Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

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2
votes
1answer
39 views

Expressing a series involving the Riemann zeta function in terms of known functions

We have the series: $$\sum_{n=1}^{\infty}\sin\left(\frac{\pi n}{2}\right)\frac{\zeta(n+1)}{(2\pi)^{n+1}}\frac{\Gamma(z)}{\Gamma(z-n)}\left[\psi^{0}(z-n)-\psi^{0}(z)\right]$$ Where $\psi^{0}(\cdot)$ ...
4
votes
0answers
20 views

Analytical approximation of integral of Bessel function

I am trying to approximate the integral: $$ \int_0^z \left(\frac{J_1(x\,\sin\theta)}{\sin\theta}\right)^2 {\rm d}\theta $$ My very naive approach was to do the Taylor series of the integrand. ...
0
votes
0answers
7 views

Is it possible to get one Jacobian matrix of a 2-part continuous piecewise function?

For a 2-part continuous piecewise function, normally we can get two Jacobian matrix for each part. Is it possible to get just one Jacobian matrix of this function?
4
votes
2answers
99 views

Is there a name for the closed form of $\sum_{n=0}^{\infty} \frac{1}{1+ a^n}$?

I hope this is not a duplicate question. If we modify the well known geometric series, with $a>1$, to $$ \sum_{n=0}^{\infty} \frac{1}{1+a^n} $$ is there a closed form with a name? I suspect ...
2
votes
0answers
26 views

Weierstrass-$\wp$ Function Asymptotics

Given the Weierstrass-$\wp$ function, $$\wp(2x+1+\tau \mid 1, \tau),$$ with half-periods $1$ and $\tau=\omega_2/ \omega_1$, I want to look at the case where $\rm{Re}(\tau) \in \mathbb{Z}$ and I want ...
2
votes
1answer
21 views

Will numerical routines for the Exponential Integral function E_n work when n is continuous?

So I am a mathematical biologist of sorts. I rely heavily on Mathematica which often provides analytic results couched in terms of special functions which I then try to go and learn about. Right now ...
5
votes
3answers
212 views

Not the toughest integral, not the easiest one

Perhaps it's not amongst the toughest integrals, but it's interesting to try to find an elegant approach for the integral $$I_1=\int_0^1 \frac{\log (x)}{\sqrt{x (x+1)}} \, dx$$ $$=4 ...
0
votes
1answer
32 views

equally spaced on circle question

Define $$\|\vec{x}\|:=\sqrt{\alpha^2+\beta^2},$$ where $\vec{x}:=(\alpha,\beta)\in \mathbb{R}^2.$ Set $$\mathbb{S}^1:=\{\vec{x}\in \mathbb{R}^2: \|\vec{x}\|=1\}\quad \quad and\quad \quad ...
0
votes
2answers
50 views

A question about Idempotent functions [on hold]

some functions are such that $f\circ f(x)=f(x)$ like these 1) $$f(x)=x \implies f\circ f(x)=x=f(x)\\$$ 2)$$f(x)=\lvert x\rvert \implies f\circ f(x)=\lVert x\rVert=\lvert x\rvert=f(x)\\$$ 3) ...
2
votes
1answer
50 views

Help on finding the closed form of the integral

Can anyone help me to find closed solution of the integral $$\int_0^{1-e^{-\lambda x}}\frac{u^{b-1}\,(1-u)^{a+c-1}}{[1-(1-e^{-\lambda_1 t_1})u]^{a+b+c}}\,{\rm d}u,$$ where ...
14
votes
5answers
2k views

When I was teaching absolute function properties, I suddenly made this question …

I was teaching absolute function properties in a K-12 class. I made this question in my mind. Suppose $f(x)$ is a one-to-one function, and its definition is $f(x)=max\left \{ x,3x\right ...
3
votes
0answers
72 views

Calculating in closed form $\int_0^{\infty} \frac{\text{PolyLog}^{(1,0)}(1,-x)}{1+x^2} \, dx$

Can you confirm the following result? Mathematica and other computational stuff I used seem unable to do anything about this result. Maybe to confirm it numerically? $$\int_0^{\infty} ...
1
vote
1answer
26 views

Prove that $T_n(x)={}_2F_1\left(-n,n;\tfrac 1 2; \tfrac{1}{2}(1-x)\right) $

Prove that, for Chebyshev polynomials of the first kind, \begin{align} T_n(x) & = \tfrac{n}{2} \sum_{k=0}^{\left \lfloor \frac{n}{2} \right \rfloor}(-1)^k \frac{(n-k-1)!}{k!(n-2k)!}~(2x)^{n-2k} ...
-1
votes
1answer
34 views

prove $\max \mathbf{a}^T \mathbf{b}_i \leq \mathbf{a}^T \mathbf{c}_i$

Can I prove the following (with or without assumptions, e.g. all the elements in $\mathbf{a}$ or $\mathbf{b}$ are positive? $\max \mathbf{a}^T \mathbf{b}_i \leq \mathbf{a}^T \mathbf{c}_i$ where ...
0
votes
0answers
15 views

Is it possible to prove $\arg\min_a f(\max(\mathbf{a}^T\mathbf{b}_i)) = \arg\min_a f(\mathbf{a}^T\max(\mathbf{b}_i))$

I have the following optimization problem, $ \arg\min_\mathbf{a} f(\max(\mathbf{a}^T\mathbf{b}_i))\;\; i=1, \dots ,N$ where $\mathbf{a}$ and $\mathbf{b}_i$ are vectors of dimension $d$. Let $B = ...
-1
votes
0answers
20 views

Confluent hypergeometric function recurrence relation

How to prove the following contiguous relation for the Kummer function $M(a,b,z)$: $$(a−1+z)M(a,b,z)+(b−a)M(a−1,b,z)+(1−b)M(a,b−1,z)=0.$$
0
votes
0answers
22 views

Are the polynomials that are orthogonal in the continuous case, still continuous in the discrete case?

One of my friends asked me this question. "Are the polynomials that are orthogonal in the continuous case, still continuous in the discrete case?" It is curious how even the most trivial questions ...
-1
votes
0answers
19 views

Fourier Transform of $|x|^\frac{7}{6} K_{-\frac{1}{6}}(|x|)$ [closed]

What is the Fourier Transform of $|x|^{\frac{7}{6}} K_{-\frac{1}{6}}(|x|)$ with $K_{-\frac{1}{6}}$ the modified bessel function of the second kind?
8
votes
0answers
85 views

A conjectured identity for tetralogarithms $\operatorname{Li}_4$

I experimentally discovered (using PSLQ) the following conjectured tetralogarithm identity: $$\begin{align}&\phantom{+\;}19\!\;\pi^4-570\ln^42-90\ln^43\\ ...
1
vote
1answer
102 views

Calculating in closed form $\int_0^1 \log(x)\left(\frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}}\right)^2 \,dx$

What real tools excepting the ones provided here Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $ would you like to recommend? I'm not against them, they might ...
0
votes
1answer
13 views

Definition of sigmoidal curve with epsilon

I want to create a sigmoidal curve $f(x)$ with the parameters $s$ and $\epsilon$ so that it has the following features: $f(0) = 0 +\epsilon$ $f(s) = 1 - \epsilon$ $f'(s/2)=1$ Is this possible? If ...
0
votes
0answers
2 views

A saturation-type function that transitions from a linear function to a step function

Let $s = g(t;\theta)$ be some saturation function for a signal in the range $[0,1]$ and a parameter $\theta\in[0,1]$. I would like $g(t;\theta)$ to have the following properties: for $\theta=0$, ...
0
votes
0answers
113 views

A double integral consisted of hypergeometric functions [closed]

Calculate in closed form $$\small\int _0^1\int _0^{\infty }\left(-\frac{9 \sqrt{\frac{3}{\pi }} \Gamma \left(\frac{4}{3}\right) \Gamma \left(\frac{5}{3}\right) \, ...
0
votes
3answers
43 views

Finding the Correct Function that fits the Scenario

i have been trying to find a function that fits the following scenario: $$ f'(c) = 1^0 $$ $$ f''(c) = 2^1 $$ $$ f^{(3)}(c) = 3^2 $$ $$ f^{(4)}(c) = 4^3 $$ and so on, the purpose is to derive a way to ...
1
vote
1answer
82 views

Calculating in closed form an integral in Airy function

Can we hope for a nice closed form for the integral below? $$\int_0^1 \frac{\displaystyle \text{Ai}\left(-\frac{t}{2^{2/3} \sqrt[3]{3-3 t}}\right)^2+\text{Bi}\left(-\frac{t}{2^{2/3} \sqrt[3]{3-3 ...
2
votes
0answers
27 views

Does the Borel-transform of the Lerch-Transcendent have a name/simple expression?

The Lerch-transcendent as given in Mathworld is $$ \Phi(z,s,a)= \sum_{k=0}^\infty {z^k\over (a+k)^s}$$ I'm fiddling with series of the form $$ f_n(z)=\sum_{k=0}^\infty {z^k\over (1+k)^n} $$ and their ...
1
vote
0answers
47 views

integral from gradshteyn and ryzhik

I'm interested in evaluating the integral $$ \int_{a}^\infty e^{-x\cosh\alpha}\,K_{\nu}(x\sinh\alpha)\,\frac{dx}{x}, $$ where $a>0$ and $\nu$ is purely imaginary. Here $K$ denotes the MacDonald ...
2
votes
2answers
16 views

Lipschitz-like behaviour of quartic polynomials

I have observed the following phenomenon: Let the biquadratic $q(x)=x^4-Ax^2+B$ have four real roots and perturb it by a linear factor $p(x)=q(x)+mx$, so that $m$ not too large with respect to ...
0
votes
1answer
38 views

Prove that $T_n$ satisfy $ \sum_{k=0}^{N-1}{T_i(x_k)T_j(x_k)} = \begin{cases} 0 &: i\ne j \\ l\neq 0 &: i=j \end{cases} \,\! $

The Chebyshev polynomials of the first kind satisfy the recurrence relation $$ \begin{cases} T_{n}(x)=2xT_{n-1}(x)-T_{n-2}(x) \qquad n \geq 2 \\ T_{0}(x)=1, \ \ T_{1}(x)=x \\ \end{cases} $$ The ...
0
votes
1answer
115 views

Solve $x^a = 1 - \exp(-x)$ for $x$

I would like to obtain a closed-form solution for the equation $x^a = 1 - \exp(-x)$, in which $x$ is the (real strictly positive) unknown and $a$ is a real positive parameter. So far, I have tried ...
1
vote
1answer
43 views

Integral that resembles an exponential integral

$$ I(y;c,\lambda) \equiv\int_{0}^\infty \frac{\lambda c}{x} \exp\left(-\lambda x\right)\exp\left(-\frac{c}{x}y\right)dx$$ where $c,\lambda>0$. Q: Can this integration be made in analytic form ...
8
votes
0answers
164 views

Help with the integral $\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$

Referring to a previous question, i want help with the integral : $$\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$$ Where $n,m$ ...
4
votes
0answers
35 views

Laplace transform of the logarithmic integral function

What is the Laplace transform of the logarithmic integral function $\text{li}(t)$. Meaning, how to compute the integral : $$\int_{0}^{\infty}\text{li}(t)e^{-st}dt$$
5
votes
0answers
40 views

Asymptotic behavior of the generalized polygamma function

The generalized polygamma function$^{[1]}$$\!^{[2]}$ is defined as $$\psi^{(\nu)}(z)=e^{-\gamma\!\;\nu}\;\partial_\nu\!\left(\frac{e^{\gamma\!\;\nu}\;\zeta(\nu+1,z)}{\Gamma(-\nu)}\right),\tag1$$ where ...
2
votes
0answers
98 views

A difficult integral $\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$

Here is an integral that I want to see a different approach: $$\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$$ Well, for someone who is deeply aware of the exponential integral function and the ...
0
votes
2answers
72 views

Another integral involving a Gaussian and a logarithm

By generalizing methods used in An integral involving a Gaussian and a logarithm. I have computed the following integral below: \begin{eqnarray} \tilde{\mathcal I}(A) &:=& ...
2
votes
0answers
86 views

The ratio of jacobi theta functions

Let $q=e^{2\pi i\tau}$. If $\theta_2$ and $\theta_3$ are jacobi theta functions , is it true that the ratio of the two functions can be expressed as a continued fraction of the form $$ ...
6
votes
0answers
68 views

Why is it that the Lambert W relation cannot be expressed in terms of elementary functions?

According to this Wikipedia page, the Lambert W relation cannot be expressed in terms of elementary functions. However, it does not explain why this is the case. An elementary function is "a ...
1
vote
3answers
31 views

Switching the order of summations of a certain function

I am looking to switch the order of the summations of the following function: $$ \lambda = -\sum_{c=1}^{n-1} \sum_{k=c}^n {k \choose c} \frac{(-1)^k}{k!} f^{k-c}U(-c,k-2c+1,-f)\phi(n,k) $$ I don't ...
6
votes
3answers
186 views

Why is the Gamma function off by 1 from the factorial? [duplicate]

Why didn't they define it as $$ \tilde \Gamma(x) = \int_0^\infty t^x e^{-t} \, dt ?$$ Then the definition would have two less characters than the standard definition of $\Gamma(x)$, and we would have ...
7
votes
1answer
138 views

Computing a double gamma-digamma-trigamma series

What are your thoughts on this series? $$\sum _{k=1}^{\infty } \sum _{n=1}^{\infty } \frac{\Gamma (k)^2 \Gamma (n) }{\Gamma (2 k+n)}((\psi ^{(0)}(n)-\psi ^{(0)}(2 k+n)) (\psi ^{(0)}(k)-\psi ^{(0)}(2 ...
1
vote
0answers
46 views

a q-continued fraction related to the octahedral group

Let $q=e^{2\pi i\tau}$. If $u(\tau)$ is Ramanujan's octic continued fraction, $$u(\tau)=\cfrac{\sqrt{2}\,q^{1/8}}{1+\cfrac{q}{1+q+\cfrac{q^2}{1+q^2+\cfrac{q^3}{1+q^3+\ddots}}}}$$ is it true that ...
0
votes
2answers
27 views

Logarithmic derivative of Polygamma functions

While studying Gamma function and related functions I noticed that its logarithmic derivative (the so-called Digamma function) is studied more than its "normal" derivative but on the other hand I ...
1
vote
0answers
65 views

Derivation of approximation of Error function

In Abramowitz and Stegun there are some formulas for approximation of Error funtion. I am intrested in the formulas $7.1.25$ to $7.1.28$, here is one of them ($7.1.26$). ...
0
votes
0answers
23 views

research in special function with Lie algebra

First of all, I don't know if this is the right place to ask about this. If not, please direct me somewhere I can get more help. I have to research in the field of special functions with a lie ...
4
votes
0answers
71 views

Is there a name for this type of integral $\int_a^b \frac{P(x)}{\sqrt{1-P(x)^2}}dx$?

Given a polynomial of arbitrary degree, $P(x)$, on $[a,b]$ is there a name for this type of integral: $$\int_a^b \frac{P(x)}{\sqrt{1-P(x)^2}}dx$$
1
vote
2answers
67 views

Looking for a function that fits a certain criteria

I am looking for a function that fits this description: $$ \frac{d^n}{dx^n}[f(x)] = n! f(x) $$ or $$ \frac{d^n}{dx^n}[f(x)] = (n-1)! f(x) $$ For all values of $n$, with this function i am looking to ...
0
votes
1answer
36 views

How to find the derivative of $|f(x)|$

The original question was to find domain of derivative of $y=|\arcsin(2x^2−1)|$. First method My attempt was to break $y$ into intervals ,i.e., where $\arcsin(2x^2−1)\geq 0$ and where ...
0
votes
2answers
37 views

Other forms for the derivative of the Gamma function

When I searched for the derivative of the Gamma function I got something of the form: $$\Gamma'(x)=\Gamma(x) \psi(x)$$ But from the definition of the Digamma function to me it's like writing: ...
-1
votes
0answers
29 views

Values of the Sudan function

I am talking about the first discovered recursive function which is not primitive recursive. I would like to know the exact values of $\ f(3,3,3), f(2,0,4), f(2,7,1), f(2,3,2)$ (where $f$ is the ...