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

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2
votes
1answer
39 views

Asymptotic series of Confluent Hypergeometric function $U(a,1,z) $ as $z \to 0$

Consider the Confluent hypergeometric function $U(a,b,z)$, which is a solution of the Kummer's Equation : $$zw''+(b-z)w'-aw=0$$ it has the following integral representation when $- \pi/2 < \arg ...
0
votes
1answer
46 views

Evaluation of Spence's function.

Spence's function is defined as $${\rm Li}_2 (z)=- \int_0^z \frac{\ln(1-u)}{u} \, du $$ where $$z \in {\mathbb C} \setminus [1, \infty )$$ For $|z|<1 $ $${\rm Li}_2 (z)= \sum_1^ \infty \frac{ ...
3
votes
0answers
64 views

About the closed form for $\lim_{y\to +\infty}\left(-\frac{2}{\pi}\log(1+y)+\int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx\right)$

Recently, when facing a baby Rudin's exercise, I proved that: $$ \int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx = \frac{2}{\pi}\log(1+y)+O(1) $$ holds by integration by parts. Now I wonder if ...
4
votes
3answers
99 views

Proving positivity of the exponential function

Question. Without using the semigroup property ($\mathrm{e}^{x}\mathrm{e}^{y}=\mathrm{e}^{x+y}$), how can we show that $\mathrm{e}^{x}>0$ for all $x\in\mathbb{R}$ only by using the series ...
0
votes
2answers
27 views

How can I show that $\left(a-n-1\right)!/\left(a-1\right)!=\left(-a\right)!\left(-1\right)^n/\left(-a+n\right)!$?

Is it possible to show that \begin{align}\frac{\left(a-n-1\right)!}{\left(a-1\right)!}\stackrel{?}{=}\frac{\left(-a\right)!\left(-1\right)^n}{\left(-a+n\right)!}\tag{1},\end{align} or, more ...
1
vote
1answer
14 views

Set of points at which a function coincides with its convexification is compact?

Let $f:[0,1]\rightarrow\ \mathbb{\bar{R}}$, and let $\tilde{f}$ be the convexification of $f.$ (i.e., $\tilde{f}$ is the pointwise supremum of all affine functions that lie everywhere below $f$.) Let ...
5
votes
1answer
74 views

Evaluation of the series $ \sum_{k=0}^{\infty}\frac{k}{2^{k+1}\left ( k+1 \right )^2}$.

The following series: $$\sum_{k=0}^{\infty}\frac{k}{2^{k+1}\left ( k+1 \right )^2}$$ came up as an intermediate step of calculating an integral. The answer according to Wolfram is $\displaystyle ...
0
votes
1answer
12 views

A low-discrepancy or quasirandom series which would guarantee all value sequences

I am trying to find a type of quasi-random sequence which would guarantee that it could produce all possible sequences of values within the possible value range, while still producing random-seeming ...
4
votes
2answers
112 views

How to find a closed form for the derivatives of $F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,dt,$ $F(0)=\frac12$?

I have been given the function $$F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,{\rm d}t$$ for $x\ne 0,$ $F(0)=\frac12,$ and charged with finding a Taylor polynomial for $F(x)$ differing from $F$ by no ...
2
votes
1answer
106 views

Limit involving incomplete gamma function

Let $\Gamma(a,x) = \int_x^\infty t^{a-1} e^{-t} dt$ be the incomplete gamma function. What is the limit $$ \lim_{p \to \infty} \frac{1}{\sqrt{p}} \Big[\Gamma(p+\tfrac12,x)\Big]^{\frac{1}{2p}} $$ as a ...
2
votes
0answers
24 views

I would like to find a generalization of the plane wave expansion to Hankel functions.

The plane wave expansion is \begin{equation} e^{i\vec{k}\cdot \vec{x}}=\sum_{\ell=0}^{\infty}i^\ell(2\ell+1)j_{\ell}(kx)P_{\ell}(\cos(\theta)) \end{equation} where $j_\ell$ is the spherical Bessel ...
1
vote
1answer
46 views

Product with $\Gamma$ function

Evaluate the product: $$\Gamma \left ( \frac{1}{n} \right )\Gamma \left ( \frac{2}{n} \right )\cdots \Gamma \left ( \frac{n-1}{n} \right )$$ A path to a solution: $$\begin{aligned}\Gamma\left ...
4
votes
2answers
72 views

How are Fox-H functions useful in math?

The Fox H-function, as far as I know, is the most general families of functions - encompassing an even larger family of functions than the already very general Meijer G-function. While I've known ...
0
votes
0answers
20 views

Upper bound for the ratio of Bessel functions

I am looking for an upper bound for the ratio of Bessel I functions $\dfrac{|I_\nu'(z)|}{|I_\nu(z)|}$ where $\nu$ is complex and $z$ is a positive real number. Do you know any results about it? Thank ...
1
vote
1answer
30 views

How do I know if a fractional linear transformation exists?

I have a feeling I'm missing another obvious point about FLTs. How do I know if a specific fractional linear transformation exists? I think I can find specific transformations by using the ...
1
vote
0answers
35 views

Question about the function ξ(s)

We define When Π(s) is the gamma function and ζ(s) is the zeta function. I can prove that ξ(s) has order one. Let be ρ the zeros of ξ(s) (the non- trivial zeros of ζ(s)). Why ∑1/|ρ| diverges? And ...
107
votes
0answers
4k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
1
vote
1answer
31 views

Quasiconvex functions

Let $\lambda$ be any real number and $f:[0,1]\times[0,1]\rightarrow\mathbb{R}$ given by $$ f(x,y) = \begin{cases} \lambda &\mbox{if } \quad0<x<1, y=1, \\ 1+\lambda y & \mbox{if } ...
21
votes
3answers
450 views

An integral $\int_0^\infty P_s(x-1)\,e^{-x}\,dx$ involving Legendre functions

Let $P_s(x)$ denote the Legendre functions of the $1^{st}$ kind, i.e. the Legendre polynomial generalized to an arbitrary (not necessarily integer) order $s$. It can be expressed using the ...
0
votes
1answer
26 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
39 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
2answers
497 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 ...
0
votes
1answer
94 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
169 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
532 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
89 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
80 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
91 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 ...
5
votes
0answers
259 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
44 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
54 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
12 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
21 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
12 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
93 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
46 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 ...
5
votes
0answers
87 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
370 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
36 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
639 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 ...