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

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0
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1answer
25 views

Polygamma reflection formula

How does one prove the polygamma reflection formula: $$\psi^{(n)}(1-z)+(-1)^{n+1}\psi^{(n)}(z)=(-1)^n \pi \frac{d^n}{d z^n} \cot \pi z $$ Do we have to invoke the power of contour integration and ...
1
vote
1answer
34 views

How to remodel sigmoid function so as to move stretch/enlarge it?

I have a question similar to this. I want the sigmoid to have asymptotes to $+1$ and $0$ in specific points $\frac{1}{A}$ and $-\frac{1}{A}$, as in the Figure (where $\frac{1}{A}=2$ and ...
-2
votes
0answers
67 views

Do we have this type of integral expression of Bessel function of the first kind?

Let $z=\lambda+i\mu$ with $\mu>0$. Then for any $r>0$, $k=1,2,3, \cdots$. Do we have the following identity $$ \int_{r}^{\infty}{\frac{t}{\sqrt{t^2-r^2}}(\frac{1}{t}\frac{d}{dt})^k ...
0
votes
0answers
22 views

Proving a property of Fractional Linear Transformations

I'm having some trouble showing that FLTs send circles and lines to circles or lines. I know that they are compositions of linear maps and inversions. Showing that the linear maps send circles to ...
2
votes
2answers
484 views

Numerical approximation of the modified Bessel function $I_0$ with radical argument for integration purposes

I have to numerically calculate the following definite integral $$\int_{\alpha}^{\beta}I_0(a\sqrt{1-x^2})dx$$ for different values of $\alpha$ and $\beta$, where $a$ has a value of, say, $30$. I'm ...
1
vote
0answers
21 views

What is the status on questions related to Bhargava's factorial function? [migrated]

In Manjul Bhargava's The Factorial Function and Generalizations he motivates a new type of factorial $n!_S$ using by generalizing a few theorems like: For $k, l \in \mathbb{Z}$, we have $k! \times ...
0
votes
1answer
89 views

A limit related to an alternating series [closed]

Show that the limit $$ \lim_{N\rightarrow\infty}\bigg(\sum_{k=1}^{2N-1}(-1)^{k+1}\frac{\sin\big(\frac{\pi}{2N}\big)}{\sin\big(\frac{k\pi}{2N}\big)}\bigg) = 2\ln2 $$ holds. Hint: I think the identity ...
2
votes
1answer
158 views

Definite integral of polynomial times exponential times hypergeometric function of imaginary argument

How would one deal with such an integral? $$I(k)\equiv \int_0^\infty r^n e^{-r(1+\mu)} e^{-{\mathrm i} kr}\:{}_1F_1({\mathrm i}/k+1;2;2{\mathrm i} kr) \, \mathrm{d} r$$ Here $n\in\{0,1\}$, $\mu\in ...
26
votes
1answer
527 views

$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$

I need help with calculating this integral: $$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx,$$ where ...
2
votes
2answers
82 views

Challenging integral

I am trying to find a close form representation for the following integral: $$ A(x;a,b,c)= \int_{0}^{x}\frac{\sin\left(a k+b k^{2}\right)+\sin\left(c k-b k^{2}\right)}{k}dk $$ for $0<x \ll ...
0
votes
0answers
22 views

What can we say about variational energies?

Suppose that $U \subset \mathbb{R}^d$ is open and let $V_{ij}^{kl}(r)$ $(1 \leq i,j,k,l \leq d)$ be functions on $V$ to $\mathbb{R}$ which are as smooth as the coming problem may require. For the ...
1
vote
2answers
79 views

Closed form of the integral $\int_0^1 \frac{x^n}{1+x}\, dx$

I am trying to evaluate the integral $$\int_0^1 \frac{x^n}{1+x}\, dx, \;\;\; n \in \mathbb{N}$$ in a closed form. I tried tackling it using Beta Form $\displaystyle \int_0^1 ...
0
votes
0answers
24 views

Taylor series of $(1-x)^b$ $_2F_1(a,b;c;x)$: when to stop?

Let $f(x)= (1-x)^b$ $_2F_1(a,b;c;x)$, where $0<x<1$ and $a=(K-1)d$, $b=K$, c=$Kd$ (with $a$, $b$ and $c$ are positive and $K>d$ ). I need to derive the Taylor series of the corresponding ...
1
vote
0answers
88 views

Compute $\int_{1}^{\infty} \frac{J^2_{n}(k)}{k^m} dk$

Question as the title showed,in which J means Bessel functions, n and m are positive integers. How to get the analytic result? Any comment is much appreciated. Many thanks in advance.How to simplify ...
0
votes
0answers
14 views

Can the regularized beta function be calculated for integer values of $a$ using identities?

I've been trying to create a cumulative Student's T Distribution calculator using javascript as fun side project. I've successfully created a gamma function approximator, and from there, a beta ...
0
votes
0answers
19 views

Proof of identity with Hermite polynomials

Let's have Hermite polynomial, $H_{n} = e^{\frac{x^{2}}{2}}\frac{d^{n}}{dx^{n}}e^{-\frac{x^{2}}{2}}$. How to prove the identity $$ \tag 1 \sum_{n = 0}^{\infty}H_{n}(x)H_{n}(y)\frac{t^{n}}{n!} = (1 - ...
5
votes
0answers
255 views

What special role plays the function $\pi^{\frac x\pi}$ in analysis?

I have tried to redefine some special functions in the most "natural" way, that is the way which allows to simplify the relations the most. I would call these functions "parelementary". The ...
1
vote
2answers
49 views

Integrals with the special functions $Ci(x)$ and $erf(x)$

I'm looking for the solutions of the following two integrals: $$I_1=\int\limits_0^\infty dx\, e^{-x^2}Ci(ax)$$ and $$I_2=\int\limits_0^\infty dx\, e^{-ax}erf(x)$$ with ...
0
votes
0answers
30 views

Solution to a particular Wave Equation

Consider the partial differential equation \begin{align} \frac{1}{c^{2}} \, \frac{ \partial^{2} U}{\partial t^{2}} &= \frac{\partial^{2} U}{\partial x^{2}} + x \, \frac{\partial U}{\partial x} + ...
1
vote
1answer
40 views

Asymptotic behavior of the zeros of the digamma function

The gamma function has just one extremum on each interval $(k,k+1)$, where $k$ is a negative integer. These extrema occur at the zeros of the derivative of the gamma function. Let $z_n$ denote the ...
3
votes
1answer
52 views

Hypergeometric function with negative $b$ and $a>c>0$

Recall the definition of the hypergeometric function $$_2F_1(a,b,c;z)=\sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{n!(c )_n}x^n$$ where $(a)_n$ is defined to be $a(a+1)\cdots(a+n-1)$. We suppose that none ...
1
vote
1answer
52 views

Can this series be expressed as a Hyper Geometric function

I am trying find a Hyper Geometric function representation of the following series. $$\sum\limits_{k=0}^{\infty} \frac{a^k}{k!}\frac{\Gamma \left(\frac{b+k}{c}\right)}{\Gamma ...
0
votes
1answer
35 views

How to explain hypergeometric $2F_1[1+m,n,2+m,-2]$?

Question as title showed. What expression it represents? Many thanks.
3
votes
1answer
95 views

Simplification of Kampé de Fériet function

I was dealing with a convolution type integral $$ \int^z_0 t^m {}_0F_1(;1;-t) \: {}_2F_3\Big( 1,1;2,m,m+1 ; -a t\Big) \:\mathrm{d}t $$ By applying one of the identities in Exton's book, the solution ...
0
votes
0answers
11 views

Separable sigmoid-like function

Is there any squashing function that is multiplicatively separable? I'm looking for a sigmoid-like function f(x) that allows me to calculate f(ax)+f(ay) given "a" and f(x)+f(y).
0
votes
1answer
19 views

Plot my own function 'in two variables'

I have matlab function in two variables say $function_{something}(t)$ where $t$ is of size $2 \times k$, this is to allow to evaluate $k$ times in two values. So my output is then a $k \times 1$ ...
0
votes
0answers
10 views

Finding an analytic form of a function that satisfies asymptotic conditions

I have a family of functions that I obtain numerically. They depend on $x$ and parametrically also upon a certain parameter $L$. I would like to find an analytical form for this family of functions so ...
0
votes
3answers
90 views

Various evaluations of the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^3}$

I recently ran into this series: $$\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^3}$$ Of course this is just a special case of the Beta Dirichlet Function , for $s=3$. I had given the following ...
1
vote
0answers
45 views

Relation between Nuttal Q-function and Gaussian Q-function

I am trying to express the famous Nuttal Q-function, given as: $$\mathcal Q_{m,n}(p,q)=\int_q^\infty t^me^{-0,5\left[p^2+t^2\right]}I_n(pt)\;dt$$ where $m$, $n$, $p$, and $q$ are constants and ...
4
votes
0answers
77 views

$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$

I have recently met with this integral: $$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$$ I want to evaluate it in a closed form, if possible. 1st functional equation: $\displaystyle f(a)=\ln^2 2-f\left ( ...
1
vote
1answer
366 views

Solution of Bessel equation

Prove that for a Bessel function in its normal form that is: $$u'' + \left(1 + \frac{1-(4*p^2)}{4x^2}\right)u=0$$ if $p > \frac12$ then every interval of length $\pi$ contains at most one zero of ...
2
votes
0answers
29 views

q-Hermite polynomials

It is well known that the q-Hermite polynomials defined by $$H_n(\theta; q)= \sum\limits_{k=0}^n \frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}e^{i(n-2k)\theta}$$ are orthogonal in $\theta \in [0, \pi]$ with ...
0
votes
1answer
11 views

generate large range function from smaller

i lies between 0 to 31, with equal probability (1/32). k is either 0 or 1.(equal probability 1/2) How can one generate i using only k?
9
votes
3answers
621 views

Can you recommend some books on elliptic function?

I plan to study elliptic function. Can you recommend some books? What is the relationship between elliptic function and elliptic curve?Many thanks in advance!
9
votes
1answer
204 views

Riemann zeta function and Bernoulli function

I encountered the following problem: Show that $$\zeta(2n+1)=\frac{(-1)^{n+1}(2\pi)^{2n+1}}{2(2n+1)!}\int_0^{1}B_{2n+1}(x)\cot({\pi}x)dx$$ where $B_{2n+1}(x)$ is the Bernoulli polynomial. This ...
1
vote
0answers
41 views

Solving Integral $\int_{0}^{x}K_{0}\left(\frac{2t^{\beta /2}}{\sigma ^{\beta }}\right)\;dt$

This question is a continuation of the question posted here. The problem here is to solve the integral with modified Bessel function of second kind, $K_0\left(u\right)$: $$F\left ( x \right ...
4
votes
3answers
544 views

Integrating Modified Bessel function of the second kind?

I need to compute the following integral: $$\int_0^\infty\;\;K_0\left(\sqrt{a(k^2+b)}\right)dk$$ where $a>0$ and $b>0$. I have tried several substitutions and played around a lot in ...
3
votes
1answer
55 views

Identities for hypergeometric functions ${}_2F_1$ with z=1/2

Is there a closed form (or approximation) for a hypergeometric function of form: $_2F_1(1,b+c;c;\frac{1}{2}) \quad \text{where} \; b,c \in \mathbb{N}$ ? I researched all identities in ...
2
votes
1answer
33 views

Proving that $\lim \limits_{b \rightarrow \infty} F(a,b,\frac{1}{2};\frac{z^2}{4ab})=\cosh z$

I am trying to prove that $\lim \limits_{a,b \rightarrow \infty} F(a,b,\frac{1}{2};\frac{z^2}{4ab})=\cosh z$ . Here $F$ is the hypergeometric function. Here because of two limits I am unable to ...
9
votes
2answers
218 views

How find that $\left(\frac{x}{1-x^2}+\frac{3x^3}{1-x^6}+\frac{5x^5}{1-x^{10}}+\frac{7x^7}{1-x^{14}}+\cdots\right)^2=\sum_{i=0}^{\infty}a_{i}x^i$

let $$\left(\dfrac{x}{1-x^2}+\dfrac{3x^3}{1-x^6}+\dfrac{5x^5}{1-x^{10}}+\dfrac{7x^7}{1-x^{14}}+\cdots\right)^2=\sum_{i=0}^{\infty}a_{i}x^i$$ How find the $a_{2^n}=?$ my idea:let ...
3
votes
3answers
101 views

Residue of $\Gamma^{2}$ and $\Gamma^{3}$

Based on wiki, the residues of $\Gamma$ at non positive integers are given by: $$\text{Res}\left ( \Gamma(z),z=-n \right )=\frac{(-1)^{n}}{n!}.$$ I have been trying to find residue for $\Gamma^{2}$ ...
0
votes
0answers
17 views

Polylogarithm and unclear statement

I am trying to solve this question which may not have an answer at all, but any clarification would be much appreciated. I also tried to explain what I have tried/thought about it below. Let ...
3
votes
0answers
47 views

About a sequence related with the complete elliptic integral of the second kind

When answering this related question I proved that if we define $B(\lambda)$ as: $$\begin{eqnarray*} ...
0
votes
0answers
37 views

Proving That Two Paths of Different Lengths Are Adjoined

In the section on 'Adjoining Paths' of its 'Topology' book's page on 'Path Connectedness,' WikiBooks shows that, for any topological space $X$ with members $a$, $b$, and $c$, the following…: ...
2
votes
2answers
45 views

Dirichlet $L$ functions at $s=2$

Let $\chi$ be a Dirichlet character and let $L(\chi,s)$ denotes its Dirichlet $L$-function. What is the value of $L(2,\chi)$ ? Or simply, is $L(2,\chi)/\pi^2$ rational ? Many thanks for your answer ...
0
votes
1answer
32 views

Proving the transform of the Q-function

I have the Gaussian Q-function, given by: and I want to prove that it can be also expressed as: Can somebody help explaining how to obtain the second integral from the first?
10
votes
2answers
426 views

Prove that $2^{2z-1}\Gamma(z)\,\Gamma(z+\frac{1}{2})=\sqrt{\pi}\,\Gamma(2z)$ using Gauss's identity.

I'm trying to derive the functional equation $2^{2z-1}\Gamma(z)\,\Gamma(z+\frac{1}{2})=\sqrt{\pi}\,\Gamma(2z)$ using Gauss's formula: ...
7
votes
6answers
5k views

Algorithm to compute Gamma function

The question is simple. I would like to implement the Gamma function in my calculator written in C; however, I have not been able to find an easy way to programmatically compute an approximation to ...
0
votes
0answers
29 views

Definite integral with Bessel functions

Show that for $Re(\lambda)>0,Re(\mu)>0$ it holds the following identity $\int_0^a x J_\lambda(2a)I_\lambda(2x) J_\mu(2 \sqrt{a^2-x^2}) I_\mu(2 \sqrt{a^2-x^2}) dx = \frac{a^{2 \lambda + 2 \mu + ...
0
votes
1answer
40 views

Does this special function exist?

Is it possible to find a non-trivial function $f(x_1,x_2)$ that has two parameters $x_1$ and $x_2$. This function should satisfy $f(x_1,x_2) = f(\frac{x_1}{1+r},x_2 +r)$, for any non-negative $r$. ...