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Weierstrass Elliptic Function

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The Weierstrass elliptic functions (or Weierstrass P-functions, voiced "p-functions") are elliptic functions which, unlike the Jacobi elliptic functions, have a second-order pole at z=0. To specify P(z) completely, its half-periods (omega_1 and omega_2) or elliptic invariants (g_2 and g_3) must be specified. These two cases are denoted P(z|omega_1,omega_2) and P(z;g_2,g_3), respectively.

The Weierstrass elliptic function is implemented in the Wolfram Language as WeierstrassP[u, {g2, g3}]. Half-periods and invariants can be interconverted using the Wolfram Language commands WeierstrassInvariants[{omega1, omega2}] and WeierstrassHalfPeriods[{g2, g3}]. The derivative of a Weierstrass elliptic function is implemented as WeierstrassPPrime[u, {g2, g3}], and the inverse Weierstrass function is implemented as InverseWeierstrassP[p, {g2, g3}]. InverseWeierstrassP[{p, q}, {g2, g3}] finds the unique value of u for which p=P(u;g_2,g_3) and q=P^'(u;g_2,g_3).

WeierstrassP

The above plots show the Weierstrass elliptic function P(z;g_2,g_3) and its derivative P^'(z;g_2,g_3) for elliptic invariants g_2=4 and g_3=0 along the real axis.

WeierstrassPReIm
WeierstrassPContours
WeierstrassPPrimeReIm
WeierstrassPPrimeContours

The plots above show the Weierstrass P-function and its derivatives for the elliptic invariants (g_2,g_3)=(4,0).

Specific cases of the elliptic invariants g_2 and g_3 are given the special names summarized in the following table (Abramowitz and Stegun 1972). The real half-period in the equianharmonic case is called the omega2-constant.

g_2g_3case name
01equianharmonic case
10lemniscate case
-10pseudolemniscate case

The Weierstrass elliptic functions are defined by

P(z)=1/(z^2)+sum^'_(m,n=-infty)^infty[1/((z-2momega_1-2nomega_2)^2)-1/((2momega_1+2nomega_2)^2)]
(1)

(Whittaker and Watson 1990, p. 434), where the prime indicated that terms in the sum giving zero denominators are omitted. Write Omega_(mn)=2momega_1+2nomega_2. Then this can be written

P(z)=z^(-2)+sum^'_(m,n)[(z-Omega_(mn))^(-2)-Omega_(mn)^(-2)].
(2)

An equivalent definition which converges more rapidly is

P(z)=(pi/(2omega_1))^2[-1/3+sum_(n=-infty)^inftycsc^2((z-2nomega_2)/(2omega_1)pi)-sum^'_(n=-infty)^inftycsc^2((nomega_2)/(omega_1)pi)]
(3)

(Whittaker and Watson 1990, p. 434). P(z) is an even function since P(-z) gives the same terms in a different order.

The series expansion of P(z) is given by

P(z)=z^(-2)+sum_(k=2)^inftyc_kz^(2k-2),
(4)

where

c_2=(g_2)/(20)
(5)
c_3=(g_3)/(28)
(6)

and

c_k=3/((2k+1)(k-3))sum_(m=2)^(k-2)c_mc_(k-m)
(7)

for k>=4 (Abramowitz and Stegun 1972, p. 635). The first few values for c_k for k>=4 in terms of c_2 and c_3 are given by

c_4=1/3c_2^2
(8)
c_5=1/(11)(3c_2c_3)
(9)
c_6=1/(39)(2c_2^3+3c_3^2)
(10)
c_7=2/(33)c_2^2c_3
(11)
c_8=5/(7293)(11c_2^4+36c_2c_3^2)
(12)
c_9=(29)/(2717)(c_2^3c_3+11c_3^3)
(13)
c_(10)=1/(240669)(242c_2^5+1455c_2^2c_3^2)
(14)

(Abramowitz and Stegun 1972, p. 636).

The Weierstrass elliptic function describes how to get from a torus giving the solutions of an elliptic curve to the algebraic form of the elliptic curve.

The differential equation from which Weierstrass elliptic functions arise can be found by expanding about the origin the function f(z)=P(z)-z^(-2).

P(z)-z^(-2)=f(0)+f^'(0)z+1/(2!)f^('')(0)z^2+1/(3!)f^(''')(0)z^3+1/(4!)f^((4))(0)z^4+....
(15)

But f(0)=0 and the function is even, so f^'(0)=f^(''')(0)=0 and

f(z)=P(z)-z^(-2)=1/(2!)f^('')(0)z^2+1/(4!)f^((4))(0)z^4+....
(16)

Taking the derivatives

f^'=-2Sigma^'(z-Omega_(mn))^(-3)
(17)
f^('')=6Sigma^'(z-Omega_(mn))^(-4)
(18)
f^(''')=-24Sigma^'(z-Omega_(mn))^(-5)
(19)
f^((4))=120Sigma^'(z-Omega_(mn))^(-6).
(20)

So

f^('')(0)=6Sigma^'Omega_(mn)^(-4)
(21)
f^((4))(0)=120Sigma^'Omega_(mn)^(-6).
(22)

Plugging in,

P(z)-z^(-2)=3Sigma^'Omega_(mn)^(-4)z^2+5Sigma^'Omega_(mn)^(-6)z^4+O(z^6).
(23)

Define the elliptic invariants

g_2=60Sigma^'Omega_(mn)^(-4)
(24)
g_3=140Sigma^'Omega_(mn)^(-6),
(25)

then

P(z)=z^(-2)+1/(20)g_2z^2+1/(28)g_3z^4+O(z^6)
(26)
P^'(z)=-2z^(-3)+1/(10)g_2z+1/7g_3z^3+O(z^5).
(27)

Now cube (26) and square (27)

P^3(z)=z^(-6)+3/(20)g_2z^(-2)+3/(28)g_3+O(z^2)
(28)
P^'^2(z)=4z^(-6)-2/5g_2z^(-2)-4/7g_3+O(z^2).
(29)

Taking (29) minus 4× (28) cancels out the z^(-6) term, giving

P^'^2(z)-4P^3(z)=(-2/5-3/5)g_2z^(-2)+(-4/7-3/7)g_3+O(z^2)
(30)
=-g_2z^(-2)-g_3+O(z^2)
(31)

giving

P^'^2(z)-4P^3(z)+g_2z^(-2)+g_3=O(z^2).
(32)

But, from (◇)

P(z)=z^(-2)+1/(2!)f^('')(0)z^2+1/4f^((4))(0)z^4+...,
(33)

so P(z)=z^(-2)+O(z^2) and (◇) can be written

P^'^2(z)-4P^3(z)+g_2P(z)+g_3=O(z^2).
(34)

But the Weierstrass elliptic function is analytic at the origin and therefore at all points congruent to the origin. There are no other places where a singularity can occur, so this function is an elliptic function with no singularities. By Liouville's elliptic function theorem, it is therefore a constant. But as z->0, O(z^2)->0, so

P^'^2(z)=4P^3(z)-g_2P(z)-g_3
(35)

(Whittaker and Watson 1990, pp. 436-437).

The solution to the differential equation

y^'^2=4y^3-g_2y-g_3
(36)

is therefore given by y=P(z+alpha), providing that numbers omega_1 and omega_2 exist which satisfy the equations defining the elliptic invariants. Writing the differential equation in terms of its roots e_1, e_2, and e_3,

y^'^2=4y^3-g_2y-g_3=4(y-e_1)(y-e_2)(y-e_3)
(37)

(Rainville 1971, p. 312),

2ln(y^')=ln4+sum_(r=1)^3ln(y-e_r)
(38)
(2y^(''))/(y^')=y^'sum_(r=1)^3(y-e_r)^(-1)
(39)
(2y^(''))/(y^'^2)=sum_(r=1)^3(y-e_r)^(-1)
(40)
2(y^'^2y^(''')-y^('')(2y^'y^('')))/(y^'^4)=-y^'sum_(r=1)^3(y-e_r)^(-2)
(41)
(2y^('''))/(y^'^3)-(4y^('')^2)/(y^'^4)=-sum_(r=1)^3(y-e_r)^(-2).
(42)

Now take (◇) divided by 4 plus [(◇) divided by 4] quantity squared,

((y^('''))/(2y^'^3)-(y^('')^2)/(y^'^4))+((y^('')^2)/(4y^'^4))=-1/4sum_(r=1)^3(y-e_r)^(-2)+1/(16)[sum_(r=1)^3(y-e_r)^(-1)]^2
(43)
(3y^('')^2)/(4y^'^4)-(y^('''))/(2y^'^3)=3/(16)sum_(r=1)^3(y-e_r)^(-2)-3/8yproduct_(r=1)^3(y-e_r)^(-1).
(44)

The term on the right is half the Schwarzian derivative.

The derivative of the Weierstrass elliptic function is given by

P^'(z)=d/(dz)P(z)
(45)
=-2sum_(m,n)1/((z-Omega_(mn))^3)
(46)
=-2z^(-3)-2sum^'_(m,n)(z-Omega_(mn))^(-3).
(47)

This is an odd function which is itself an elliptic function with pole of order 3 at z=0. The integral is given by

z=int_(P(z))^infty(4t^3-g_2t-g_3)^(-1/2)dt.
(48)

The second derivative satisfies

P^('')(1/2omega_1)=2(e_1-e_2)(e_1-e_3)
(49)

(Apostol 1997, p. 23).

A duplication formula is obtained as follows.

P(2z)=lim_(y->z)P(y+z)=1/4lim_(y->z)[(P^'(z)-P^'(y))/(P(z)-P(y))]^2-P(z)-lim_(y->z)P(y)
(50)
=1/4lim_(h->0)[(P(z)-P^'(z+h))/(P(z)-P(z+h))]^2-2P(z)
(51)
=1/4{[lim_(h->0)(P^'(z)-P^'(z+h))/h][lim_(h->0)h/(P(z)-P(z+h))]}^2-2P(z)
(52)
=1/4[(P^('')(z))/(P^'(z))]^2-2P(z)
(53)

(Apostol 1997, p. 24).

A general addition theorem is obtained as follows. Given

P^'(z)=AP(z)+B
(54)
P^'(y)=AP(y)+B
(55)

with zero y and z where z≢+/-y(mod 2omega_1,2omega_2), find the third zero zeta. Consider P^'(zeta)-AP(zeta)-B. This has a pole of order three at zeta=0, but the sum of zeros (=0) equals the sum of poles for an elliptic function, so z+y+zeta=0 and zeta=-z-y.

P^'(-z-y)=AP(-z-y)+B
(56)
-P^'(z+y)=AP(z+y)+B.
(57)

Combining (◇), (◇), and (◇) gives

[P(z) P^'(z) 1; P(y) P^'(y) 1; P(z+y) -P^'(z+y) 1][A; -1; B]=[0; 0; 0],
(58)

so

|P(z) P^'(z) 1; P(y) P^'(y) 1; P(z+y) -P^'(z+y) 1|=0.
(59)

Defining u+v+w=0 where u=z and v=y gives the symmetric form

|P(u) P^'(u) 1; P(v) P^'(v) 1; P(w) P^'(w) 1|=0
(60)

(Whittaker and Watson 1990, p. 440). To get the expression explicitly, start again with

P^'(zeta)-AP(zeta)-B=0,
(61)

where zeta=z,y,-z-y.

P^'^2(zeta)-[AP(zeta)+B]^2=0.
(62)

But from (◇), P^('2)(zeta)=4P^3(zeta)-g_2P(zeta)-g_3, so

4P^3(zeta)-A^2P^2(zeta)-(2AB+g_2)P(zeta)-(B^2+g_3)=0.
(63)

The solutions P(zeta)=z are given by

4z^3-A^2z^2-(2AB+g_2)z-(B^2+g_3)=0.
(64)

But the sum of roots equals the coefficient of the squared term, so

P(z)+P(y)+P(z+y)=1/4A^2
(65)
P^'(z)-P^'(y)=A[P(z)-P(y)]
(66)
A=(P^'(z)-P^'(y))/(P(z)-P(y))
(67)
P(z+y)=1/4[(P^'(z)-P^'(y))/(P(z)-P(y))]^2-P(z)-P(y)
(68)

(Whittaker and Watson 1990, p. 441).

Half-period identities include

x=P(1/2omega_1)
(69)
=P(-homega_1+omega_1)
(70)
=e_1+((e_1-e_2)(e_1-e_3))/(P(-1/2omega_1)-e_1)
(71)
=e_1+((e_1-e_2)(e_1-e_3))/(x-e_1).
(72)

Multiplying through,

x^2-e_1x=e_1x-e_1^2+(e_1-e_2)(e_1-e_3)
(73)
x^2-2e_1x+[e_1^2-(e_1-e_2)(e_1-e_3)]=0,
(74)

which gives

P(1/2omega_1)=1/2{2e_1+/-sqrt(4e_1^2-4[e_1^2-(e_1-e_2)(e_1-e_3)])}
(75)
=e_1+/-sqrt((e_1-e_2)(e_1-e_3)).
(76)

From Whittaker and Watson (1990, p. 445),

P^'(1/2omega_1)=-2sqrt((e_1-e_2)(e_1-e_3))(sqrt(e_1-e_2)+sqrt(e_1-e_3)).
(77)

The function is homogeneous,

P(lambdaz|lambdaomega_1,lambdaomega_2)=lambda^(-2)P(z|omega_1,omega_2)
(78)
P(lambdaz;lambda^(-4)g_2,lambda^(-6)g_3)=lambda^(-2)P(z;g_2,g_3).
(79)

To invert the function, find 2omega_1 and 2omega_2 of P(z|omega_1,omega_2) when given P(z;g_2,g_3). Let e_1, e_2, and e_3 be the roots such that (e_1-e_2)/(e_1-e_3) is not a real number >1 or <0. Determine the half-period ratio tau from

(e_1-e_2)/(e_1-e_3)=(theta_4^4(0|tau))/(theta_3^4(0|tau)).
(80)

Now pick

A=(sqrt(e_1-e_2))/(theta_4^2(0|tau)).
(81)

As long as g_2^3!=27g_3, the periods are then

2omega_1=piA
(82)
2omega_2=(pitau)/A.
(83)

Weierstrass elliptic functions can be expressed in terms of Jacobi elliptic functions by

P(u;g_2,g_3)=e_3+(e_1-e_3)ns^2(usqrt(e_1-e_3),sqrt((e_2-e_3)/(e_1-e_3))),
(84)

where

P(omega_1)=e_1
(85)
P(omega_2)=e_2
(86)
P(omega_3)=-P(-omega_1-omega_2)=e_3,
(87)

and the elliptic invariants are

g_2=60sum^'_(m,n)Omega_(mn)^(-4)
(88)
g_3=140sum^'_(m,n)Omega_(mn)^(-6).
(89)

Here, Omega_(mn)=2momega_1-2nomega_2.

An addition formula for the Weierstrass elliptic function can be derived as follows (Whittaker and Watson 1990, p. 444).

P(z+omega_1)+P(z)+P(omega_1)=1/4[(P^'(z)-P^'(omega_1))/(P(z)-P(omega_1))]^2=1/4(P^'^2(z))/([P(z)-e_1]^2).
(90)

Use

P^('2)(z)=4product_(r=1)^3[P(z)-e_r],
(91)

so

P(z+omega_1)=-P(z)-e_1+1/4(4product_(r=1)^(3)[P(z)-e_r])/([P(z)-e_1]^2)
(92)
=-P(z)-e_1+([P(z)-e_2][P(z)-e_3])/(P(z)-e_1).
(93)

Use sum_(r=1)^(3)e_r=0,

P(z+omega_1)=e_1+([-2e_1-P(z)][P(z)-e_1])/(P(z)-e_1)+(P^2(z)-P(z)(e_2+e_3)+e_2e_3)/(P(z)-e_1)
(94)
=e_1+(-P(z)(e_1+e_2+e_3)+e_2e_3+2e_1^2)/(P(z)-e_1).
(95)

But sum_(r=1)^(3)e_r=0 and

2e_1^2+e_2e_3=e_1^2-e_1(e_2+e_3)+e_2e_3=(e_1-e_2)(e_1-e_3),
(96)

so

P(z+omega_1)=e_1+((e_1-e_2)(e_1-e_3))/(P(z)-e_1).
(97)

The periods of the Weierstrass elliptic function are given as follows. When g_2 and g_3 are real and g_2^3-27g_3^2>0, then e_1, e_2, and e_3 are real and defined such that e_1>e_2>e_3.

omega_1=int_(e_1)^infty(4t^3-g_2t-g_3)^(-1/2)dt
(98)
omega_3=-iint_(-infty)^(e_3)(g_3+g_2t-4t^3)^(-1/2)dt
(99)
omega_2=-omega_1-omega_3.
(100)

The roots of the Weierstrass elliptic function satisfy

e_1=P(omega_1)
(101)
e_2=P(omega_2)
(102)
e_3=P(omega_3),
(103)

where omega_3=-omega_1-omega_2. The e_is are roots of 4t^3-g_2t-g_3 and are unequal so that e_1!=e_2!=e_3. They can be found from the relationships

e_1+e_2+e_3=-a_2=0
(104)
e_2e_3+e_3e_1+e_1e_2=a_1=-1/4g_2
(105)
e_1e_2e_3=-a_0=1/4g_3.
(106)

See also

Elliptic Curve, Elliptic Function, Eisenstein Series, Elliptic Invariants, Equianharmonic Case, Half-Period, Half-Period Ratio, Jacobi Elliptic Functions, Lemniscate Case, omega2 Constant, Pseudolemniscate Case, Weierstrass Sigma Function, Weierstrass Zeta Function

Related Wolfram sites

http://functions.wolfram.com/EllipticFunctions/WeierstrassP/, http://functions.wolfram.com/EllipticFunctions/WeierstrassPPrime/, http://functions.wolfram.com/EllipticFunctions/WeierstrassHalfPeriods/, http://functions.wolfram.com/EllipticFunctions/InverseWeierstrassP/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Weierstrass Elliptic and Related Functions." Ch. 18 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 627-671, 1972.Apostol, T. M. "The Weierstrass P Function," "The Laurent Expansion of P Near the Origin," "Differential Equation Satisfied by P," "The Eisenstein Series and the Invariants g_2 and g_3," "The Numbers e_1, e_2, and e_3," and "The Discriminant Delta." §1.6-1.11 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 9-14, 1997.Brezhnev, Y. V. "Uniformisation: On the Burnside Curve y^2=x^5-x." 9 Dec 2001. http://arxiv.org/abs/math.CA/0111150.Eichler, M. and Zagier, D. "On the Zeros of the Weierstrass P-Function." Math. Ann. 258, 399-407, 1982.Fischer, G. (Ed.). Plates 129-131 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 126-128, 1986.Huang, J. "Integral Representation of Harmonic Lattice Sums." J. Math. Phys. 40, 5240-5246, 1999.Rainville, E. D. Special Functions. New York: Chelsea, 1971.Tölke, F. "Spezielle Weierstraßsche P-Funktionen." Ch. 4 in Praktische Funktionenlehre, zweiter Band: Theta-Funktionen und spezielle Weierstraßsche Funktionen. Berlin: Springer-Verlag, pp. 115-244, 1966.Tölke, F. Praktische Funktionenlehre, fünfter Band: Allgemeine Weierstraßsche Funktionen und Ableitungen nach dem Parameter. Integrale der Theta-Funktionen und Bilinear-Entwicklungen. Berlin: Springer-Verlag, 1968.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.Woods, F. S. "The Function p(u)." §160 in Advanced Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied Mathematics. Boston, MA: Ginn, pp. 381-382, 1926.

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Weierstrass Elliptic Function

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Weisstein, Eric W. "Weierstrass Elliptic Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WeierstrassEllipticFunction.html

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