WeierstrassP - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Special Functions : Weierstrass

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+omega1zomega21omega2

WeierstrassPPrimez,g2,g3=zWeierstrassPz,g2,g3

=2z32omega1zomega3

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

=1z+omega1zomega+1omega+zomega2

WeierstrassSigmaz,g2,g3=ⅇ∫0zWeierstrassZetat,g2,g3ⅆt

=zomega1zomegaⅇzomega+12z2omega2

  

where sums and products range over omega=2m1omega1+2m2omega2 such that m1,m2 is in ZxZ0,0. WeierstrassP and WeierstrassPPrime are elliptic functions (also known as doubly periodic functions) with periods 2omega1 and 2omega2.

• 

Quantities g2 and g3 are known as the invariants and are related to omega1 and omega2 by

g2=60omega1omega4

g3=140omega1omega6

  

where sums range over omega=2m1omega1+2m2omega2 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=12t+32tcsc12z6t2

• 

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

1.31740619529101

(2)

WeierstrassZeta1.0,2.0,3.0

0.944344946052752

(3)

WeierstrassSigma1.0,2.0,3.0

0.988067433302645

(4)

See Also

EllipticF

EllipticK

EllipticPi

JacobiSN