)^2 + 1} \,dx = \frac{\...
2
votes
0
answers
141
views
closed form for limit?
Consider the function
$$ f(x)=\lim_{k \to \infty}\bigg(\int_0^x \sum_{n=1}^k e^{\frac{\log n}{\log r}}~dr \bigg)\bigg( \int_0^1 \sum_{n=1}^k e^{\frac{\log n}{\log r}}~dr \bigg)^{-1} $$
I want to find ...
- 756
2
votes
1
answer
164
views
Another weird limit involving gamma and digamma function via continued fraction
Context :
I want to find a closed form to :
$$\lim_{x\to 0}\left(\frac{f(x)}{f(0)}\right)^{\frac{1}{x}}=L,f(x)=\left(\frac{1}{1+x}\right)!×\left(\frac{1}{1+\frac{1}{1+x}}\right)!\cdots$$
Some ...
2
votes
1
answer
151
views
For $J=\{1,2,\dots,n \}$ is there an easy way to compute $\prod\limits_{i\in J | i \ne k} (k-i)$?
When I studied calculus at my university there is one question that I hated the most which is given a finite number of terms for some sequence find the $n-$th term. I hated this type of question ...
- 6,620
3
votes
1
answer
221
views
Value of $ {_3F_2}{\left(1,n+\frac32,1-r;\frac32,n+2;1\right)} $
This sum can be calculated using a CAS as a product of Gamma and Generalized Hypergeometric functions
$$\displaystyle \sum_{k=0}^n \frac {\dbinom{n}k \dbinom{n+r}k}{\dbinom{2n}{2k}} = \dfrac{(2n+1) \...
- 923
2
votes
1
answer
124
views
Could this closed form for negative integer values of a series involving $\zeta(s)$ be proven?
The following series expression:
$$f(s) = \sum_{n=2}^\infty \frac{\zeta(n)-1}{n^s} \tag{1}$$
has well known closed forms for $s=0$ and $s=1$, however also seems to converge for $\Re(s) < 0$. After ...
- 3,191
3
votes
4
answers
276
views
$\int_{-1}^u\text{exp}(\frac{1}{x^2-1})dx$
I'm trying to compute $\int_{-1}^u\text{exp}(\frac{1}{x^2-1})dx$ where $u\in[-1,1]$.
This is a crucial element of this paper and I need to be able to compute it quickly in Mathematica thousands of ...
- 303
1
vote
3
answers
165
views
What are some unique integral representations of Apery's constant - $\zeta(3)$?
I've been playing around with some integrals, and I started looking at Apery's constant.
There are some integral representation I've found online, such as:
$$\zeta(3)=\frac{16}{3}\int_{0}^{1}\frac{x\...
- 71
1
vote
0
answers
29
views
Solution to Hyperbolic PDE
Let $A\in C^{\infty}(\mathbb{R}^2)$ be Lebesgue integrable, and $c_1,c_2\in C^{\infty}(\mathbb{R})$ also be Lebesuge integrable. Consider the hyperbolic PDE
$$
\begin{cases}
\partial_{x,y}u & = A\...
- 6,779
0
votes
2
answers
83
views
Prove optimality of constrained convex optimization problem analitically (using KKT conditions)
I'm trying to prove that the constrained convex minimization problem with decision variable $\boldsymbol{x} \in \mathbb{R}^{n}$ given by
$$ \min_{\boldsymbol{x}} \Vert \boldsymbol{x} \Vert_{2} \text{ ...
0
votes
0
answers
21
views
Deriving MLE & asymptotic variance manually - for which distributions/cases is it possible?
I am interested whether you know any distributions or special cases where the maximum likelihood estimator, the theoretical and estimated asymptotic variance can be fully derived manually, i.e. which ...
- 213
2
votes
0
answers
67
views
Closed form for $\psi^{1/k}(1)$, where $k$ is an integer
I have proven the identity
$$
\sum_{k=1}^{\infty} \dfrac{\operatorname{_2F_1}(1, 2, 2-1/t,-1/k)}{{k}^{2}} = Γ(2-\dfrac{1}t){\psi^{1/t}(1)}+\psi(-\dfrac{1}t)(\dfrac{1}t(1-\dfrac{1}t))+\gamma(1-\dfrac{1}...
6
votes
6
answers
378
views
Show that $\int_0^1\int_{1-y}^1\sqrt{(x-1)(y-1)(x+y-1)}\mathrm dx\mathrm dy=\frac{2\pi}{105}$.
How can we show that $\int_0^1\int_{1-y}^1\sqrt{(x-1)(y-1)(x+y-1)}\mathrm dx\mathrm dy=\frac{2\pi}{105}$ ?
Desmos says it's true.
The inner indefinite integral is not nice. And strangely, when I plug ...
- 25.7k
0
votes
1
answer
510
views
Closed form of integral $\int_{0}^{\infty}\frac{e^{-x}}{1+x/\cdots}dx$ with continued fraction
Problem: Let
$$f(x)=\frac{1}{1+\frac{x}{1+\frac{x}{\cdots}}}$$
Then does
$$\int_{0}^{\infty}e^{-x}f(x)dx=:I$$
have a closed form ?
The first step in the Continued Fraction involves not surprisingly ...
1
vote
0
answers
42
views
Analytical Solution for a Double Integral Involving Logistic Functions and Gaussian Distributions
I am working on a mathematical problem involving the evaluation of a double integral, and I am seeking an analytical solution or techniques to solve it. The integral I'm dealing with is as follows:
$$...
- 23