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WeierstrassP

The Weierstrass P function, P(z,g2,g3)

WeierstrassPPrime

The Derivative of the Weierstrass P function, P'(z,g2,g3)

WeierstrassZeta

The Weierstrass zeta function, zeta(z,g2,g3)

WeierstrassSigma

The Weierstrass sigma function, sigma(z,g2,g3)

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

WeierstrassP(z, g2, g3)

WeierstrassPPrime(z, g2, g3)

WeierstrassZeta(z, g2, g3)

WeierstrassSigma(z, g2, g3)

Parameters

z

-

algebraic expression

g2, g3

-

algebraic expressions (invariants)

Description

• 

WeierstrassP (Weierstrass elliptic function), WeierstrassPPrime, WeierstrassZeta, and WeierstrassSigma are defined by

WeierstrassPz,g2,g3=1z2+ω1zω21ω2

WeierstrassPPrimez,g2,g3=zWeierstrassPz,g2,g3

=2z32ω1zω3

WeierstrassZetaz,g2,g3=0zWeierstrassPt,g2,g3ⅆt

=1z+ω1zω+1ω+zω2

WeierstrassSigmaz,g2,g3=ⅇ0zWeierstrassZetat,g2,g3ⅆt

=zω1zωⅇzω+z22ω2

  

where sums and products range over ω=2m1ω1+2m2ω2 such that m1,m2 is in ZxZ0,0. WeierstrassP and WeierstrassPPrime are elliptic functions (also known as doubly periodic functions) with periods 2ω1 and 2ω2.

• 

Quantities g2 and g3 are known as the invariants and are related to ω1 and ω2 by

g2=60ω1ω4

g3=140ω1ω6

  

where sums range over ω=2m1ω1+2m2ω2 such that m1,m2 is in ZxZ0,0.

• 

An important property of the invariants g2 and g3 is that WeierstrassP satisfies the differential equation

WeierstrassPPrimez,g2,g32=4WeierstrassPz,g2,g33g2WeierstrassPz,g2,g3g3

• 

A special case of WeierstrassP happens when the discriminant g2327g32 is equal to zero, in which case g2 and g3 are related, can be expressed in terms of a single parameter, say t, and the function is given by

WeierstrassPz,3t2,t3=t2+3tcscz6t222

• 

Refer to Chapter 18, "Weierstrass Elliptic and Related Functions" of Handbook of Mathematical Functions edited by Abramowitz and Stegun for more extensive information.

Examples

WeierstrassP1.0,2.0,3.0

1.21443370936522

(1)

WeierstrassPPrime1.0,2.0,3.0

−1.317406197

(2)

WeierstrassZeta1.0,2.0,3.0

0.944344946514663

(3)

WeierstrassSigma1.0,2.0,3.0

0.9880674327

(4)

See Also

EllipticF

EllipticK

EllipticPi

JacobiSN