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
0answers
71 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 ...
8
votes
4answers
152 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
27 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
52 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 ...
6
votes
3answers
435 views

Evaluating $ \int_{-\infty}^{\infty}x\exp\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}^{2}\left(a\left(x-d\right)\right)\,\mathrm{d}x $

I have big difficulties solving the following integral: $$ \int_{-\infty}^{\infty}x\exp\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}^{2}\left(a\left(x-d\right)\right)\,\mathrm{d}x $$ I tried to ...
12
votes
2answers
148 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) &= ...
8
votes
1answer
116 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 ...
2
votes
1answer
65 views

Is it possible to express $\Gamma\!\left(\tfrac{1}{50}\right)$ through values of the $\Gamma$-function at rational points with smaller denominators?

Sometimes it is possible to express a value of the $\Gamma$-function at a rational point through values of the $\Gamma$-function at rational points with smaller denominators, e.g. ...
1
vote
2answers
65 views

a limit of the digamma function

Numerical calculations show that: $$ \lim_{x\to 0^+}\left(\frac{1}{x}-\psi(x)+\psi(x/2)\right) = 0.$$ Can anybody help to find a formal proof of the above limit?
7
votes
1answer
918 views

Median of the F-distribution

Is the median of the F-distribution with m and n degrees of freedom decreasing in n, for any m? From experiments it looks like it might be, but I have been unable to prove it.
28
votes
2answers
681 views

On Shanks' quartic approximation $\pi \approx \frac{6}{\sqrt{3502}}\ln(2u)$

In Mathworld's "Pi Approximations", (line 58), Weisstein mentions one by the mathematician Daniel Shanks that differs by a mere $10^{-82}$, $$\pi \approx ...
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
38 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 ...
5
votes
2answers
531 views

On the absolute integrability of Bessel functions

Reading "How do you integrate a Bessel function", it didn't seem like it was an easy task. Thinking more about Bessel functions, speficially $J_0(x)$, it occurred that it looked a lot like the sinc ...
3
votes
1answer
70 views

Equality with dilogarithms

During some calculations with definite integrals I happened to get the equality \begin{eqnarray} 2\, \textrm{Li}_2(-\frac{1}{2}) - 2 \, \textrm{Li}_2(\frac{1}{4})+ 2\, \textrm{Li}_2(\frac{2}{3})= 3 ...
1
vote
1answer
52 views

Derive recurrence relations for Bessel functions from the generating function

I'm going through practice questions for my exams but this question has left me confused: The Bessel functions of integer order, $J_n(x)$, are described by the generating function: Derive the ...
0
votes
0answers
37 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 ...
12
votes
1answer
246 views

The log integrals $\int_{0}^{1/2} \frac{\log(1+2x) \log(x)}{1+x} \, dx $ and $ \int_{0}^{1/2} \frac{\log(1+2x) \log(1-x)}{1+x} \, dx$

Using M.N.C.E.'s suggestion in the comments, both integrals can be expressed in terms of integrals that can be evaluated by integrating by parts. In attempting to evaluate $ \displaystyle ...
5
votes
3answers
381 views

Proof that $Γ'(1) = -γ$?

I know that $Γ'(1) = -γ$, but how does one prove this? Starting from the basics, we have that: $$Γ(x) = \int_0^\infty e^{-t} t^{x-1} dt$$ How do we differentiate this? How do we then find that ...
-1
votes
1answer
37 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 ...
30
votes
1answer
616 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
0answers
56 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 ...
13
votes
1answer
319 views

Show the equivalence of two infinite series over Bessel functions

The following sums pop up in diffraction theory and are related to Lommel's function of two variables. Let $u,v\in\mathbb{R}$. I claim that $$\sum_{n=0}^\infty i^n \left ( \frac{u}{v} \right )^n ...
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
108 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. ...
1
vote
0answers
130 views

How to compute the definite integrals of special functions?

How can these integrals be solved: $${1\over \pi} \int_{0}^{\infty}\left({{\sqrt{x}(a-bx)}\over {x^{3}+(a-bx)^{2}}}\right)\cos(\sqrt{\alpha x}) \exp(-xt)\,\mathrm{d} x, $$ and $${1\over \pi} ...
4
votes
1answer
54 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 ...
3
votes
0answers
393 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 ...
2
votes
3answers
43 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 ...
16
votes
0answers
202 views

Definite integral of arcsine over square-root of quadratic

For $a,b\in\mathbb{R}\land0<a\le1\land0\le b$, define $\mathcal{I}{\left(a,b\right)}$ by the integral ...
16
votes
2answers
620 views

Closed form for $\int_{-\infty}^0\operatorname{Ei}^3x\,dx$

Let $\operatorname{Ei}x$ denote the exponential integral: $$\operatorname{Ei}x=-\int_{-x}^\infty\frac{e^{-t}}tdt.\tag1$$ It's not difficult to find that ...
1
vote
0answers
29 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 ...
14
votes
2answers
295 views

Integral ${\large\int}_0^1\left(-\frac{\operatorname{li} x}x\right)^adx$

Let $\operatorname{li} x$ denote the logarithmic integral $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Consider the following parameterized integral: $$I(a)=\int_0^1\left(-\frac{\operatorname{li} ...
1
vote
0answers
66 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
28 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
12 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
13 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 ...
109
votes
1answer
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
27 views

Can anyone come up with an example of a monotonously not-falling function whose breakpoints are everywhere dense set on $[0,1]$?

$$f: \displaystyle{R\to[0,1]}$$ I can't come up with one, can anyone else ? :D How about : $$f(x)=\begin{cases} x, x \in Q \\ 0, x\in I \end {cases}$$ ??
0
votes
0answers
32 views

Orthogonality of Spherical Bessel Function

How to prove Orthogonality of Spherical Bessel Function?
3
votes
0answers
51 views

Double integral of symmetric polylogarithmic function over rectangular region

This question was inspired by M.N.C.E.'s wonderful response here. While exploring the possibility of generalizing his result, I found that a significant part of the problem reduced to evaluating the ...
1
vote
0answers
366 views

Approximations of the incomplete elliptic integral of the second kind

For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I know that this is a ...
0
votes
0answers
50 views

Some expectations of psi (digamma) function

I want to derive an Expectation-Maximization algorithm for my model. But some expectations of psi (digamma) function is needed in the procedure. Assuming I have a Gamma distributed random Variable ...
4
votes
0answers
35 views

Identification of a function

I recently came across the following function $$\sum_{k=1}^\infty(\log(k))^n\frac{z^k}{k}$$ I found it while dealing with the polylogarithm function, $Li_n (z)$ (Notice that if instead of ...
9
votes
2answers
1k views

Proving and deriving a Gamma function

I'm having a hard time trying to prove this Gamma function and trying to derive the duplication formula: a.) Prove that $$\frac{\Gamma (p)\Gamma (p)}{\Gamma (2p)} = ...
3
votes
0answers
60 views

Integral involving Whittaker function

Consider the following integral: $$ \int_1^{\infty} \frac{e^{u/2}}{u}[-\mathrm{Ei}(-u)]\,W_{1,\imath p}(u)\,du, $$ where $\imath=\sqrt{-1}$ and $p>0$ selected so that $W_{1,\imath p}(1)=0$; here ...
7
votes
1answer
115 views

Can derivative of Hurwitz Zeta be expressed in Hurwitz Zeta?

Can the derivative of Hurwitz Zeta function by the first argument be expressed in terms of Hurwitz Zeta and elementary fuctions? There is a formula which expresses Hurwitz Zeta through its ...
4
votes
0answers
110 views

Solving integral with spherical bessel functions

I would like to find if possible a solution (closed form) for the following integral: $$\frac{1}{2 \pi}\cdot\int\limits_0^{2\pi}\exp\bigg[-ia(\cos x+\sin x)\bigg]\,j_{0}(b\cos x)\,j_{0}(b\sin ...