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

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6
votes
2answers
118 views

Given a Fourier series $f(x)$: What's the difference between the value the expansion takes for given $x$ and the value it converges to for given $x$?

The question given to me was: Find the Fourier series for $f(x) = e^x$ over the range $-1\lt x\lt 1$ and find what value the expansion will have when $x = 2$? The Fourier series for $f(x)=e^x$ is ...
4
votes
2answers
244 views

$(\partial_{tt}-\nabla^2+\partial_t)f=g,\quad (\partial_t-\nabla^2+b)g=\partial_t f$

Hi I am looking for complete solutions for $f(r,t),g(r,t)$ given in the coupled linear partial differential equations below: $$ (\partial_{tt}-a\nabla^2+b\partial_t)f(r,t)=bg(r,t) $$ $$ (\partial_t-c ...
3
votes
3answers
147 views

Prove that $\int_0^\infty\frac{x\cos(x)-\sin(x)}{x\left({e^x}-1\right)}\,dx = \frac{\pi}2+\arg\left(\Gamma(i)\right)-\Re\left(\psi_0(i)\right)$

While I was working on this question, I've found that $$ I=\int_0^\infty\frac{x\cos(x)-\sin(x)}{x\left({e^x}-1\right)}\,dx = \frac{\pi}2+\arg\left(\Gamma(i)\right)-\Re\left(\psi_0(i)\right), $$ where ...
1
vote
1answer
60 views

Derive the Differential Equation for Laguerre polynomials

How can i derive this linear differential equation $$ xD^2L^{(\alpha)}_n (x)+(1+\alpha -x)DL^{(\alpha)}_n (x)+nL^{(\alpha)}_n (x)=0 $$ from the following recurrence relations ...
0
votes
3answers
21 views

Seeking a 3 variable function with a specific property

I'm looking for a function $f(x_1,x_2,y)$ which has the following property. $y \in [0,1]$, $x_1,x_2>0$. The minimum value of $f(x_1,x_2,y)$ is $max(x_1,x_2)$ at $y=1$; The maximum value of ...
10
votes
1answer
4k views

Solution of an integral with strange imprecision of gamma functions

Trying to solve the following integral, with $n,m \in \mathbb{Z^+}$, $\alpha>1$, $0 < \epsilon < 1$, and $\Gamma(.)$ and $\Gamma(.,.)$ the gamma and incomplete gamma functions, respectively: ...
0
votes
1answer
48 views

Legendre functions $Q_n(x)$ of the second kind

Legendre functions $Q_n(x)$ of the second kind \begin{equation*} Q_n(x)=P_n(x) \int \frac{1}{(1-x^2)\cdot P_n^2(x)}\, \mathrm{d}x \end{equation*} what to do after this step? how can I complete ? I ...
0
votes
1answer
37 views

complex infinity

Hi I am interested in a question regarding complex infinities. For example, consider the function $$ q(x)=\frac{1}{x}\sqrt{i-1},\quad x\in \mathbb{R} $$ where $i=\sqrt{-1}$. Now let's take the limit ...
0
votes
0answers
12 views

Functions that sum to zero under cyclic index permutations?

Consider $n$ real vectors of arbitrary dimension $x_i$ with $i\in \{1,2,...,n\}$. Furthermore, consider a function $f(x_1,x_2,x_3)$ where actually the function can only depend on scalars $x_i\cdot ...
0
votes
1answer
21 views

Have the derivative of a function as a recurrence, looking for a closed form expression of the function.

I have a recurrence relation for the derivative of a function. Can anyone give me a closed form expression for the function? The is the derivative: $${\partial \over \partial x} f(x,z,a) = ...
5
votes
0answers
110 views

Simplifying a certain polylogarithmic sum in two variables

This question is related to my previous question here. While tinkering around for a solution I found that the integral there can be reduced to the problem of solving the following basic logarithmic ...
2
votes
1answer
143 views

PDE $(\partial_{tt}+a\partial_t-b\nabla^2)f(r,t)=0$

I am interested in solving the linear PDE for $f(r,t)$ $$ (\partial_{tt}+a\partial_t-b\nabla^2)f(r,t)=0 $$ $$ \nabla^2\equiv ...
1
vote
2answers
109 views

complex conjugate of Bessel function

This is probably a simple question, however, how does one take a complex conjugate of the Bessel function,$$ J_1(z),\quad z\in \mathbb{C}$$ I am asking because I am interested in calculating $$ ...
0
votes
1answer
32 views

Sigmoid function increasing for large values of variable x

I am looking for a function that involves the sigmoid function but for large values of variable $x$ increases. Maybe sth like this, but there is a part missing: $f(x)= A +B\ \frac{1}{(1+e^{-x})}\ +\ ...
0
votes
0answers
27 views

Identities of Lambert W

I am interested in $y\in \mathcal R$: $$ y_n = \frac{-1}{\lambda} W_n \left(-\exp\left(-\frac{\lambda}{a}\right)\frac{\lambda}{a}\right) - \frac{1}{a}$$ Where $\lambda, a > 0$, and $x \equiv ...
0
votes
1answer
27 views

Restrict solutions using lambert W to reals

I have an equation of the form $$ x a + b = \exp(xc)$$ where I, in fact, know that $b =1$. Which implies that one solution to the equation is always at $x = 0$. I'm now searching for the other ...
0
votes
1answer
58 views

Inverse Function of sum of exponential function

What is the inverse function for $$y=a^x+b^x+...+z^x$$ where $a, b, .. , z$ are positive constant and $x>0$ Thanks in advance!
1
vote
2answers
41 views

A function that is tangent to both axes

I am looking to design a function in the plane, $y=f(x)$, which is tangent to both axes $x$ and $y$ at certain points. Say, for example: $$ f(0)=\alpha>0, \; f(\beta)=0,\; f'(0)=-\infty, \; ...
12
votes
0answers
169 views

Dilogarithm identity containing the tribonacci constant

The motivation of this question is the brilliant conjecture by @Tito Piezas III. In $(4)$ of his question the equation seems to be true for all $n > 1$ real numbers. The case $n=2$ leads us to a ...
0
votes
0answers
56 views

Finding the radius of convergence of a power series involving legendre polynomials

I am trying to find the radius of convergence for $f(x,t)=\sum P_n(x)t^n$ considered as a power series in $t$ as a function of $x$. Here $P_n(x)$ are the legendre polynomials on the interval $[-1,1]$. ...
3
votes
0answers
33 views

Proving that $ \frac {\sin (n+1) \theta}{\sin \theta}=\sum \limits _{l=0}^{n} P_l(cos \theta)P_{n-l}(cos \theta)$

I am trying to prove that $ \frac {\sin (n+1) \theta}{\sin \theta}=\sum \limits _{l=0}^{n} P_l(cos \theta)P_{n-l}(cos \theta)$. I found this identity on wikipedia but I am unable to prove it. If ...
4
votes
0answers
66 views

Simplification of an expression involving the dilogarithm with complex argument

Do you think there is a way to get a nice form of the expression below $$\Im{\left( \text{Li}_2\left(\frac{3}{5}+\frac{4 i}{5}\right)- \text{Li}_2\left(-\frac{3}{5}+\frac{4 i}{5}\right)+ ...
3
votes
1answer
34 views

Define a particular function

Does anybody know how you can define a function $\eta \in C_c^1(B_R(0))$ such that $\eta = 1$ on $B_{\frac{R}{2}}(0)$ cause I need such a function in a particular proof, so I would really like to know ...
2
votes
0answers
26 views

expansion of Bessel function $J_1$

this is a general question. Is there a general way to expand the Bessel Function $J_1(z)$ when $z\in \mathbb{C}$ and when z is large? Or in other words, what is the asymptotic expansion of $J_1(z)$? ...
8
votes
2answers
142 views

Closed form of $\lim\limits_{n\to\infty}\left(\int_0^{n}\frac{{\rm d}k}{\sqrt{k}}-\sum_{k=1}^n\frac1{\sqrt k}\right)$

Show that $$ L=\lim_{s\rightarrow\infty}\left(\int_0^s\frac{ds'}{\sqrt{s'}}-\sum_{s'=1}^s\frac{1}{\sqrt{s'}}\right) = 1.460\ldots $$ My attempts: To begin, rewriting the limit of the ...
2
votes
0answers
59 views

Jacobi Theta Functions?

For the Jacobi theta function $\vartheta_3(z|\tau)$ there exists an equality (by Whittaker & Watson) \begin{equation} \vartheta_3(z|\tau) = \sum_{n=-\infty}^{\infty} e^{n^2 \pi i \tau + 2 n i z} ...
0
votes
1answer
22 views

How to plot $ y(t)=-1000(\ r(t)-r(\ t-2\cdot 10^{-3})) $

Where $r(t)$ is the ramp function and i consider $t$ to be the independent variable (time). Wolfram gives me this result. However i'm confused why this is true. Could someone explain to me? Thanks in ...
4
votes
1answer
95 views

Evaluating a logarithmic integral in terms of trilogarithms

For $a,c\in\mathbb{R}\land-1\le a\land-1<c$, define the function $J{\left(a,c\right)}$ to be the value of the dilogarithmic integral ...
1
vote
0answers
22 views

Embedding a Torus into Plane with Cuts

I have a complex plane with two horizontal cuts $[-\alpha \pm i/2, \alpha \pm i/2]$ for real $\alpha$. We can imagine gluing the two cuts to get a torus with complex parameter $\tau$. Thinking of ...
0
votes
1answer
22 views

Derivation of an identity with beta function.

Beta function is defined as: $$B(x,y)=\int_0^1 t^{x-1}(1-t)^{y-1}dt$$ for $\Re(x) , \Re(y)>0$, I want to show that:$\frac{B(x,y)}{c^y}=\int_0^\infty \frac{t^{x-1}dt}{(c+t)^{x+y}}$. I thought of ...
2
votes
1answer
71 views

Solution of second order linear ODE

I consider a second order linear ODE : $$ x^{2\beta+2}\frac{\partial^2 V}{\partial x^2}+(a+x^{2\beta})x\frac{\partial V}{\partial x}+(b+x^{2\beta})V=0. $$ I am expecting that the above equation can ...
1
vote
0answers
33 views

Rules for Partial Fraction decomposition in functions of two variables

I am interested in using the method of partial fractions to decompose a function of two variables. I have seen a few examples, but was interested in the specific rules. from the two examples on stack ...
1
vote
2answers
55 views

How does one calculate the following limit using analysis?

I want to evaluate the limit: $$\lim_{ n \rightarrow +\infty} \frac{1}{\Gamma(n)} \int_0^n x^{n-1}e^{-x}\, {\rm d}x$$ Well $\Gamma$ here stands for the gamma function hence that $\Gamma(n)=(n-1)!$ ...
4
votes
0answers
52 views

Can anyone identify the function that represents this infinite product?

$$\lim_{\omega \to \infty} \prod_{N=1}^{\omega} {{1+e^{b \cdot c^{-N}}} \over 2}$$ For instance, the Lerch Transcendent is a analogous example of a special function that defines the sum of a useful ...
1
vote
0answers
25 views

Comprehensive account of the Barnes G function?

I am looking for a comprehensive account of the properties and applications of the Barnes G-Function. Everything from recurrence relations, proof of the infinite product representation, functional ...
4
votes
1answer
166 views

How to solve the Brioschi quintic in terms of elliptic functions?

Given the Brioschi quintic $$w^{5}-10cw^{3}+45c^{2}w-c^2=0$$ I'm interested in seeing different ways of solving it in terms of elliptic functions or theta functions.
5
votes
1answer
113 views

Rogers-Ramanujan continued fraction in terms of theta functions?

The Rogers-Ramanujan cfrac is, $$r = r(\tau)= \cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\ddots}}}$$ If $q = \exp(2\pi i \tau)$, then it is known that, $$\frac{1}{r}-r ...
5
votes
2answers
94 views

Transformation of second order ODE

I am a beginner for a ODE theory. There is a ODE: $$ x^{2\beta+2} \frac{d^2 V}{dx^2}+x\frac{dV}{dx}+bV=0. $$ In the paper, the authors introduce some transformation $$ ...
16
votes
1answer
198 views

Is there any proof that the Riemann Zeta function is not elementary?

I'm just curious, has anyone ever proved that the Riemann Zeta function is not an elementary function? Here I am using the term "elementary" in the sense of Liouville or as defined in this paper. ...
0
votes
0answers
18 views

Possible to find roots of a function involving heaviside?

Consider the function $f(t) = 3+t-(t-3)^2\theta(t-3)$, where $\theta(t)$ is the heaviside step function. Can i solve the equation $f(t) = 0$ somehow?
1
vote
1answer
32 views

Is $f(g)$ homogeneous? If so, of what degree?

Given $f$ and $g$ are homogeneous functions of degree $k$. I have to show if $f(g)$ is homogeneous or not, and if so, of what degree. Definition (Homogeneous function). Let ...
2
votes
0answers
183 views

Contour Integral solution to differential equations, Euler transformation?

In Spain's book, Functions of mathematical physics he introduces the contour integral method of solving ODEs. The baseic idea is: given an ODE $\sum_0^m a_r(t) \frac {d^rf}{dt^r} = 0$, a solution may ...
8
votes
1answer
134 views

What is a mock theta function?

We define a mock theta function as follows: A mock theta function is a function defined by a $q$-series convergent when $|q|<1$ for which we can calculate asymptotic formulae when $q$ tends ...
3
votes
0answers
35 views

What do the Stirling numbers of the first kind have to do with polylogarithms?

On a whim, I had decided to look into ways of evaluating series of the form $$\sum_{n\ge1}\frac{1}{n^k2^n}$$ which I learned has a more general form in terms of polylogarithms: ...
9
votes
2answers
237 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
92 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}= ...
7
votes
3answers
172 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
43 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} ...
1
vote
0answers
100 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
39 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} ...