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
0answers
80 views

Another integral related to Fresnel integrals

How would we prove this result by real methods ? $$\int_0^{\infty } \frac{\sin \left(\pi x^2\right)}{x+2} \, dx=\frac{1}{4} \left(\pi-2 \pi C\left(2 \sqrt{2}\right)-2 \pi S\left(2 ...
4
votes
2answers
62 views

Closed form of a series (dilogarithm)

We are all aware of the dilogarithm function (Spence's function): $$\sum_{n=1}^{\infty} \frac{x^n}{n^2}, \;\; x \in (-\infty, 1]$$ Also it is known that: $$\sum_{n=1}^{\infty} \frac{\cos n x}{n^2}= ...
6
votes
3answers
119 views

Closed-form of $\operatorname{Li}_2\left(1 \pm i\sqrt{3}\right)$

I've found the following identity while I was going through a quite difficult path. $$ \Re\operatorname{Li}_2\left(1 \pm i\sqrt{3}\right) = \frac{\pi^2}{24} -\frac{1}{2}\ln^2 2 - ...
1
vote
1answer
14 views

Various forms of the Confluent Heun Equation

The Confluent Heun equation is expressed in various forms. It's non-symmetrical canonical form is: \begin{equation} ...
0
votes
0answers
90 views

Riemann zeta function, functional equation, what completes this analogy?

What completes this analogy? This: $$\zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s)\;\;\;\;\;\;\;\;\;\;(1)$$ is to: $$\chi(s)=\pi ^{-\frac{s}{2}} \Gamma ...
1
vote
1answer
25 views

Question Regarding a Second Order Ordinary Differential Equation

I was wondering if the solution to the following differential equation belongs to a class of special functions. If not, is it exactly solvable? \begin{equation} ...
-3
votes
0answers
26 views

clarification domain and range of functions [closed]

Want a clarification on the domains and ranges of addition, subtraction, multiplication, division and enhancement of functions, if possible with an example with roots and absolute value for a ...
4
votes
2answers
67 views

Does the elliptic function $\operatorname{cn}\left(\frac{2}{3}K\left(\frac{1}{2}\right)\big|\frac{1}{2}\right)$ have a closed form?

Given the complete elliptic integral of the first kind $K(k)$ for the modulus $k$, can the elliptic function $$\text{cn}\left(\frac{2}{3}K\left(\frac{1}{2}\right)\bigg|\frac{1}{2}\right)$$ be ...
3
votes
1answer
39 views

Prove or disprove that Weierstrass $\wp$ function is holonomic

Recall that a holonomic function $f$ (say over $\mathbb C$) is one that is a solution to a differential equation of the form: $$p_0(z)f(z)+p_1(z)f'(z)+p_2(z)f''(z)+\dots+p_k(z)f^{(k)}(z) = 0$$ where ...
4
votes
1answer
61 views

closed-form of an integral similar to Bessel function

The integral form of the $n$-th modified Bessel function of the first kind is $$ I_n(z)=\frac{1}{\pi}\int_0^{\pi}e^{z\cos\theta}\cos(n\theta)\;d\theta. $$ However, I found an integral $$ ...
3
votes
1answer
56 views

Is the functional equation for $\zeta (s) \left(1-\frac{1}{3^{s-1}}\right)$ known?

It says in wikipedia that Hardy gave a simple proof of the functional equation for: $$\eta(s)=\zeta (s) \left(1-\frac{1}{2^{s-1}}\right)$$ and that it is: $$\eta(-s) = 2 ...
2
votes
1answer
27 views

Confusion with Heaviside Step Function and Ramp Function.

I want to know how to represent the following graph with Heaviside step functions and ramp functions. My guess is that this is represented as $r(t) -r(t-2) +2u(t-2)$, where $r(t)$ is the unit ramp ...
0
votes
0answers
35 views

Can this relation be made into a functional equation?

I am trying to find the functional equation for this: $$\zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ Therefore I let: $$x_1=\left(1-\frac{1}{n^{s-1}}\right)$$ which I substitute with ...
4
votes
3answers
139 views

Closed-form of $\int_0^1 \operatorname{Li}_3\left(1-x^2\right) dx$

By using dilogarithm functional equations we can show that $$ \int_0^1 \operatorname{Li}_2\left(1-x^2\right)\,dx = \frac{\pi^2}{2}-4, $$ where $\operatorname{Li}_2$ is the dilogarithm function. Could ...
1
vote
1answer
41 views

Show that the expression holds for any $x\gt 1$

Let $f(u)$ be a continuous function and, for any real number $u$, let $[u]$ denote the greatest integer less than or equal to $u$. Show that for any $x \gt 1$, $$\int \limits_1 ^x ...
1
vote
0answers
19 views

Generalized Hyperbolic and Circular Functions

I have recently posted a couple of questions in regards to Generalized Hyperbolic and Circular Functions and I was hoping to find a couple more papers available on the particular subject. The papers ...
2
votes
2answers
42 views

Inequality with decay of modified Bessel functions of the second kind

I think that the following inequality holds for all $x > 0$ and all $\nu$ above some constant that is somewhere around 0.2: $$ K_\nu(2 x) \le \frac{2^{2 - 2 \nu}}{\Gamma(\nu)} x^\nu K_\nu^2(x) $$ ...
3
votes
1answer
105 views

Hypergeometric function values and the Baxter constant

While I was working on this question by @Vladimir Reshetnikov, I've found the following relations between Gaussian hypergeometric function values and the Baxter constant: ...
1
vote
1answer
25 views

Caculation of involving Hermite polynomial

I have a trouble with this problem involving Hermite polynomial(probability version!). The problem is $$ \frac {(-1)^{r-1}H_{2r-1}(x)}{2^{r-1}(r-1)!x}=\sum_{s=0}^{r-1}\frac{(-1)^s}{2^ss!}H_{2s}(x) $$ ...
0
votes
0answers
24 views

functional equation from a recurrence relation

Hoe can we get a functional equation from a recurrence relation? Lets say I have a recurrence relation $P_n(x)=a\cdot P_{n-1}(x)-b\cdot P_{n-1}(x)$. We let $\sum P_m(x) t^n=P(x,t)$ and now we have to ...
1
vote
3answers
54 views

Exponential integral representation

According to exponential integral eqn. (8) $\; E_{1}(x) \;$ can be represented by: $$ E_1(x)= - \gamma - \ln(x) - \sum _{n=1}^{\infty } \frac{(-1)^n x^n}{n n!} $$ where $\gamma$ is the ...
7
votes
0answers
98 views

Bear of an integral

I have a pretty ferocious integral to solve, and would be over the moon if I were able to get some sort of analytic expression / insight for it. $$ I = \int_{r}^{\infty} r_0^{-5/2} W_{-i\alpha'/2, ...
6
votes
1answer
143 views

Closed-form of the hypergeometric function ${_4F_3}\left(\begin{array}c1,1,\tfrac54,\tfrac74\\\tfrac32,2,2\end{array}\middle|\,-t\right)$

Inspired by this question and by using Mathematica the following conjecture seems to be true for all nonzero complex $t$ number: ...
5
votes
1answer
197 views

Integral $\int_0^1 \ln(x)^n \operatorname{Ei}(x) \, dx$

I've conjectured the following identity for $n\geq0$ integers: $$ \int_0^1 \ln(x)^n \operatorname{Ei}(x) \, dx = (-1)^{n+1}n! \cdot \left(-\operatorname{Ei}(1)+\sum_{k=1}^{n+1} ...
0
votes
0answers
37 views

How to solve the general sextic equation with Kampé de Fériet functions?

It is frequently stated, for example on Wolfram Mathworld, that the general sextic equation $$x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x^1 + a_0 = 0$$ can be solved in terms of Kampé de ...
3
votes
1answer
77 views

Question about Meijer-G definition and identity

I'm trying to wrap my mind around computation involving the Meijer $G$ function, as defined here. (Edit: I'm actually using a somewhat mixed notation using the definitions from MathWorld and the ...
2
votes
0answers
67 views

What do engineers do when they confront special integrals?

Suppose in a real life situation engineers have to calculate $\int{{2^2}^2}^xdx$ or $\int\sqrt{4-\sin^2x} dx$. The first one doesn't have an integral at all and the second one is an elliptic one. A ...
1
vote
1answer
39 views

Is this a correct proof that $\tau(n)$ is eventually smaller than any $n^{\delta}$, $\delta>0$?

Is this a correct proof that $\tau(n)$ is eventually smaller than any $n^{\delta}$, $\delta>0$? I'm not even sure about the statement, let alone the proof. Let's first proof this result: $\tau(n) ...
2
votes
0answers
37 views

Integral Representation of Terminating Hypergeometric function

For Hypergeometric function ${}_2F_1(a_1,a_2;b_1;z)$, if $a_1$, $a_2$ are negative integers, it will be terminated and convergence is not a problem. Under this circumstances, does anyone know the ...
1
vote
0answers
30 views

Complicated recurrence relation

I would like to know if the following recurrence relation is solvable \begin{equation} (\alpha_{1}n^{2}+\beta_{1}n+\gamma_{1})\ c(n+1)+(\alpha_{0}n^{2}+\beta_{0}n+\gamma_{0})\ ...
3
votes
2answers
84 views

Sum of Legendre function

I'm currently trying to solve the sum $$ f(x)=\sum\limits_{n=0}^\infty\frac{x^{n+1}}{n+1}P_n(x), $$ where $P_n(x)$ is the Legendre function of order n. I also named the sum $f(x)$ since I'm the ...
1
vote
0answers
29 views

Why $\lim_{a\to\infty} \frac{ Q_0(a,b)}{ \sqrt{a e}}e^{\frac{(a-1)^2}{2}}\neq Q (b)$?

I’m trying to find a connection between Marcum-Q function, which is defined as: $$Q_M(a,b)=a^{1-M}e^{-\frac{a^2}{2}}\int_{b}^{\infty} x^M \exp^{-\frac{x^2}{2}} \mathrm I_{M-1}(a x)\mathrm dx$$ where ...
1
vote
1answer
25 views

Erroneous reasoning over equivalence of hankel function with logarithm

We have the following equation/solution pairs: $$(\nabla^2+k^2)G(\mathbf{x},\mathbf{x'}) = \delta(\mathbf{x}-\mathbf{x'}) \to G(\mathbf{x},\mathbf{x'}) = ...
1
vote
1answer
41 views

what is the upper bound of $\max \mathbf{w}^T\mathbf{x}_i$

I need to find an equation for the upper bound of $\max \mathbf{w}^T\mathbf{x}_i, \; i=1, \dots N$. where $\mathbf{w}$ and $\mathbf{x}_i$ are two vectors. I need to find a function $f$ which holds ...
2
votes
4answers
83 views

If $f(\alpha x, \alpha y) = f(x,y)$ is $f$ some special function?

Like the title reads, if $f(\alpha x, \alpha y) = f(x,y)$ is $f$ some special function? Assume $x,y,\alpha\in\mathbb{R}$.
2
votes
2answers
162 views

A theta function around its natural boundary

Let $q = e^{2\pi i\tau}$, if $$\psi(q^2)=\sum_{n=0}^{\infty} q^{n(n+1)}$$ is one of ramanujan theta functions,is it possible to evaluate the limit $$\lim_{q\rightarrow 1} (1-q){\psi^2(q^2)}$$ In fact ...
2
votes
1answer
92 views

What is the limit $\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$

If $\vartheta_{2}(q)$ is jacobi's theta function, what is the limit $$\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$$ for the nome $q$. I would like to know whether the limit exists or not. If it does, ...
0
votes
1answer
53 views

counter examples for Riemann and Lebesgue integrabilities

I'm seeking interesting examples that are not mentioned in usual real analysis texts. It seems that in general there is no relationship between Riemann integrability and Lebesgue integrability when ...
1
vote
0answers
20 views

$t^{\nu}K_{\nu}(\beta t)$ The solution to an unknown ODE.

Good day to you all, This question is fairly open, so I hope that I'm not in breach of any of the Stack Exchange rules. Clearly $f(t)=K_{\nu}(\beta t)$, where K is the modified Bessel function of ...
2
votes
1answer
30 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 ...
4
votes
2answers
152 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 ...
2
votes
1answer
29 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
29 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 ...
5
votes
0answers
105 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
0answers
31 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 ...
4
votes
0answers
86 views

Integral $\int z^2\Re(J_1(z))dz$=$\int y^{3/2} \Re \left[\frac{1}{\sqrt y} (1-e^{-y})\right]dy$

Hi I am trying to simplify and calculate the integral below. $$ I=\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)} ...
7
votes
5answers
410 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 ...
0
votes
0answers
44 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 ...
5
votes
0answers
40 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
4answers
38 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 $$ D^{-1}_{h,x}=\frac{h}{e^{hD}-1}, $$ ...