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
54 views

Why it is necessary for copula functions to be grounded?

I know what the properties "grounded" and "2-increasing" means in copula functions definition but actually I can't understand the reason behind these two! I mean why it is necessary for copulas to be ...
2
votes
1answer
30 views

Is this function defined in terms of elliptic $\mathrm{K}$ integrals even?

Let $R,z > 0$ be positive real constants, and consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $$ f(v) = \frac{1}{\sqrt{(R+v)^2+z^2}}\ \mathrm{K}\!\left( \frac{4 R v}{(R+v)^2+z^2} ...
1
vote
0answers
37 views

Proof for the Rodrigues formula for Neumann's Spherical functions.

I've been trying to prove the Rodrigues formula for Neumann's Spherical functions. The Neumann's Spherical functions are: $$N_n(x))=-(-x)^n\left[\frac{1}{x}\frac{d}{dx}\right]^n\frac{\cos{(x)}}{x}$$ ...
5
votes
2answers
254 views

Solution of a Lambert W function

The question was : (find x) $6x=e^{2x}$ I knew Lambert W function and hence: $\Rightarrow 1=\dfrac{6x}{e^{2x}}$ $\Rightarrow \dfrac{1}{6}=xe^{-2x}$ $\Rightarrow \dfrac{-1}{3}=-2xe^{-2x}$ ...
0
votes
0answers
13 views

$K$ which is of second category in itself.let $H = K \cap ( - K)$. Why $H$ is non empty interior

Let $X$ be topological vector space.Let $K$ be closed, convex, dense subset of $X$ and $K$ which is of second category in itself. Put $H = K \cap ( - K)$. Why does $H$ is nonempty interior?
1
vote
2answers
73 views

A special modular function: $ j $-invariant.

It is known that j invariant $$j(\tau)= 1728 \frac{g_2^3(\tau)}{\Delta(\tau)} $$ $\tau \in \mathbb{H}$ attains every complex value , Can someone guide me its proof.?? where $L(\tau ) = \{\tau m ...
0
votes
1answer
61 views

Heaviside function & Integral Limits

When considering integration, how does one use the Heaviside function in order to alter the limits of integration. For example If i have $$ \int_a^b f(x) dx $$ But want to change this integral to be ...
2
votes
0answers
34 views

Barnes' double gamma function versus q-gamma function

According to wikipedia, the q-analog of the gamma function is closely related to a multiple gamma function defined by Barnes. Besides the fact that they are both generalizations of the Gamma function, ...
0
votes
1answer
26 views

Difference Between Lyapunov and Strong Lyapunov Function.

Good Day everyone. I was assigned to show that given an autonomous system of Differential Equations and a function $V$, I need to show that $V$ is Lyapunov function. To show that $V$ is Lyapunov. I ...
2
votes
1answer
77 views

Name of function $(1+x)^n-1$

Is there any name for this formula $$(1+x)^n-1$$ When working with floating point numbers this can be calculated with much better precision for very small $|x|<1$ values using Taylor series ...
1
vote
1answer
83 views

How to prove this problem about supermodularity function?

The problem is as follows, and I have solved the subproblem (a), but haven't solved (b) yet. And for (b) the method I think about is proof by contradiction, but I get stuck before I could solve this. ...
2
votes
2answers
72 views

Derivative of the Gamma function

How do you prove that $$ \Gamma'(1)=-\gamma, $$ where $\gamma$ is the Euler-Mascheroni constant?
1
vote
2answers
37 views

Integrals involving exponential functions and the gamma function [duplicate]

I'm having trouble evaluating this integral $$\int_0^\infty {e^{-ax^2}} \,dx $$ My guess is that it would evaluate into something like $$\int_0^\infty \frac 12e^{-s}s^{\frac 12} \ldots \,dx = \frac ...
0
votes
1answer
58 views

Best aproximation to an numerical solution using two aproximated functions

I want to find the best aproximation to a numerical solution. For that I want to use two aproximated functions (that I already know). If I plot them I see that one of them underestimates the original ...
1
vote
1answer
40 views

Formula for the Beta function for natural m, n

Using only the definition $$B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1}dt$$ for the Beta function $B(x, y)$, it's symmetry $B(x,y) = B(y,x)$ aswell as the fact that $(x + y)B(x + 1, y) = xB(x, y) ...
2
votes
2answers
56 views

Evaluate $-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$ in $\gamma$.

Evaluate $\gamma$ expressed, involving Lambert function, by $$-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$$ where $\gamma<1$. I doubt that it is possible to find a value for ...
0
votes
1answer
20 views

Logical comparison of two values with algebra

Suppose I have two real numbers A and B (A $\wedge$ B $\subset$ $\mathbb{R}$). I want to do some algebra over these number and get 1 if they are equal and get 0 if not. For example: In this ...
1
vote
1answer
63 views

Primitive of the function $(\sin x)/x$

I know that for some functions, for instance $f(x) = e^{-x^2}$, there does not exist a primitive. Does there is a primitive for the function $f(x) = \frac{\operatorname{sin}(x)}{x}$?
0
votes
1answer
22 views

Logarithmic algorithm performance

If I have an algorithm that on $T$ iterations gets me within $O(\log(T)/T)$ accuracy, what is a (preferably concise, closed form) lower bound on $T$ that gets me within $\epsilon$ accuracy? In other ...
13
votes
1answer
317 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 ...
2
votes
1answer
42 views

An integral related to the derivative of Legendre polynomials

I want to calculate the integral $$ I=\int_{-1}^{1} \Big(\frac{\mathrm{d}P_{n+1}(t)}{\mathrm{d}t}\Big) \Big(\frac{\mathrm{d}P_{m+1}(t)}{\mathrm{d}t}\Big) \mathrm{d}t $$ where $P_n(t)$ is Legendre ...
1
vote
2answers
72 views

Eliminate the arbitrary funcion - PDE first order

I'm heading the book Elements Of Partial Differential Equations -Sneddon 1957. At chapter two exists this exercise "Eliminate the arbitrary function $f$ fron the equation $$ z= ...
0
votes
2answers
51 views

How to evaluate the length of the perimeter of a low eccentricity ellipse?

Given that $ e= \frac{a^2-b^2}{b^2} $ , and $L$ is the length of the perimeter, which equals $4aE(e, \pi/2)$, find the length of the perimeter up to $e^2$ in terms of $a$ and $b$. How does one begin ...
0
votes
1answer
61 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{ ...
2
votes
1answer
70 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 ...
3
votes
0answers
75 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 ...
0
votes
2answers
28 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
15 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
77 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
21 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 ...
2
votes
0answers
27 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
55 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 ...
0
votes
0answers
21 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
0answers
45 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 ...
4
votes
3answers
109 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
1answer
38 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
62 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 ...
0
votes
0answers
18 views

transformation involving elliptic integrals

I have two expressions which I know are equivalent but I just can't see how to go from one to another. I'm sure it involves properties of elliptic integrals however I am not very familiar with the ...
1
vote
1answer
37 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
1answer
36 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 } ...
0
votes
1answer
99 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
2answers
118 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 ...
1
vote
2answers
84 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 ...
0
votes
0answers
31 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 - ...
0
votes
0answers
23 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
61 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
33 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} + ...
0
votes
1answer
36 views

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

Question as title showed. What expression it represents? Many thanks.
1
vote
0answers
92 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 ...