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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 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

$\mathrm{WeierstrassP}\left(z,\mathrm{g2},\mathrm{g3}\right)=\frac{1}{{z}^{2}}+\sum _{\mathrm{\omega }}\left(\frac{1}{{\left(z-\mathrm{\omega }\right)}^{2}}-\frac{1}{{\mathrm{\omega }}^{2}}\right)$

$\mathrm{WeierstrassPPrime}\left(z,\mathrm{g2},\mathrm{g3}\right)=\frac{\partial }{\partial z}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\mathrm{WeierstrassP}\left(z,\mathrm{g2},\mathrm{g3}\right)$

$=-\frac{2}{{z}^{3}}-2\left(\sum _{\mathrm{\omega }}\frac{1}{{\left(z-\mathrm{\omega }\right)}^{3}}\right)$

$\mathrm{WeierstrassZeta}\left(z,\mathrm{g2},\mathrm{g3}\right)=-\left({\int }_{0}^{z}\mathrm{WeierstrassP}\left(t,\mathrm{g2},\mathrm{g3}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt\right)$

$=\frac{1}{z}+\sum _{\mathrm{\omega }}\left(\frac{1}{z-\mathrm{\omega }}+\frac{1}{\mathrm{\omega }}+\frac{z}{{\mathrm{\omega }}^{2}}\right)$

$\mathrm{WeierstrassSigma}\left(z,\mathrm{g2},\mathrm{g3}\right)={ⅇ}^{{\int }_{0}^{z}\mathrm{WeierstrassZeta}\left(t,\mathrm{g2},\mathrm{g3}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt}$

$=z\left(\prod _{\mathrm{\omega }}\left(1-\frac{z}{\mathrm{\omega }}\right){ⅇ}^{\frac{z}{\mathrm{\omega }}+\frac{{z}^{2}}{2{\mathrm{\omega }}^{2}}}\right)$

 where sums and products range over $\mathrm{\omega }=2{m}_{1}{\mathrm{\omega }}_{1}+2{m}_{2}{\mathrm{\omega }}_{2}$ such that ${m}_{1},{m}_{2}$ is in $\left(ZxZ\right)-\left(0,0\right)$. WeierstrassP and WeierstrassPPrime are elliptic functions (also known as doubly periodic functions) with periods $2{\mathrm{\omega }}_{1}$ and $2{\mathrm{\omega }}_{2}$.
 • Quantities g2 and g3 are known as the invariants and are related to ${\mathrm{\omega }}_{1}$ and ${\mathrm{\omega }}_{2}$ by

$\mathrm{g2}=60\left(\sum _{\mathrm{\omega }}\frac{1}{{\mathrm{\omega }}^{4}}\right)$

$\mathrm{g3}=140\left(\sum _{\mathrm{\omega }}\frac{1}{{\mathrm{\omega }}^{6}}\right)$

 where sums range over $\mathrm{\omega }=2{m}_{1}{\mathrm{\omega }}_{1}+2{m}_{2}{\mathrm{\omega }}_{2}$ such that ${m}_{1},{m}_{2}$ is in $\left(ZxZ\right)-\left(0,0\right)$.
 • An important property of the invariants g2 and g3 is that WeierstrassP satisfies the differential equation

${\mathrm{WeierstrassPPrime}\left(z,\mathrm{g2},\mathrm{g3}\right)}^{2}=4{\mathrm{WeierstrassP}\left(z,\mathrm{g2},\mathrm{g3}\right)}^{3}-\mathrm{g2}\mathrm{WeierstrassP}\left(z,\mathrm{g2},\mathrm{g3}\right)-\mathrm{g3}$

 • A special case of WeierstrassP happens when the discriminant ${\mathrm{g2}}^{3}-27{\mathrm{g3}}^{2}$ is equal to zero, in which case $\mathrm{g2}$ and $\mathrm{g3}$ are related, can be expressed in terms of a single parameter, say $t$, and the function is given by

$\mathrm{WeierstrassP}\left(z,3{t}^{2},{t}^{3}\right)=-\frac{t}{2}+\frac{3t{\mathrm{csc}\left(\frac{z\sqrt{6}\sqrt{t}}{2}\right)}^{2}}{2}$

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

Examples

 > $\mathrm{WeierstrassP}\left(1.0,2.0,3.0\right)$
 ${1.21443370936522}$ (1)
 > $\mathrm{WeierstrassPPrime}\left(1.0,2.0,3.0\right)$
 ${-1.317406197}$ (2)
 > $\mathrm{WeierstrassZeta}\left(1.0,2.0,3.0\right)$
 ${0.944344946514663}$ (3)
 > $\mathrm{WeierstrassSigma}\left(1.0,2.0,3.0\right)$
 ${0.9880674327}$ (4)