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

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7
votes
1answer
140 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
75 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
29 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
67 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 ($7.1.26$). ...
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 ...
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 ...
0
votes
0answers
28 views

Re-Expressing the Digamma

I was reading some articles on the digamma function, and I was wondering if anyone knows how to express the digamma function $\psi^{(0)}(n)$ in terms of a trigonometric function or a logarithmic ...
4
votes
1answer
53 views

$\int x J_0(k x)e^{-x^2/2}dx$ Bessel function decomposition of a gaussian

$$\int ^\infty _0 x J_0(k x)e^{-x^2/2}dx$$ The integral above corresponds to fourier transform in radial coordinates. The fourier transform of a 2D gaussian is still a 2D gaussian. So the integral ...
14
votes
3answers
239 views

Computing $\sum _{k=1}^{\infty } \frac{\Gamma \left(\frac{k}{2}+1\right)}{k^2 \Gamma \left(\frac{k}{2}+\frac{3}{2}\right)}$ in closed form

What tools other than beta function you might like to use here? $$\sum _{k=1}^{\infty } \frac{\displaystyle \Gamma \left(\frac{k}{2}+1\right)}{\displaystyle k^2 \Gamma ...
6
votes
1answer
112 views

Does $ \sum_{(m,n) \neq (0,0)} \frac{(-1)^{m+n}}{m^2 + n^2} $ have an exact value?

I am looking for an $\mathbb{Z}[i]$ analogue of the alternating harmonic series: $L(1,\chi)=\sum_{n=0}^\infty (-1)^n \frac{1}{n} = \frac{\pi}{4}$. If we try adding the reciprocals of the Gaussian ...
0
votes
2answers
50 views

Finding the inverse of the function $f(k, x) = k^{x}x.$

Recently, I have been looking at the function $f(x) = e^{x}x,$ where its inverse is the Lambert W function. I was intrigued by the fact that it is rather hard to calculate its solution, in comparison ...
3
votes
1answer
126 views

A new $q$-continued fraction of order $12$

I think I may have discovered a $q$-continued fraction of order $12$ with a form different from that established by Mahadeva Naika. Let $q=e^{2i \pi \tau}=\exp(2i \pi \tau)$, then, $$\begin{aligned} ...
1
vote
1answer
43 views

Verify $\frac {\partial B} {\partial T} =$ $\frac{c}{(e^\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}e^\frac{hf}{kT}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(e^\frac{hf}{kT}-1\right)}$$ The correct answer (I believe) ...
0
votes
0answers
16 views

Functions to manipulate (increase) probability exponentially or logaritmically?

Very simple. I want a function to manipulate a probability in order increase it without getting out of the range of 0 to 1. Basically a function similar to the blue lines in the following sketch: ...
1
vote
0answers
38 views

An integral involving the Gamma function

By utilizing the results of two previously asked questions, Series involving Laguerre polynomials and Integral of binomial coefficients, what is a resulting value of the integral \begin{align} ...
0
votes
0answers
15 views

Laplace Transform of Kelvin functions

What is the value of the Laplace transform, in terms of the G-function, \begin{align} \int_{0}^{\infty} e^{-st} \, t^{m} \, \left(ber_{\nu}^{2}(t) + bei_{\nu}^{2}(t)\right) \, dt \hspace{5mm} ? ...
2
votes
1answer
72 views

Autocorrelated, discrete, bounded and symmetric random walk with no edge attraction

I need to move over a discrete set of linearly organized.. let's say "Japan steps" $S=\{0,\dots,c\}, c \in \mathbb{N}^*$. My current position is given by $d \in S$. On each time step, I need to draw ...
0
votes
2answers
45 views

Show that $\frac {\partial B} {\partial T} =$ $\frac{c}{(\exp\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(\exp\frac{hf}{kT}-1\right)}$$ The correct answer is $\frac ...
1
vote
1answer
64 views

A differential equation I

Consider the second order differential equation \begin{align} 2 t^{3} y'' + (5 t^{2} - t) y' + (t^{2} - t + 1) y = 0 \end{align} with the conditions $y(0) = 0$ and $y'(0) = 1$. A solution is known in ...
0
votes
1answer
21 views

Properties of greatest integer function

I am curious to know some properties of the floor functions, for instance, $\lfloor a \cdot x \rfloor$, $\lfloor a1\cdot x1+a2\cdot x2 \rfloor$, etc. Is there any book that contains such properties ?
11
votes
0answers
124 views

Reducing multi-variable functions to a composition of 1- or 2-variable functions

There are some special functions of 3 or more complex variables that are analytic in some domain (a region in $\mathbb C^n$) with respect to each variable. To give some examples: the incomplete beta ...
1
vote
0answers
30 views

Series involving Laguerre polynomials

Given the series \begin{align} S_{x}(a) = \sum_{k=1}^{\infty} (-1)^{k+1} \, \binom{x-1}{k} \, L_{k+n-1}(a) \end{align} where $L_{m}(x)$ is the Laguerre polynomial. By using \begin{align} L_{n}(z) = ...
4
votes
1answer
52 views

Simplifying real part of hypergeometric function with complex parameters

I am looking for a simpler representation of the following hypergeometric function with complex parameters in terms of more basic functions and manifestly real parameters: ...
9
votes
2answers
119 views

Integral of binomial coefficients

Let the integral in question be given by \begin{align} f_{n}(x) = \int_{1}^{x} \binom{t-1}{n} \, dt. \end{align} The integral can also be seen in the form \begin{align} f_{n}(x) = \frac{1}{n!} \, ...
8
votes
4answers
151 views

Is there an integral representation for $\frac{1}{n!}$?

I know that $n!$ has various integral representations, for instance the $\Gamma$ function. I was wondering if $\frac{1}{n!}$ has an integral representation.
0
votes
0answers
24 views

Scaling properties of Airy functions.

I'm specifically intereseted in how to rewrite $(\alpha\in \mathbb{R})$ $$\mathrm{Ai}(\alpha\cdot z)$$ as some constant times $\mathrm{Ai}(z).$ Edit: oops sorry I thought I had posted the question ...
1
vote
1answer
48 views

Fourier transform of cosine with square root

In relativistic mechanics, i came across the Fourier transform of the following function : $\cos \left(t \sqrt{x^2+m^2} \right)$ or $e^{it \sqrt{x^2+m^2}}$ ($t$ and $m$ are constants). Is there a way ...
8
votes
1answer
115 views

Cosh and Sinh analogs

We know that $$\cosh{x}+\sinh{x}=e^x$$ and that his can be expressed as $$\frac{e^x+e^{-x}}{2}+\frac{e^x-e^{-x}}{2}=\frac{(e^x+e^x)+(e^{-x}-e^{-x})}{2}=e^x$$ and this works out nicely because the ...
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 ...
12
votes
2answers
142 views

What is the exact value of $\eta(6i)$?

Let $\eta(\tau)$ be the Dedekind eta function. In his Lost Notebook, Ramanujan played around with a related function and came up with some of the nice evaluations, $$\begin{aligned} \eta(i) &= ...
2
votes
0answers
60 views

Conway's box function generalized as a hierarchy of nested sets of real numbers

Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). When the ...
1
vote
1answer
42 views

Function with special behavior near zero

I'm looking for functions that have the following behaviors: $f(x) \to 0$ as $|x| \to 0$, as $x \to 0^+$, $\alpha < \frac{{df(x)}}{{d(x)}}$ for any $0<\alpha<\infty $. One example of ...
1
vote
2answers
35 views

Slow decreasing function that exhibits asymptotic behaviour.

I am currently doing some work on modelling the effects of treated nets usage on mosquito populations. Nets do not retain their maximum efficacy forever. They lose their chemical efficacy after about ...
0
votes
0answers
32 views

Quasi concavity and quasi convexity of a max function

Consider the production function: Q = max {K, L}. We have to find out whether this function is quasi concave or quasi convex. According to me, since this function is concave, it should automatically ...
1
vote
1answer
23 views

Solving $\int_0^{\infty}\,dk\,\exp{(-\delta^2k^2)}\,\frac{J_1(kR)}{k^2}$

I would like to understand if there is a closed formula for this integral: $$\int_0^{\infty}\,dk\,\exp{(-\delta^2k^2)}\,\frac{J_1(kR)}{k^2}$$ where $R,\delta>0$ and $J_1(\cdot)$ is the bessel ...
-1
votes
1answer
36 views

Need Help Evaluating This Indefinite Integral

I would appreciate any help finding a possible closed form solution of this integral. $$\int\sqrt{\cosh(u)-\cos(v)}\cdot e^\frac{u}{2}~du$$ Any help would be greatly appreciated! The solution for ...
2
votes
0answers
53 views

Evaluate of $\int \frac{(a+bx)^k}{x} dx$

Consider the integral $$ \int \frac{(a+bx)^k}{x} dx. $$ Does this have a representation using (preferably) elementary or special functions? Edit: $0 <k<1$. Wolfram alpha produces an ...
1
vote
0answers
23 views

An equality involving the lower Gamma incomplete function

I'm trying to prove the following equality: $$ \gamma(r,x)=\int_0^x \Gamma(r,y) \sum_{m=1}^\infty\frac{e^{-(x-y) } (x-y) ^{mr-1}}{\Gamma(mr)}dy, $$ where $\Gamma(\cdot)$ is the Gamma function, ...
2
votes
2answers
80 views

Proving Bonnets' Recursion with Rodrigues' Formula

I would like to show that $(n+1)P_{n+1}(x)+nP_{n-1}(x)=(2n+1)xP_{n}(x)$ using Rodrigues' formula, not the generating function. I got to this point, but have not been able to progress further. ...
4
votes
1answer
50 views

Iteration of $\log(z) / \sqrt{z}$

The complex function $\log(z) / \sqrt{z}$ is a curiosity that I find interesting since one can express $e^{i\pi}+1=0$ as $\log(-1) / \sqrt{-1} = \pi$. My question is, what is the significance of the ...
1
vote
0answers
28 views

How do I visualize an $n$-dimensional function?

How do I visualize the concept of a function that has more than three dimensions in the context of noise functions? I understand that a one-dimensional function as a function that can be plotted in ...
1
vote
0answers
65 views

What's is the name of this function?

A function, $f:\mathbb{N}\to\mathbb{N}$, is defined in the following way, \begin{equation} f(n)=\#\{m\mid m\leq n\text{ and there does not exists any integer }m'>m\text{ such that }m\text{ divides ...
0
votes
0answers
26 views

How to learn Digamma Function and how to take derivative of Gamma function?

How can I learn polygamma function?(More precisely digamma function)As I was learning Bessel Function of Second kind expressed in terms of power series digamma function is used.I have firm-grasp in ...
0
votes
0answers
11 views

Graph of Bessel Function 2nd kind

When I was graphing Bessel function of 1st kind using my 'Microsoft Mathematics' using the series and adding terms up to 0 to 20. I was getting very good approximation of the function x ranges from 0 ...
0
votes
0answers
12 views

Solution of Bessel function of 2nd kind positive integer order

How to derive series expansion of Bessel function of 2nd kind positive integer order? I am not asking about integral expression or any other crazy things like this.I want power series expansion ...
2
votes
3answers
41 views

Function to express a time interval with results between 0 and 1

Its my first time on here and my maths is poor so please be kind. I am working on a Masters dissertation focused on document clustering methods in which I would like to apply a weight based on the ...