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

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4
votes
1answer
39 views

Help with C is Euler's constant and $\Gamma(0)=\infty$ in paper

I am referring to a paper by S. Nadarajah & S. Kotz. The notation is simple enough to understand, however i having trouble with $C$ is Euler’s constant and $\Gamma(0)=\infty$ by equation (2.3) I ...
1
vote
1answer
12 views

Hilbert Curve and Spatial properties

I'm trying to understand the following proposition about the Hilbert Curves: If (x,y) are the coordinates of a point within the unit square, and d is the distance along the curve when it ...
2
votes
1answer
21 views

Degree of Polynomial in Centered Moments of Gamma$(n,1)$

I'm interested in the degree of the polynomial in $n$ of the expression for the $k$-th central moment $$ E((X_n - n)^k) $$ where $X_n$ is a Gamma$(n,1)$ random variable, that is, the sum of $n$ ...
1
vote
0answers
23 views

Help with the integral $\int_{0}^{\infty}\frac{x^{y}}{\Gamma(y)}\cos(y)dy$

We have the integral : $$\int_{0}^{\infty}\frac{x^{y}}{\Gamma(y)}\cos(y)dy$$ We have: $$\frac{1}{\Gamma(y)}=\frac{i}{2\pi}\int_{C}(-t)^{-y}e^{-t}dt$$ Where the path $C$ encircling 0 in the positive ...
12
votes
5answers
820 views

What are special functions for?

If you read enough mathematics, you eventually come across several so-called "special functions". I'm always left wondering what on Earth these things are actually for. We have the Euler Gamma ...
3
votes
0answers
52 views

Integral $\int z^2\Re(J_1(z))dz$

$$ \int x^2 \, \Re\left[{J_1(a x)}\right]dx=\frac{1}{a^3}\int z^2 \Re\left[\frac{z}{2}\sum_{k\geq 0} \frac{(-1)^k}{k!\Gamma(k+2)} \left(\frac{z}{2}\right)^{2k}\right]dz $$ where $a\in \mathbb{C}$ and ...
5
votes
0answers
85 views

The number $\sum\limits_{n=-\infty}^{\infty} \frac{1}{2^{n^2}}$ is transcendental

Prove that the number: $$\sum_{n=-\infty}^{\infty} \frac{1}{2^{n^2}}$$ is transcendental. I don't have a direct proof but a round one. The series can be expressed in terms of $\vartheta_3$ ...
1
vote
1answer
30 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 ...
1
vote
0answers
25 views

simplifying complex expression

Hi I am trying to simplify the following expression:$$ \left|\frac{1}{a+ib}\left(\frac{J_1(c x)}{J_1(c b)}-x\right)\right|^2,\quad a,b,x\in \mathbb{R}, \ c\in \mathbb{C} $$ Is there a simple way of ...
7
votes
5answers
401 views

New idea to solve this equation

I was teaching $\left \lfloor x \right \rfloor$ function properties and equation . I solved this equation in my class . My works are show below. Some students ask me for new Idea...,and now I am ...
4
votes
0answers
33 views

Generalized Trigonometric Functions in terms of exponentials and roots of unity

I am trying to come up with generalized trigonometric functions using the exponential definition that we use today for the trig functions sine and cosine $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}; \cos x ...
0
votes
0answers
41 views

Does my derivation work?

I've been totally engaged with exponential integrals for a while. I came across to this limit in my work. I started to calculate the limit as below: currently, I am not sure about my handouts. would ...
30
votes
3answers
653 views

A closed form for $\int_0^1{_2F_1}\left(-\frac{1}{4},\frac{5}{4};\,1;\,\frac{x}{2}\right)^2dx$

Is it possible to evaluate in a closed form integrals containing a squared hypergeometric function, like in this example? ...
0
votes
4answers
36 views

Find inverse operator

Let $D=\dfrac{d}{dx}.$ Consider the operator $$ D_{h,x}=\frac{e^{hD}-1}{h}. $$ Question. What is explicit form of the operator $D^{-1}_{h,x}?$ I think that $$ D^{-1}_{h,x}=\frac{h}{e^{hD}-1}, $$ ...
-1
votes
0answers
27 views

Confluent hypergeometric function recurrence relation

How to prove the following contiguous relation for the Kummer function $M(a,b,z)$: $$(a−1+z)M(a,b,z)+(b−a)M(a−1,b,z)+(1−b)M(a,b−1,z)=0.$$
6
votes
1answer
70 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. ...
6
votes
0answers
46 views

How to solve this definite Integral containing $E_{1}${.}!

The integral is: $$\int_{N}^{\infty}\frac{E_{1}(cz+d)}{az+b}e^{-pz}dz$$ where, $E_{1}${.} is the exponential integral, and $$a>0,\ b>0,\ c>0,\ d>0,\ p>0,\ N>0.$$ This is similar ...
2
votes
1answer
51 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 $\psi^{0}(\cdot)$ ...
0
votes
1answer
40 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 ...
16
votes
1answer
349 views

Other challenging logarithmic integral $\int_0^1 \frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx$

How can we prove that: $$\int_0^1\frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx=\frac{\pi^2}{8}\zeta(3)-\frac{27}{16}\zeta(5) $$
0
votes
2answers
74 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) &:=& ...
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 ...
6
votes
3answers
240 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 ...
8
votes
5answers
187 views

The maximal size of between $\varphi(n)$ divided by $\lambda(n)$.

I want to find $$f(n) = \max\left\{\frac{\varphi(k)}{\lambda(k)} : 1 \leq k \leq n\right\}$$ In other words, I want to find the maximal value of $\frac{\varphi(k)}{\lambda(k)}$ when $k$ is ...
4
votes
2answers
105 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 ...
3
votes
2answers
246 views

Is there any proof for this formula $\lim_{n \to ∞} \prod_{k=1}^n \left (1+\dfrac {kx}{n^2} \right) =e^{x⁄2}$

Some times ago, In a mathematical problem book I sow that this formula. I don't no whether it is true or not. But now I'm try to prove it. I have no idea how to begin it. Any hint or reference would ...
2
votes
1answer
22 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 ...
0
votes
0answers
8 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?
20
votes
2answers
337 views

Does $\int_{-1}^1\frac{\arctan x}{\text{arctanh}\,x}\,dx$ have a closed form?

Mathematica gives an approximate result of $1.581949621806183890451628...$, but no exact form. I predict it's a function of $e$ and $\pi$, and perhaps even the Golden Ratio $\phi$ (It certainly ...
2
votes
0answers
29 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 ...
0
votes
2answers
53 views

A question about Idempotent functions [closed]

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) ...
3
votes
1answer
51 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 ...
3
votes
0answers
74 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
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 ...
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} ...
2
votes
1answer
326 views

Solve equation with lower gamma function: $A \gamma(2;x/B)=x$ for $x$

I need to find an expression for $x$ given: $A \gamma(2;x/B)=x$ where $\gamma(a,x)=\int\limits_0^x t^{a-1} e^{-t} \mathrm{d}t$ is the lower incomplete gamma function. $A$ and $B$ are real, positive ...
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 = ...
8
votes
4answers
493 views

Integral $\int_0^1 \log \left(\Gamma\left(x+\alpha\right)\right)\,{\rm d}x=\frac{\log\left( 2 \pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha$

Hi I am trying to prove$$ I:=\int_0^1 \log\left(\,\Gamma\left(x+\alpha\right)\,\right)\,{\rm d}x =\frac{\log\left(2\pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha\,,\qquad \alpha \geq 0. $$ I am ...
-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?
0
votes
0answers
119 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
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 ...
8
votes
0answers
166 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 $n,m$ ...
12
votes
0answers
193 views

Relations connecting values of the polylogarithm $\operatorname{Li}_n$ at rational points

The polylogarithm is defined by the series $$\operatorname{Li}_n(x)=\sum_{k=1}^\infty\frac{x^k}{k^n}.$$ There are relations connecting values of the polylogarithm at certain rational points in the ...
9
votes
0answers
92 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
108 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 ...
11
votes
3answers
295 views

Evaluating $\int_0^1 \frac{1}{\sqrt{\Gamma(x)}} dx$

What is the value of the following integral? $$\int_0^1 \frac{1}{\sqrt{\Gamma(x)}} \,dx$$ Here $\Gamma(x)$ is Euler's gamma function. EDIT: Can we improve the upper bound strictly smaller than $1$? ...
0
votes
1answer
14 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 ...
7
votes
1answer
112 views

Essential Selfadjointness of Quantum Harmonic Oscillator Hamiltonian

The Hamiltonian for the Quantum Harmonic Oscillator is (disregarding constants) the Hermite operator $$ Hf = -f''+x^{2}f, $$ where $\mathcal{D}(H)$ consists of all twice absolutely ...
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
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 ...