OpenMath Content Dictionary: Weierstrass

Canonical URL:
http://www.openmath.org/CDs/Weierstrass.ocd
CD File:
Weierstrass.ocd
CD as XML Encoded OpenMath:
Weierstrass.omcd
Defines:
WeierstrassP, WeierstrassPPrime, WeierstrassSigma, WeierstrassZeta
Date:
2001-30-08
Version:
1
Review Date:
Status:
experimental

2002-01-01

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


WeierstrassP

This symbol represents the WeierstrassP function, it takes three arguments and is defined as follows:

WeierstrassP(z,g2,g3) = 1/z^2+sum(1/(z-w)^2-1/w^2,w)

where g2 = 60*sum(1/w^4,w), g3 = 140*sum(1/w^6,w)

and the sums range over w=2*m1*omega1+2*m2*omega2 such that (m1,m2) is in (Z x Z) - (0,0)

it is defined as in Abromowitz and Stegun 18.1

Signatures:
sts


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WeierstrassPPrime

This symbol represents the WeierstrassPPrime function, it takes three arguments and is defined as follows:

WeierstrassPPrime(z,g2,g3) = diff(WeierstrassP(z,g2,g3),z) = -2/z^3-2*sum(1/(z-w)^3,w)

where g2 = 60*sum(1/w^4,w), g3 = 140*sum(1/w^6,w)

and the sums range over w=2*m1*omega1+2*m2*omega2 such that (m1,m2) is in (Z x Z) - (0,0)

it is defined as in Abromowitz and Stegun 18.1.6

Commented Mathematical property (CMP):
WeierstrassPPrime(z,g2,g3) = diff(WeierstrassP(z,g2,g3),z)
Signatures:
sts


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WeierstrassZeta

WeierstrassZeta(z,g2,g3) = -int(WeierstrassP(t,g2,g3),t=0..z) = 1/z+sum(1/(z-w)+1/w+z/w^2,w)

where g2 = 60*sum(1/w^4,w), g3 = 140*sum(1/w^6,w)

and the sums range over w=2*m1*omega1+2*m2*omega2 such that (m1,m2) is in (Z x Z) - (0,0)

it is defined as in Abromowitz and Stegun 18.1

Commented Mathematical property (CMP):
WeierstrassZeta(z,g2,g3) = -int(WeierstrassP(t,g2,g3),t=0..z)
Signatures:
sts


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WeierstrassSigma

WeierstrassSigma(z,g2,g3) = exp(int(WeierstrassZeta(t,g2,g3),t=0..z)) = z*product((1-z/w)*exp(z/w+z^2/(2*w^2)),w)

where g2 = 60*sum(1/w^4,w), g3 = 140*sum(1/w^6,w)

and the sums range over w=2*m1*omega1+2*m2*omega2 such that (m1,m2) is in (Z x Z) - (0,0)

it is defined as in Abromowitz and Stegun 18.1

Commented Mathematical property (CMP):
WeierstrassSigma(z,g2,g3) = exp(int(WeierstrassZeta(t,g2,g3),t=0..z))
Signatures:
sts


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