Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).
0
votes
1answer
33 views
Why is this function homogenous to the specified degree?
I have this function
$$ w(q) = (1 - \alpha)q^nBk^\alpha + c $$
The paper I'm reading says that w is homogenous of degree
$$ n/(1-\alpha) $$
and so small differences in q cause large differences ...
0
votes
0answers
106 views
About the definition of Bessel functions of the second kind
Why Bessel functions of the second kind does not define from the second linearly independent solution of the Bessel equation that solved by Frobenius method?
For example about the Bessel function of ...
2
votes
2answers
49 views
$\frac{\mathrm{d} g(x)}{\mathrm{d}x}=h(x)$ and $\frac{\mathrm{d} h(x)}{\mathrm{d}x}=g(x)$ where $h(x)\neq g(x)$
Is there any other solution to :
$$\frac{\mathrm{d} g(x)}{\mathrm{d}x}=h(x)$$
$$\frac{\mathrm{d} h(x)}{\mathrm{d}x}=g(x)$$
other than $h(x)=g(x)=e^x$?
By varying $\alpha,\beta$ in
$$\frac{\mathrm{d} ...
3
votes
0answers
88 views
Saddle point and stationary point approximation of the Airy equation
Happy New Year to you all.
Let $$\tag 1 J(N)=\int_a^b e^{Nf(x)}dx$$ where $N\in\mathbb R$ and $N>>1$ and $f(x)$ has a global maximum at $x=x_0$ with Taylor expansion $$f(x) \approx ...
1
vote
1answer
43 views
Is a Macdonald function a Bessel function with imaginary argument??
I mean that
$$ K_{a} (x)= CJ_{a}(ix).$$
Here $C$ is a complex number, and $a$ is real.
So is the Macdonald function a Bessel function in disguise (or proportional) of complex argument??
0
votes
0answers
39 views
Summing equally spaced samples of a periodic function
I'm a little stuck at the moment and wondered if someone could point me in the direction of the theory I need to read.
I have a $2\pi$-periodic function, $f:\mathbb{R}\rightarrow\mathbb{R}$ which I ...
11
votes
1answer
227 views
The Gamma function and the Pi function
I have been studying differential equation, in particular special functions.
Euler's Gamma function, and Gauss's Pi function are essentially the same, differing only by an offset of one unit.
for ...
2
votes
1answer
70 views
About the values of the $\Gamma$ function
The $\Gamma$ function is defined by
$$\Gamma(z)=\int_{0}^{+\infty}t^{z-1}e^{-t}dt$$
where $z$ is a complex number.
We know that if $z$ is real then the values of $\Gamma$ are also real. I am ...
1
vote
0answers
79 views
A Thue-Morse Zeta function ( Generalized Riemann Zeta function and new GRH )
Consider $t_n$ as the Thue-Morse sequence.
Let $m$ be a positive integer and $s$ a complex number.
Odiuos Number
Now consider the sequence of functions below
$f(1,s)=1+2^{-s}+3^{-s}+4^{-s}+...$
...
0
votes
0answers
53 views
An infinite product. Is it the value of a special function?
Let E(k) be the Euler numbers. Then
$$
\prod _{k=1}^{\infty }{ \exp\left({\frac {4-{E}
\left( 2\,k \right) }{4\,k \left( 4\,x+3 \right) ^{2\,k}}} \right)} =
\frac {\Gamma \left( x+1/2 \right) \left( ...
3
votes
2answers
292 views
Prove that $\int_0^1 \psi{(x) \sin(2 n \pi x)} \space\mathrm{dx}=-\frac{\pi}{2}$
Prove that
$$\int_0^1 \psi{(x) \sin(2 n \pi x)} \space\mathrm{dx}=-\frac{\pi}{2}, \space n\ge1$$
where $\psi(x)$ - digamma function
2
votes
1answer
163 views
error function (erf) with better precision
Currently I'm using this C++ routine to approximate the error function
...
4
votes
2answers
208 views
Solving the integral of a Modified Bessel function of the second kind
I would like to find the answer for the following integral
$$\int x\ln(x)K_0(x) dx $$
where $K_0(x)$ is the modified Bessel function of the second kind and $\ln(x)$ is the natural-log. Do you have ...
5
votes
1answer
125 views
Finding x in $\frac{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,1-x)}{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,x)} = \sqrt{n}$
I was trying to find a closed-form for $0<x<1$ in,
$$\frac{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,1-x)}{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,x)} = \sqrt{n}$$
where $\,_2F_1(a,b,c,z)$ ...
-1
votes
1answer
152 views
Integral of the square of the Bessel Function
Has anyone ever solved the integral of square of Bessel and Neumann functions? This is for standing wave analysis on a cylindrical object.
I need to integrate $(AJ(kr) + BY(kr))^2$ from $r=a$ to ...
2
votes
0answers
47 views
asymptotics of $ J_{iu} (ia)$ for a Bessel function
Let $J_{iu}(ia)$ be the Bessel function of imaginary order. ($a$ is a real number (positive or negative) and $u$ is also real.)
In the limit $u \to \infty $ how does the function $J_{iu} (ia)$ ...
3
votes
2answers
44 views
Showing integrability (Riemann)
I was trying to show whether or not the function:
$f: [0,1 ] \rightarrow \mathbb{R}$
$f(x)= \frac {1}{n}$ for $x = \frac {1}{n}$ $(n \in \mathbb{N})$
and
$f(x) = 1$ if the condition isn't ...
2
votes
1answer
70 views
What is the “name” of this function?
There is a function I met in complex analysis.
$$f(\lambda) = \int \limits_{-\infty}^{\infty}\frac{e^{i\lambda x}}{\sqrt{1 + x^{2n}}}dx$$
0
votes
0answers
38 views
A different version of Thomae's function and differentiability
So, you have a function
$f(x) = x^2$ if $x \in \mathbb{Q}$
and $f(x) = 0 $ if $x \in \mathbb{R} \setminus \mathbb{Q}$
I'm almost certain it is differentiable at $0$ and nowhere else and I was ...
7
votes
2answers
202 views
An inequality from the handbook of mathematical functions (by Abramowitz and Stegun)
Prove that
$$\frac{1}{x+\sqrt{x^2+2}}<e^{x^2}\int\limits_x^{\infty}e^{-t^2} \, \text dt \le\frac{1}{x+\sqrt{x^2+\displaystyle\tfrac{4}{\pi}}}, \space (x\ge 0)$$
0
votes
1answer
71 views
Identity concerning $e^{ia\sin{x}}$ as a series of bessel functions
Prove the following identity:
\begin{equation}
e^{ia\sin{x}}=\sum_{-\infty}^{+\infty}J_k(a) e^{ikx}
\end{equation}
, where $a$ is a real constant and $J_k$ is the Bessel function of the first type ...
1
vote
0answers
40 views
Does this series converge (squares of associated Legendre polynomials)?
Consider the following series (where $l,\,m\in\mathrm{Z}\,$):
$S = \displaystyle\sum^{\infty}_{l\,=\,2} \frac{2l+1}{(l-1)(l+2)(1+l^2)}\sum^{l}_{m\,=\,-l}\frac{(l-m)!}{(l+m)!}\Big(P^m_l(x)\,\Big)^2$,
...
0
votes
2answers
112 views
$e^x-x-4$equating with zero
I want to find out the values of x where the $f(x) = e^x-x-4$ will equal zero.
My problem by solving this myself is that I cannot use logarithm natural (ln) because I have a normal x:
$f(x) = e^x - ...
1
vote
2answers
98 views
Scale modified Bessel functions to then unscale later
So I have some variables $\,x_{1},\, x_{2},\, \nu\, =\, 12.654,\, 13.487,\, 0\,$ and the following function:
$\dfrac{(x_{1}\cdot(-BesselK(\nu,x_{1}\cdot125))\cdot ...
1
vote
1answer
49 views
Bounds on geometric sum
Consider the sum $\sum_{x=1}^{\infty} \frac{\log{x}}{z^x}$. We can assume that $z\geq1$ (and is real). Mathematica gives this sum as
-PolyLog^(1, 0)[0,1/z]
...
2
votes
1answer
130 views
Inequality for Gamma functions
Let $k, n ,m \in N$ and such that $0\leq k \leq n \leq m$. When the following ineuality is true?
$$
...
3
votes
2answers
58 views
Advice on an integral involving the error function
I'd like to calculate the following integral:
$$\int^{\infty}_{0} \mathrm{erf}\left(\frac{\alpha}{\sqrt{1+x}} - \frac{\sqrt{1+x}}{\beta}\right) \exp\left(-\frac{x}{\gamma}\right)\, dx,$$
where ...
2
votes
1answer
103 views
Problem with the Dirichlet Eta Function
I was doing a bit of self-study of sequences, and I considered
$$\sum_{n=1}^{\infty}\frac {(-1)^n \ln(n)}{n} $$
which I then found out is ${\eta}'(1)$, the derivative of the Dirichlet Eta Function ...
2
votes
2answers
120 views
Quotient of Gamma functions
I am trying to find a clever way to compute the quotient of two gamma functions whose inputs differ by some integer. In other words, for some real value $x$ and an integer $n < x$, I want to find a ...
0
votes
1answer
47 views
reference needed for Gamma function
Please help me to find a reference (book) for the following upper bound of Gamma function
For $x \geq 1$
$$
\Gamma(x)\leq x^{x-1}.
$$
Thank you.
1
vote
2answers
171 views
Meaning of function with circle and cross
I've seen this function M2 = tmp ⊕ Pi. What does the circle with cross do?
0
votes
0answers
63 views
When inequality for binomial coefficients is true?
I've asked similar question here Inequality for binomial coefficients, but with slightly different assumptions. I am curious what happend if $m, k$ are fixed.
Let $m \leq n, n \leq N$ and $0\leq k ...
2
votes
1answer
138 views
Conformal mapping from triangle to upper half plane in terms of Weierstrass $\wp$
I'm trying to explicitly compute a conformal map $f:\Delta \rightarrow \mathbb{H}$ where $\Delta$ is a triangle and $\mathbb{H}$ is the upper half plane, in terms of the Weierstrass $\wp$ function. I ...
2
votes
1answer
74 views
Show that the series representation of the Bessel function works
For the following series representation of the Bessel function:
$$w = J_n = \sum_{k=0}^{\infty} \frac{(-1)^k z^{n+2k}}{k!(n+k)!2^{n+2k}}.$$
I want to show that w is indeed the Bessel function, such ...
0
votes
1answer
92 views
Riemann's Zeta function [duplicate]
Possible Duplicate:
Riemann Zeta Function and Analytic Continuation
Calculating the Zeroes of the Riemann-Zeta function
It is stated that Riemann's Zeta function has zeros at negative ...
2
votes
2answers
141 views
Adding imaginary number to exponential of Euler Gamma function
This is gamma function:
$\Gamma (n) = \int_0^\infty x^{n-1}e^{-x}\,dx$
What will be Result if I add Imaginary Number to Exponential of Euler Gamma Function?
$$? = \int_0^\infty x^{n-1}e^{-ix}\,dx$$
...
1
vote
1answer
45 views
weird bessel zero question
given 'a' and 'b' fixed i define the function
$$ f(t)= bJ_{2t}(a) $$
here $ J_{n} $ is a Bessel function but in this cases i would be interested in getting the solutions (?? are there any ? ) for
...
3
votes
0answers
155 views
Questions about the Fourier expansion of $e^{iz\cot(x)}$
By analogy with Jacobi–Anger expansion, one expects that $e^{iz\cot(x)}$ has a Fourier expansion of the form :
$$e^{iz\cot(\theta)}=\sum_{n=-\infty}^{\infty}\Lambda_{n}(z)e^{in\theta}$$
...
16
votes
0answers
658 views
Prove that sum is finite
Let $j \in \mathbb{N}$. Set
$$
a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!}
$$
and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$.
Please help me to prove that the following sum is ...
1
vote
1answer
78 views
Upper bound for a gamma function
Let $n \in N$. How to find a non-asymptotic upper bound for $\Gamma(n)$ and $\Gamma(\frac n2+1)$?
Thank you
0
votes
1answer
57 views
Zernike and Legendre polynomials
The even and odd Zernike polynomials are defined as follows:
$$Z^{m}_n(\rho,\varphi) = R^m_n(\rho)\,\cos(m\,\varphi) \!$$
and:
$$Z^{-m}_n(\rho,\varphi) = R^m_n(\rho)\,\sin(m\,\varphi), \!$$
with:
...
15
votes
1answer
312 views
Intuition why the volume and surface area of the unit sphere eventually decrease
The volume formula for a unit sphere, $$\frac{\pi^{n/2}}{\Gamma{(1 + n/2)}},$$
and the surface area formula, $$\frac{2\pi^{n/2}}{\Gamma{(n/2)}},$$
both attain maximum values for finite $n$. We can ...
2
votes
1answer
174 views
solution of Lagrange differential equation are square integrable
I was recently posing myself this question. Given the Lagrange DE $$[(1-x^2)u']'+\lambda u=0,$$ where $\lambda$ is a real parameter and $x\in[-1,1]$, it is well known that, if $\lambda=n(n+1)$ for ...
0
votes
1answer
65 views
Integral of Scaled Bessel Function With Linear Phase
I am trying to solve a problem part of which includes the following integral ($j=\sqrt{-1}$):
$$\int_{k_1}^{k_2} k e^{-jk\sigma} J_n(\rho k) \, \mathrm{d}k$$
The $e^{-jk\sigma}$ term is making my ...
0
votes
0answers
45 views
if system is localized
Consider function $g=e^{-|x|}, x \in R$.
Let $\psi_n$ be a Hermite function (see http://en.wikipedia.org/wiki/Hermite_polynomials for definition). Consider system $\Psi=\{\psi_n\}, n \in N$.
Let ...
0
votes
0answers
36 views
Integral Logarithm [duplicate]
Possible Duplicate:
Proof involving the logarithmic integral
I am having a problem with the following exercise.
A function, called the integral Logarithm and denoted by Li, is defined as ...
0
votes
0answers
58 views
inequality with gamma function
Help me please to prove the following inequality
For $x,y>1, x \neq y$.
$$
\frac{1}{\Gamma(x)\Gamma(y)}\leq 2\sqrt{2\pi}\frac{\sqrt{x+y}}{\Gamma(x+y)}.
$$
Thank you.
5
votes
1answer
129 views
Error Function limit
$$\prod_{n=1}^{\infty}{\frac{2}{\sqrt{\pi}}\int_0^n e^{-x^{2}} \mathrm{d}x} \approx 0.83874 $$
Is it a known constant? I couldn't find anything about it. Do you know ways to calculate the value ...
2
votes
0answers
76 views
Solutions of legendre equation for $\vert x\vert \leq 1$
Why books say that is necessary in Legendre equation to have $l$ integer if you want regular solutions in $\vert x\vert \leq 1$. It seems not necessary.
Thanks in advance.
5
votes
1answer
134 views
Did Euler have an alpha function
I've heard of Euler Gamma function: $\Gamma(x)$, and Euler's beta function: $\text{B}(x,y)$. Did Euler have an alpha function?
