Weierstrass's elliptic functions

In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form; they are named for Karl Weierstrass. This class of functions are also referred to as p-functions and generally written using the symbol ℘. The ℘ functions constitute branched double coverings of the Riemann sphere by the torus, ramified at four points. They can be used to parametrize elliptic curves over the complex numbers, thus establishing an equivalence to complex tori. Genus one solutions of differential equations can be written in terms of Weierstrass elliptic functions. Notably, the simplest periodic solutions of the Korteweg–de Vries equation are often written in terms of Weierstrass p-functions.

Symbol for Weierstrass P function

Model of Weierstrass p-function

Contents

• 
  1 Definitions
  


• 
  2 Invariants
  
  • 
    2.1 Special cases
    
  
  


• 
  3 Differential equation
  


• 
  4 Integral equation
  


• 
  5 Modular discriminant
  


• 
  6 The constants e₁, e₂ and e₃
  


• 
  7 Addition theorems
  


• 
  8 The case with 1 a basic half-period
  


• 
  9 General theory
  


• 
  10 Relation to Jacobi elliptic functions
  


• 
  11 Typography
  


• 
  12 Footnotes
  


• 
  13 References
  


• 
  14 External links
  

Definitions

Weierstrass P function defined over a subset of the complex plane using a standard visualization technique in which white corresponds to a pole, black to a zero, and maximal saturation to |f(z)|=|f(x+iy)|=1.f(x+iy)right Note the regular lattice of poles, and two interleaving lattices of zeros.

The Weierstrass elliptic function can be defined in three closely related ways, each of which possesses certain advantages. One is as a function of a complex variable z and a lattice Λ in the complex plane. Another is in terms of z and two complex numbers ω₁ and ω₂ defining a pair of generators, or periods, for the lattice. The third is in terms of z and a modulus τ in the upper half-plane. This is related to the previous definition by τ = ω₂/ω₁, which by the conventional choice on the pair of periods is in the upper half-plane. Using this approach, for fixed z the Weierstrass functions become modular functions of τ.

In terms of the two periods, Weierstrass's elliptic function is an elliptic function with periods ω₁ and ω₂ defined as

℘(z;ω1,ω2)=1z2+∑n2+m2≠01(z+mω1+nω2)2−1(mω1+nω2)2.displaystyle wp (z;omega _1,omega _2)=frac 1z^2+sum _n^2+m^2neq 0leftfrac 1(z+momega _1+nomega _2)^2-frac 1left(momega _1+nomega _2right)^2right.

Then Λ=mω1+nω2:m,n∈Zdisplaystyle Lambda =momega _1+nomega _2:m,nin mathbb Z are the points of the period lattice, so that

℘(z;Λ)=℘(z;ω1,ω2)displaystyle wp (z;Lambda )=wp (z;omega _1,omega _2)

for any pair of generators of the lattice defines the Weierstrass function as a function of a complex variable and a lattice.

If τdisplaystyle tau is a complex number in the upper half-plane, then

℘(z;τ)=℘(z;1,τ)=1z2+∑n2+m2≠01(z+m+nτ)2−1(m+nτ)2.displaystyle wp (z;tau )=wp (z;1,tau )=frac 1z^2+sum _n^2+m^2neq 0left1 over (z+m+ntau )^2-1 over (m+ntau )^2right.

The above sum is homogeneous of degree minus two, from which we may define the Weierstrass ℘ function for any pair of periods, as

℘(z;ω1,ω2)=℘(zω1;ω2ω1)ω12.displaystyle wp (z;omega _1,omega _2)=frac wp (frac zomega _1;frac omega _2omega _1)omega _1^2.

We may compute ℘ very rapidly in terms of theta functions; because these converge so quickly, this is a more expeditious way of computing
℘ than the series we used to define it. The formula here is

℘(z;τ)=π2ϑ2(0;τ)ϑ102(0;τ)ϑ012(z;τ)ϑ112(z;τ)−π23[ϑ4(0;τ)+ϑ104(0;τ)]displaystyle wp (z;tau )=pi ^2vartheta ^2(0;tau )vartheta _10^2(0;tau )vartheta _01^2(z;tau ) over vartheta _11^2(z;tau )-pi ^2 over 3left[vartheta ^4(0;tau )+vartheta _10^4(0;tau )right]

There is a second-order pole at each point of the period lattice (including the origin). With these definitions, ℘(z)displaystyle wp (z) is an even function and its derivative with respect to z, ℘′, is an odd function.

Further development of the theory of elliptic functions shows that Weierstrass's function is determined up to addition of a constant and multiplication by a non-zero constant by the position and type of the poles alone, amongst all meromorphic functions with the given period lattice.

Invariants

The real part of the invariant g₃ as a function of the nome q on the unit disk.

The imaginary part of the invariant g₃ as a function of the nome q on the unit disk.

In a punctured neighborhood of the origin, the Laurent series expansion of ℘displaystyle wp is

℘(z;ω1,ω2)=z−2+120g2z2+128g3z4+O(z6)displaystyle wp (z;omega _1,omega _2)=z^-2+frac 120g_2z^2+frac 128g_3z^4+O(z^6)

where

g2=60∑(m,n)≠(0,0)(mω1+nω2)−4displaystyle g_2=60sum _(m,n)neq (0,0)(momega _1+nomega _2)^-4

g3=140∑(m,n)≠(0,0)(mω1+nω2)−6.displaystyle g_3=140sum _(m,n)neq (0,0)(momega _1+nomega _2)^-6.

The numbers g₂ and g₃ are known as the invariants. The summations after the coefficients 60 and 140 are the first two Eisenstein series, which are modular forms when considered as functions G₄(τ) and G₆(τ), respectively, of τ = ω₂/ω₁ with Im(τ) > 0.

Note that g₂ and g₃ are homogeneous functions of degree −4 and −6; that is,

g2(λω1,λω2)=λ−4g2(ω1,ω2)displaystyle g_2(lambda omega _1,lambda omega _2)=lambda ^-4g_2(omega _1,omega _2)

g3(λω1,λω2)=λ−6g3(ω1,ω2).displaystyle g_3(lambda omega _1,lambda omega _2)=lambda ^-6g_3(omega _1,omega _2).

Thus, by convention, one frequently writes g2displaystyle g_2 and g3displaystyle g_3 in terms of the period ratio τ=ω2/ω1displaystyle tau =omega _2/omega _1 and take τdisplaystyle tau to lie in the upper half-plane. Thus, g2(τ)=g2(1,ω2/ω1)displaystyle g_2(tau )=g_2(1,omega _2/omega _1) and g3(τ)=g3(1,ω2/ω1)displaystyle g_3(tau )=g_3(1,omega _2/omega _1).

The Fourier series for g2displaystyle g_2 and g3displaystyle g_3 can be written in terms of the square of the nome q=exp⁡(iπτ)displaystyle q=exp(ipi tau ) as

g2(τ)=43π4[1+240∑k=1∞σ3(k)q2k]displaystyle g_2(tau )=frac 43pi ^4left[1+240sum _k=1^infty sigma _3(k)q^2kright]

g3(τ)=827π6[1−504∑k=1∞σ5(k)q2k]displaystyle g_3(tau )=frac 827pi ^6left[1-504sum _k=1^infty sigma _5(k)q^2kright]

where σa(k)displaystyle sigma _a(k) is the divisor function. This formula may be rewritten in terms of Lambert series.

The invariants may be expressed in terms of Jacobi's theta functions. This method is very convenient for numerical calculation: the theta functions converge very quickly. In the notation of Abramowitz and Stegun, but denoting the primitive periods by ω1,ω2displaystyle omega _1,omega _2, the invariants satisfy

g2(ω1,ω2)=43(πω1)4(a8−a4b4+b8)=23(πω1)4(a8+b8+c8)displaystyle g_2(omega _1,omega _2)=frac 43left(frac pi omega _1right)^4(a^8-a^4b^4+b^8)=frac 23left(frac pi omega _1right)^4(a^8+b^8+c^8)

g3(ω1,ω2)=827(πω1)6(a12−32a8b4−32a4b8+b12)displaystyle g_3(omega _1,omega _2)=frac 827left(frac pi omega _1right)^6(a^12-frac 32a^8b^4-frac 32a^4b^8+b^12)^{[citation needed]}

where

a=θ2(0;q)=ϑ10(0;τ)displaystyle a=theta _2(0;q)=vartheta _10(0;tau )

b=θ3(0;q)=ϑ00(0;τ)displaystyle b=theta _3(0;q)=vartheta _00(0;tau )

c=θ4(0;q)=ϑ01(0;τ)displaystyle c=theta _4(0;q)=vartheta _01(0;tau )

and τ=ω2/ω1displaystyle tau =omega _2/omega _1 is the period ratio, q=eπiτdisplaystyle q=e^pi itau is the nome, and θmdisplaystyle theta _m and ϑmndisplaystyle vartheta _mn are alternative notations.

Special cases

If the invariants are g₂ = 0, g₃ = 1, then this is known as the equianharmonic case; g₂ = 1, g₃ = 0 is the lemniscatic case.

Differential equation

With this notation, the ℘ function satisfies the following differential equation:

[℘′(z)]2=4[℘(z)]3−g2℘(z)−g3,displaystyle [wp '(z)]^2=4[wp (z)]^3-g_2wp (z)-g_3,,

where dependence on ω1displaystyle omega _1 and ω2displaystyle omega _2 is suppressed.

This relation can be quickly verified by comparing the poles of both sides, for example, the pole at z = 0 of lhs is

[℘′(z)]2|z=0∼4z6−24z2∑1(mω1+nω2)4−80∑1(mω1+nω2)6displaystyle [wp '(z)]^2Big _z=0sim frac 4z^6-frac 24z^2sum frac 1(momega _1+nomega _2)^4-80sum frac 1(momega _1+nomega _2)^6

while the pole at z = 0 of

[℘(z)]3|z=0∼1z6+9z2∑1(mω1+nω2)4+15∑1(mω1+nω2)6.displaystyle [wp (z)]^3Big _z=0sim frac 1z^6+frac 9z^2sum frac 1(momega _1+nomega _2)^4+15sum frac 1(momega _1+nomega _2)^6.

Comparing these two yields the relation above.

Integral equation

The Weierstrass elliptic function can be given as the inverse of an elliptic integral. Let

u=∫y∞ds4s3−g2s−g3.displaystyle u=int _y^infty frac dssqrt 4s^3-g_2s-g_3.

Here, g₂ and g₃ are taken as constants. Then one has

y=℘(u).displaystyle y=wp (u).

The above follows directly by integrating the differential equation.

Modular discriminant

The real part of the discriminant as a function of the nome q on the unit disk.

The modular discriminant Δ is defined as the quotient by 16 of the discriminant of the right-hand side of the above differential equation:

Δ=g23−27g32.displaystyle Delta =g_2^3-27g_3^2.,

This is studied in its own right, as a cusp form, in modular form theory (that is, as a function of the period lattice).

Note that Δ=(2π)12η24displaystyle Delta =(2pi )^12eta ^24 where ηdisplaystyle eta is the Dedekind eta function.

The presence of 24 can be understood by connection with other occurrences, as in the eta function and the Leech lattice.

The discriminant is a modular form of weight 12. That is, under the action of the modular group, it transforms as

Δ(aτ+bcτ+d)=(cτ+d)12Δ(τ)displaystyle Delta left(frac atau +bctau +dright)=left(ctau +dright)^12Delta (tau )

with τ being the half-period ratio, and a,b,c and d being integers, with ad − bc = 1.

For the Fourier coefficients of Δdisplaystyle Delta , see Ramanujan tau function.

The constants e₁, e₂ and e₃

Consider the cubic polynomial equation 4t³ − g₂t − g₃ = 0 with roots e₁, e₂, and e₃. Its discriminant is 16 times the modular discriminant Δ = g₂³ − 27g₃². If it is not zero, no two of these roots are equal. Since the quadratic term of this cubic polynomial is zero, the roots are related by the equation

e1+e2+e3=0.displaystyle e_1+e_2+e_3=0.,

The linear and constant coefficients (g₂ and g₃, respectively) are related to the roots by the equations (see Elementary symmetric polynomial).^[1]

g2=−4(e1e2+e1e3+e2e3)=2(e12+e22+e32)displaystyle g_2=-4left(e_1e_2+e_1e_3+e_2e_3right)=2left(e_1^2+e_2^2+e_3^2right)

g3=4e1e2e3displaystyle g_3=4e_1e_2e_3

The roots e₁, e₂, and e₃ of the equation 4x3−g2x−g3=0displaystyle 4x^3-g_2x-g_3=0 depend on τ and can be expressed in terms of theta functions. As before, let,

a=θ2(0;eπiτ)=ϑ10(0;τ)displaystyle a=theta _2(0;e^pi itau )=vartheta _10(0;tau )

b=θ3(0;eπiτ)=ϑ00(0;τ)displaystyle b=theta _3(0;e^pi itau )=vartheta _00(0;tau )

c=θ4(0;eπiτ)=ϑ01(0;τ)displaystyle c=theta _4(0;e^pi itau )=vartheta _01(0;tau )

then

e1(τ)=π23(b4+c4)displaystyle e_1(tau )=tfrac pi ^23(b^4+c^4)

e2(τ)=π23(−a4−b4)displaystyle e_2(tau )=tfrac pi ^23(-a^4-b^4)

e3(τ)=π23(a4−c4)displaystyle e_3(tau )=tfrac pi ^23(a^4-c^4)

Since g2=2(e12+e22+e32)displaystyle g_2=2left(e_1^2+e_2^2+e_3^2right) and g3=4e1e2e3displaystyle g_3=4e_1e_2e_3, then these can also be expressed as theta functions. In simplified form,

g2(τ)=23π4(a8+b8+c8)displaystyle g_2(tau )=tfrac 23pi ^4(a^8+b^8+c^8)

g3(τ)=427π6(a8+b8+c8)3−54(abc)82displaystyle g_3(tau )=tfrac 427pi ^6sqrt frac (a^8+b^8+c^8)^3-54(abc)^82

Δ=g23−27g32=16π12a8b8c8=(2π)12η24(τ)displaystyle Delta =g_2^3-27g_3^2=16pi ^12a^8b^8c^8=(2pi )^12eta ^24(tau )

Where ηdisplaystyle eta is the Dedekind eta function. In the case of real invariants, the sign of Δ = g₂³ − 27g₃² determines the nature of the roots. If Δ>0displaystyle Delta >0, all three are real and it is conventional to name them so that e1>e2>e3displaystyle e_1>e_2>e_3. If Δ<0displaystyle Delta <0, it is conventional to write e1=−α+βidisplaystyle e_1=-alpha +beta i (where α≥0displaystyle alpha geq 0, β>0displaystyle beta >0), whence e3=e1¯displaystyle e_3=overline e_1, and e2displaystyle e_2 is real and non-negative.

The half-periods ω₁/2 and ω₂/2 of Weierstrass' elliptic function are related to the roots

℘(ω12)=e1℘(ω22)=e2℘(ω32)=e3displaystyle wp left(frac omega _12right)=e_1qquad wp left(frac omega _22right)=e_2qquad wp left(frac omega _32right)=e_3

where ω3=−(ω1+ω2)displaystyle omega _3=-(omega _1+omega _2). Since the square of the derivative of Weierstrass' elliptic function equals the above cubic polynomial of the function's value, ℘′(ωi2)2=℘′(ωi2)=0displaystyle wp 'left(frac omega _i2right)^2=wp 'left(frac omega _i2right)=0 for i=1,2,3displaystyle i=1,2,3. Conversely, if the function's value equals a root of the polynomial, the derivative is zero.

If g₂ and g₃ are real and Δ > 0, the e_i are all real, and ℘()displaystyle wp () is real on the perimeter of the rectangle with corners 0, ω₃, ω₁ + ω₃, and ω₁. If the roots are ordered as above (e₁ > e₂ > e₃), then the first half-period is completely real

ω12=∫e1∞dz4z3−g2z−g3displaystyle frac omega _12=int _e_1^infty frac dzsqrt 4z^3-g_2z-g_3

whereas the third half-period is completely imaginary

ω32=i∫−e3∞dz4z3−g2z+g3.displaystyle frac omega _32=iint _-e_3^infty frac dzsqrt 4z^3-g_2z+g_3.

Addition theorems

The Weierstrass elliptic functions have several properties that may be proved:

det[℘(z)℘′(z)1℘(y)℘′(y)1℘(z+y)−℘′(z+y)1]=0displaystyle det beginbmatrixwp (z)&wp '(z)&1\wp (y)&wp '(y)&1\wp (z+y)&-wp '(z+y)&1endbmatrix=0

A symmetrical version of the same identity is

det[℘(u)℘′(u)1℘(v)℘′(v)1℘(w)℘′(w)1]=0 if u+v+w=0.displaystyle det beginbmatrixwp (u)&wp '(u)&1\wp (v)&wp '(v)&1\wp (w)&wp '(w)&1endbmatrix=0text if u+v+w=0.

Also

℘(z+y)=14℘′(z)−℘′(y)℘(z)−℘(y)2−℘(z)−℘(y)displaystyle wp (z+y)=frac 14leftfrac wp '(z)-wp '(y)wp (z)-wp (y)right^2-wp (z)-wp (y)

and the duplication formula

℘(2z)=14℘″(z)℘′(z)2−2℘(z),displaystyle wp (2z)=frac 14leftfrac wp ''(z)wp '(z)right^2-2wp (z),

unless 2z is a period.

The case with 1 a basic half-period

If ω1=1displaystyle omega _1=1, much of the above theory becomes simpler; it is then conventional to
write τdisplaystyle tau for ω2displaystyle omega _2. For a fixed τ in the upper half-plane, so that the imaginary part of τ is positive, we define the
Weierstrass ℘ function by

℘(z;τ)=1z2+∑(m,n)≠(0,0)1(z+m+nτ)2−1(m+nτ)2.displaystyle wp (z;tau )=frac 1z^2+sum _(m,n)neq (0,0)1 over (z+m+ntau )^2-1 over (m+ntau )^2.

The sum extends over the lattice n+mτ : n and m in Z with the origin omitted.
Here we regard τ as fixed and ℘ as a function of z; fixing z and letting τ vary leads into the area of elliptic modular functions.

General theory

℘ is a meromorphic function in the complex plane with a double pole at each lattice point. It is doubly periodic with periods 1 and τ; this means that
℘ satisfies

℘(z+1)=℘(z+τ)=℘(z).displaystyle wp (z+1)=wp (z+tau )=wp (z).

The above sum is homogeneous of degree minus two, and if c is any non-zero complex number,

℘(cz;cτ)=℘(z;τ)/c2displaystyle wp (cz;ctau )=wp (z;tau )/c^2

from which we may define the Weierstrass ℘ function for any pair of periods. We also may take the derivative (of course, with respect to z) and obtain a function algebraically related to ℘ by

℘′2=4℘3−g2℘−g3displaystyle wp '^2=4wp ^3-g_2wp -g_3

where g2displaystyle g_2 and g3displaystyle g_3 depend only on τ, being modular forms. The equation

Y2=4X3−g2X−g3displaystyle Y^2=4X^3-g_2X-g_3

defines an elliptic curve, and we see that (℘,℘′)displaystyle (wp ,wp ') is a parametrization of that curve. The totality of meromorphic doubly periodic functions with given periods defines an algebraic function field associated to that curve. It can be shown that this field is

C(℘,℘′),displaystyle mathbb C(wp ,wp '),

so that all such functions are rational functions in the Weierstrass function and its derivative.

One can wrap a single period parallelogram into a torus, or donut-shaped Riemann surface, and regard the elliptic functions associated to a given pair of periods to be functions defined on that Riemann surface.

℘ can also be expressed in terms of theta functions; because these converge very rapidly, this is a more expeditious way of computing ℘ than the series used to define it.

℘(z;τ)=π2ϑ2(0;τ)ϑ102(0;τ)ϑ012(z;τ)ϑ112(z;τ)+e2(τ).displaystyle wp (z;tau )=pi ^2vartheta ^2(0;tau )vartheta _10^2(0;tau )vartheta _01^2(z;tau ) over vartheta _11^2(z;tau )+e_2(tau ).

The function ℘ has two zeros (modulo periods) and the function ℘′ has three. The zeros of ℘′ are easy to find: since ℘′ is an odd function they must be at the half-period points. On the other hand, it is very difficult to express the zeros of ℘ by closed formula, except for special values of the modulus (e.g. when the period lattice is the Gaussian integers). An expression was found, by Zagier and Eichler.^[2]

The Weierstrass theory also includes the Weierstrass zeta function, which is an indefinite integral of ℘ and not doubly periodic, and a theta function called the Weierstrass sigma function, of which his zeta-function is the log-derivative. The sigma-function has zeros at all the period points (only), and can be expressed in terms of Jacobi's functions. This gives one way to convert between Weierstrass and Jacobi notations.

The Weierstrass sigma-function is an entire function; it played the role of 'typical' function in a theory of random entire functions of J. E. Littlewood.

Relation to Jacobi elliptic functions

For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of Jacobi's elliptic functions. The basic relations are^[3]

℘(z)=e3+e1−e3sn2⁡w=e2+(e1−e3)dn2⁡wsn2⁡w=e1+(e1−e3)cn2⁡wsn2⁡wdisplaystyle wp (z)=e_3+frac e_1-e_3operatorname sn ^2w=e_2+(e_1-e_3)frac operatorname dn ^2woperatorname sn ^2w=e_1+(e_1-e_3)frac operatorname cn ^2woperatorname sn ^2w

where e_1–3 are the three roots described above and where the modulus k of the Jacobi functions equals

k≡e2−e3e1−e3displaystyle kequiv sqrt frac e_2-e_3e_1-e_3

and their argument w equals

w≡ze1−e3.displaystyle wequiv zsqrt e_1-e_3.

Typography

The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘.^{[footnote 1]}

In computing, the letter ℘ is available as wp in TeX. In Unicode the code point is U+2118, and the name is "script capital p", with the more correct alias "weierstrass elliptic function".^{[footnote 2]} In HTML, it can be escaped as ℘.

Footnotes

1. 
   ^
   
   This sybmol was used already at least in 1890. The first edition of A Course of Modern Analysis by E. T. Whittaker in 1902 also used it.^[4]
   


2. 
   ^
   
   The original name "script capital p" is bad in two ways. First, it is wrong, because the letter is in fact lowercase. The second reason is it is not a "script" class letter, like .mw-parser-output .monospacedfont-family:monospace,monospace
   U+1D4C5 𝓅 .mw-parser-output .smallcapsfont-variant:small-caps
   MATHEMATICAL SCRIPT SMALL P, but the letter for the Weierstrass's elliptic function.
   
   Unicode added the alias as a correction, but it is not yet the best; Unicode.org says it should have been called Weierstrass elliptic function symbol (or calligraphic small p.)^[5][6]
   

References

1. 
   ^ Abramowitz and Stegun, p. 629
   


2. 
   ^ Eichler, M.; Zagier, D. (1982). "On the zeros of the Weierstrass ℘-Function". Mathematische Annalen. 258 (4): 399–407. doi:10.1007/BF01453974..mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
   


3. 
   ^ Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw–Hill. p. 721. LCCN 59014456.
   


4. 
   ^ teika kazura (2017-08-17), The letter ℘ Name & origin?, MathOverflow, retrieved 2018-08-30
   


5. 
   ^ "Known Anomalies in Unicode Character Names". Unicode Technical Note #27. version 4. Unicode, Inc. 2017-04-10. Retrieved 2017-07-20.
   


6. 
   ^ "NameAliases-10.0.0.txt". Unicode, Inc. 2017-05-06. Retrieved 2017-07-20.
   

• 
  Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 18". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 627. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
  


• 
  N. I. Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island
  ISBN 0-8218-4532-2
  


• 
  Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York
  ISBN 0-387-97127-0 (See chapter 1.)


• K. Chandrasekharan, Elliptic functions (1980), Springer-Verlag
  ISBN 0-387-15295-4
  


• 
  Konrad Knopp, Funktionentheorie II (1947), Dover Publications; Republished in English translation as Theory of Functions (1996), Dover Publications
  ISBN 0-486-69219-1
  


• 
  Serge Lang, Elliptic Functions (1973), Addison-Wesley,
  ISBN 0-201-04162-6
  


• 
  E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge University Press, 1952, chapters 20 and 21

External links

Wikimedia Commons has media related to Weierstrass's elliptic functions.

• 
  Hazewinkel, Michiel, ed. (2001) [1994], "Weierstrass elliptic functions", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  


• 
  Weierstrass's elliptic functions on Mathworld.


• Chapter 23, Weierstrass Elliptic and Modular Functions in DLMF (Digital Library of Mathematical Functions) by W. P. Reinhardt and P. L. Walker.

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