OpenMath Content Dictionary: misc

Canonical URL:
http://www.openmath.org/CDs/misc.ocd
CD File:
misc.ocd
CD as XML Encoded OpenMath:
misc.omcd
Defines:
BernoulliNumber, BernoulliPolynomial, Dawson, eulerNumber, eulerPolynomial, GaussAGM, harmonic, lambertW, lambertWi, MeijerG, ReimannZeta
Date:
2001-26-08
Version:
1
Review Date:
Status:
experimental

2002-01-01

This content dictionary contains symbols to describe miscellaneous special functions. They are defined as in Abromowitz and Stegun (ninth printing on).


MeijerG

The MeijerG symbol represents Meijer's G-Function, it is defined as in maple 7 and section 9.3 of Table of Integrals, Series and Products by I.S. Gradshteyn and I.M. Ryzhik (Corrected and Enlarged edition, Sixth printing).

- The Meijer G function is defined by the inverse Laplace transform

$MeijerG([a_s,b_s],[c_s,d_s],z) = \frac{1}{2\pi i}\int_L \frac{\Gamma (1-a_s+y)\Gamma (c_s-y)}{\Gamma(b_s-y) \Gamma(1-d_s+y)}z^y dy$

where

$a_s = [a_1,...,a_m], \Gamma(1-a_s+y) = \Gamma(1-a_1+y)\cdots\Gamma(1-a_m+y)$

$b_s = [b_1,...,b_n], \Gamma(b_s-y) = \Gamma(b_1-y)\cdots\Gamma(b_n-y)$

$c_s = [c_1,...,c_p], \Gamma(c_s-y) = \Gamma(c_1-y)\cdots\Gamma(c_p-y)$

$d_s = [d_1,...,d_q], \Gamma(1-ds+y) = \Gamma(1-d_1+y)\cdots\Gamma(1-d_q+y)$

and L is one of three types of integration paths L[$\gamma+i*\infty$], L[$\infty$], and L[$-\infty$].

Contour L[$\infty$] starts at $\infty+i*\phi_1$ and finishes at $\infty+i*\phi_2 (\phi_2 gt \phi_1)$.

Contour L[$-\infty$] starts at $-\infty+i*\phi_1$ and finishes at $-\infty+i*\phi_2$ ($\phi_2 gt \phi_1$).

Contour L[$\gamma+I*\infty$] starts at $\gamma-\infty$ and finishes at $\gamma+i*\infty$.

All the paths L[$\infty$], L[$-\infty$], and L[$\gamma+i*\infty$] put all $c_j+k$ poles on the right and all other poles of the integrand (which must be of the form $a_j-1+k$) on the left.

Commented Mathematical property (CMP):
The MeijerG function satisfies the following qth-order linear differential equation $\left ( (-1)^{(p-m-n)} x\prod^p_{j=1}\left ( x\frac{d}{dx}-a_j+1\right ) - \prod^q_{j=1}\left ( x\frac{d}{dx} - b_j\right )\right ) y = 0$ where p is less than or equal to q.
Signatures:
sts


[Next: harmonic] [Last: ReimannZeta] [Top]

harmonic

The harmonic symbol represents the unary harmonic function which is defined as follows: harmonic(x) = digamma(x+1) + gamma

Commented Mathematical property (CMP):
for integer n harmonic(n) = \sum_{i=1}^n \frac{1}{i}
Signatures:
sts


[Next: GaussAGM] [Previous: MeijerG] [Top]

GaussAGM

The symbol GaussAGM represents the binary function which computes the limit of the iteration $a_0 := a$ $b_0 := b$ $a_{n+1} := (a-n+b_n)/2$ $b_{n+1} := (a_n+b_n)*\sqrt(\frac{a_n*b_[n]}{(a_n+b_n)^2})$ where $a$ is the first and $b$ is the second argument.

Signatures:
sts


[Next: eulerNumber] [Previous: harmonic] [Top]

eulerNumber

The symbol eulerNumber takes one argument, it represents the function which returns the n'th Eulerian number, where n is the argument. The nth Euler number E(n) is defined by the exponential generating function:

2/(exp(t)+exp(-t)) = sum(E(n)/n!*t^n, n = 0..\infty)

Signatures:
sts


[Next: eulerPolynomial] [Previous: GaussAGM] [Top]

eulerPolynomial

The symbol eulerPoly takes two arguments n,x, it represents the function which returns the n'th Euler polynomial in x. The nth Euler polynomial E(n,x) is defined by the exponential generating function:

2*exp(x*t)/(exp(t)+1) = sum(E(n,x)/n!*t^n, n = 0..\infty)

Signatures:
sts


[Next: BernoulliNumber] [Previous: eulerNumber] [Top]

BernoulliNumber

The symbol BernoulliNumber represents the function which takes one argument n, and returns the nth Bernoulli number. Bernoullinumbers come from the coefficients in the Taylor expansion of x/(e^x -1). We shall denote the nth Bernoulli number as B(n). That is: if the absolute value of x less than 2*pi then x/(e^x-1)= sum^{\infty}_{x=0}(\frac{B(n)*x_n}{n!})

Signatures:
sts


[Next: BernoulliPolynomial] [Previous: eulerPolynomial] [Top]

BernoulliPolynomial

The symbol BernoulliPolynomial represents the function which takes two arguments n and x, and returns the nth Bernoulli polynomial in the expression x. We shall denote the nth Bernoulli polynomial in x as B(n,x). The nth Bernoulli polynomial is defined by the exponential generating function:

t*exp(x*t)/(exp(t)-1) = sum(B(n,x)/n!*t^n, n=0..\infty).

Signatures:
sts


[Next: Dawson] [Previous: BernoulliNumber] [Top]

Dawson

The symbol Dawson represents Dawson's integral. It takes one argument and is defined as:

dawson(x) = exp(-x^2) * int(exp(t^2), t=0..x)

Signatures:
sts


[Next: lambertW] [Previous: BernoulliPolynomial] [Top]

lambertW

This symbol represents Lamberts W function, it is denoted as $W$. It is commonly known as the Lambert-W function. The Lambert W function is defined to be the multivalued inverse of the function $w \rightarrow w e^w$.

It is defined as in the paper "On Lamberts W function: R.Corless, G. Gonnet, D. Hare and D. Jeffrey"

Signatures:
sts


[Next: lambertWi] [Previous: Dawson] [Top]

lambertWi

The symbol lambertWi represents the ith branch of the Lambert W function. It is denoted as $W_i$, and is commonly known as the ith branch of the Lambert-W function. It's first argument is the number of the branch. The Lambert W_i function is defined to return the value of the ith branch of the inverse of the function $w \rightarrow w e^w$. It is defined as in the paper "On Lamberts W function: R.Corless, G. Gonnet, D. Hare and D. Jeffrey"

Signatures:
sts


[Next: ReimannZeta] [Previous: lambertW] [Top]

ReimannZeta

The symbol ReimannZeta represents the Riemann Zeta Function, it is denoted as $\zeta (s)$. The Riemann Zeta Function is defined by $\zeta (s)= \sum_{k=1}^\infty k^{-s}$ $R(s) gt 1$ It is also related to the distribution of primes, in the following way: $\zeta (s) = \prod_{p} {(1-p^{-s}}^{-1}$

It is defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [23.2]

Commented Mathematical property (CMP):
$\zeta (s)= \sum_{k=1}^\infty k^{-s}$ $R(s) gt 1$
Commented Mathematical property (CMP):
$\zeta (s) = \prod_{p} {(1-p^{-s}}^{-1}$
Signatures:
sts


[First: MeijerG] [Previous: lambertWi] [Top]