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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0answers
6 views

Creating an evenly distributed function $B=B(p)$ over the range $p=p_{min}$ to $p=p_{max}$

In some notes on statistical thermodynamics, I encountered this: The momentum distribution function $B(p)$ is evenly distributed over the allowed range: ...
0
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0answers
34 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
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1answer
19 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 ?
2
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0answers
23 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
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0answers
18 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
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1answer
42 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
103 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!} \, ...
7
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4answers
139 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.
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0answers
20 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 ...
0
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1answer
44 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 ...
7
votes
1answer
100 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
164 views
+50

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 ...
10
votes
2answers
120 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
45 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
41 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
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2answers
30 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
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0answers
26 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 ...
0
votes
1answer
21 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
49 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
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0answers
22 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
64 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
47 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
27 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
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0answers
64 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
25 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
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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
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0answers
11 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
38 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 ...
2
votes
1answer
58 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 ...
12
votes
1answer
222 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 ...
1
vote
1answer
26 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 : $$f(x)=\begin{cases} x, x \in Q \\ 0, x\in I \end {cases}$$ ??
1
vote
0answers
128 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} ...
0
votes
0answers
19 views

Orthogonality of Spherical Bessel Function

How to prove Orthogonality of Spherical Bessel Function?
14
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0answers
154 views

Does $\int_{-1}^1\frac{\arctan x}{\text{arctanh}\,x}\,dx$ have a closed form?

Mathematica gives an approximate result of $1.581949621806183890451628...$, but no exact form. I predict it's a function of $e$ and $\pi$, and perhaps even the Golden Ratio $\phi$ (It certainly ...
3
votes
0answers
39 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
votes
0answers
18 views

I want to find solution of Mathieu Equation

Can you please help me to find the coefficients associated with the solution of Mathieu Equation in the form of AMathieuC[a,q,z]+BMathieuS[a,q,z]? How they can be find? Please help if u can Thanks.
0
votes
0answers
35 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 ...
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votes
0answers
38 views

I want to solve Mathieu Equation $y''(x)+(a−2q \cos(2x))y(x)=0$. How to solve it using Floquet solution?

I want to solve Mathieu Equation $$y''(x)+(a−2q \cos(2x))y(x)=0.$$ How to solve it using Floquet solution? In Floquet solution for integer order of $v$ and $π$ periodicity We have Solution ...
20
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2answers
447 views
16
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0answers
175 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 ...
4
votes
0answers
32 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 ...
2
votes
0answers
40 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 ...
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votes
0answers
17 views

Finding zeros of a function involving Gamma function.

I am looking for the zeros of following function ($a$ and $b$ are real): $$ F(a,b) = 4^{a+ib} \Gamma(a+ib) \Gamma(-a) \Gamma(-ib) + \Gamma(-a-ib) \Gamma(ib)\Gamma(a) $$ and I have no idea on ...
12
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1answer
149 views

Evaluate the double sum $\sum_{m=1}^{\infty}\sum_{n=1}^{m-1}\frac{ 1}{m n\left(m^2-n^2\right)^2}$

As a follow up of this nice question I am interested in $$ S_1=\sum_{m=1}^{\infty}\sum_{n=1}^{m-1}\frac{ 1}{m n\left(m^2-n^2\right)^2} $$ Furthermore, I would be also very grateful for a solution ...
4
votes
0answers
104 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 ...
4
votes
0answers
72 views

integrate $\int \frac{1}{e^{x}+e^{ax}+e^{a^{2}x}} \, dx$

I've been trying to integrate $$ \int \frac{1}{e^{x}+e^{\omega x}+e^{\omega^{2}x}} \, dx $$ where $\omega=e^{2i\pi/3}$ but to no avail. I've tried substituting in $u=e^{(1+\omega)x}$ but ended up ...
7
votes
6answers
200 views

Solve the following equation: $\sqrt {x + \sqrt {4x + \sqrt {16x + \sqrt {64x + 5}}}} - \sqrt x= 1$

A past examination paper had the following question that I found interesting. I tried having a go at it but haven't come around with any solutions. How would one go about tackling it? $$\sqrt {x + ...
2
votes
2answers
37 views

Finding all values of $\theta$ which describes a straight line

I am having quite a bit of trouble understanding the below question; my assumption is that I should bring the right-hand side in terms of $\sin \theta$ or $\cos \theta$ however am not able to proceed ...
2
votes
0answers
44 views

Conjecture of the general form of a power series

Relcently I met a power series(Source Link-Eq(4.1)) of the type $$ f(x)=1-x+\frac{1}{2}x^2+\frac{1}{4}x^3-\frac{1}{8}x^4-\frac{35}{128}x^5-\frac{157}{1024}x^6+\cdots $$ where $x$ is supposed to be a ...