View source for Weierstrass's elliptic functions

In [[mathematics]], '''Weierstrass's elliptic functions''' are [[elliptic function]]s 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 ℘ (a calligraphic lowercase p). The ℘ functions constitute [[Branched covering|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 torus|complex tori]]. [[Genus_(mathematics)|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.
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[[Image:Weierstrass p.svg|100px|Symbol for Weierstrass P function]]<div class="thumbcaption">
Symbol for Weierstrass P function
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[[File:Modell der Weierstraßschen p-Funktion -Schilling, XIV, 7ab, 8 - 313, 314-.jpg|thumb|right|Model of Weierstrass p-function]]

==Definitions==
[[File:Weierstrass elliptic function P.png|thumb|200px|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 (color theory)|saturation]] to <math>\left|f(z)\right|=\left|f(x+iy)\right|=1\;.</math>  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 (group)|lattice]] Λ in the complex plane.  Another is in terms of ''z'' and two complex numbers ω<sub>1</sub> and ω<sub>2</sub> defining a pair of generators, or periods, for the lattice.  The third is in terms of ''z'' and a modulus {{mvar|&tau;}} in the [[upper half-plane]]. This is related to the previous definition by {{math|1=''τ'' = ''ω''<sub>2</sub>/''ω''<sub>1</sub>}}, 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 function]]s of {{mvar|&tau;}}.

In terms of the two periods, '''Weierstrass's elliptic function''' is an elliptic function with periods {{math|''ω''<sub>1</sub>}} and {{math|''ω''<sub>2</sub>}} defined as

:<math>\wp(z;\omega_1,\omega_2)=\frac{1}{z^2} + \sum_{n^2+m^2 \ne 0} \left\{\frac{1}{(z+m\omega_1+n\omega_2)^2} - \frac{1}{\left(m\omega_1+n\omega_2\right)^2}\right\}.</math>

Then <math>\Lambda=\{m\omega_1+n\omega_2:m,n\in\mathbb{Z}\}</math> are the points of the '''[[period lattice]]''', so that

:<math>\wp(z;\Lambda)=\wp(z;\omega_1,\omega_2)</math>

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

If <math>\tau</math> is a complex number in the upper half-plane, then

:<math>\wp(z;\tau) = \wp(z;1,\tau) = \frac{1}{z^2} + \sum_{n^2+m^2 \ne 0}\left\{
{1 \over (z+m+n\tau)^2} - {1 \over (m+n\tau)^2}\right\}.</math>

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

:<math>\wp(z;\omega_1,\omega_2) = \frac{\wp(\frac z {\omega_1}; \frac{\omega_2}{\omega_1})}{\omega_1^2}.</math>

We may compute ℘ very rapidly in terms of [[theta function]]s; 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

:<math>\wp(z; \tau) = \pi^2 \vartheta^2(0;\tau) \vartheta_{10}^2(0;\tau){\vartheta_{01}^2(z;\tau) \over \vartheta_{11}^2(z;\tau)}-{\pi^2 \over 3} \left[\vartheta^4(0;\tau) + \vartheta_{10}^4(0;\tau)\right] </math>

There is a second-order [[pole (complex analysis)|pole]] at each point of the period lattice (including the origin).  With these definitions, <math>\wp(z)</math> 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 function]]s with the given period lattice.

==Invariants==
[[Image:Gee three real.jpeg|thumb|The real part of the invariant ''g''<sub>3</sub> as a function of the nome ''q'' on the unit disk.]]
[[Image:Gee three imag.jpeg|thumb|The imaginary part of the invariant ''g''<sub>3</sub> as a function of the nome ''q'' on the unit disk.]]

In a punctured neighborhood of the origin, the [[Laurent series]] expansion of <math>\wp</math> is

:<math>
\wp(z;\omega_1,\omega_2)=z^{-2}+\frac{1}{20}g_2z^2+\frac{1}{28}g_3z^4+O(z^6)
</math>

where

:<math>g_2= 60\sum_{(m,n) \neq (0,0)} (m\omega_1+n\omega_2)^{-4} </math>

:<math>g_3=140\sum_{(m,n) \neq (0,0)} (m\omega_1+n\omega_2)^{-6}.</math>

The numbers ''g''<sub>2</sub> and ''g''<sub>3</sub> 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 {{math|''G''<sub>4</sub>(''&tau;'')}} and {{math|''G''<sub>6</sub>(''&tau;'')}}, respectively, of {{math|1=''&tau;'' = ''ω''<sub>2</sub>/''ω''<sub>1</sub>}} with {{math|Im(''&tau;'') > 0}}.

Note that ''g''<sub>2</sub> and ''g''<sub>3</sub> are [[homogeneous function]]s of degree &minus;4 and &minus;6; that is,

:<math>g_2(\lambda \omega_1, \lambda \omega_2) = \lambda^{-4} g_2(\omega_1, \omega_2)</math>

:<math>g_3(\lambda \omega_1, \lambda \omega_2) = \lambda^{-6} g_3(\omega_1, \omega_2).</math>

Thus, by convention, one frequently writes <math>g_2</math> and <math>g_3</math> in terms of the [[nome (mathematics)|period ratio]] <math>\tau=\omega_2/\omega_1</math> and take <math>\tau</math> to lie in the [[upper half-plane]]. Thus, <math>g_2(\tau)=g_2(1, \omega_2/\omega_1)</math> and <math>g_3(\tau)=g_3(1, \omega_2/\omega_1)</math>.

The [[Fourier series]] for <math>g_2</math> and <math>g_3</math> can be written in terms of the square of the [[nome (mathematics)|nome]] <math>q=\exp(i\pi\tau)</math> as

:<math>g_2(\tau)=\frac{4}{3}\pi^4 \left[ 1+ 240\sum_{k=1}^\infty \sigma_3(k) q^{2k} \right] </math>

:<math>g_3(\tau)=\frac{8}{27}\pi^6 \left[ 1- 504\sum_{k=1}^\infty \sigma_5(k) q^{2k} \right] </math>

where <math>\sigma_a(k)</math> is the [[divisor function]].  This formula may be rewritten in terms of [[Lambert series]].

The invariants may be expressed in terms of [[theta functions|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 <math>\omega_1,\omega_2</math>, the invariants satisfy

:<math> g_2(\omega_1,\omega_2)=\frac{4}{3}\left(\frac{\pi}{\omega_1}\right)^4(a^8-a^4b^4+b^8)= \frac{2}{3}\left(\frac{\pi}{\omega_1}\right)^4(a^8+b^8+c^8)</math>

:<math>g_3(\omega_1,\omega_2)= \frac{8}{27}\left(\frac{\pi}{\omega_1}\right)^6(a^{12}-\frac32a^8b^4-\frac32a^4b^8+b^{12})</math>{{cite web |url=http://functions.wolfram.com/EllipticFunctions/WeierstrassInvariants/27/02/02/ |title=Wolfram Functions}}

where

:<math>a=\theta_{2}(0; q)=\vartheta_{10}(0; \tau)</math>

:<math>b=\theta_{3}(0; q)=\vartheta_{00}(0; \tau)</math>

:<math>c=\theta_{4}(0; q)=\vartheta_{01}(0; \tau)</math>

and <math>\tau=\omega_2/\omega_1</math> is the [[half-period ratio|period ratio]], <math>q=e^{\pi i\tau}</math> is the nome, and <math>\theta_{m}</math> and <math>\vartheta_{mn}</math> are alternative notations.

=== Special cases ===

If the invariants are ''g''<sub>2</sub> = 0, ''g''<sub>3</sub> = 1, then this is known as the [[equianharmonic]] case; ''g''<sub>2</sub> = 1, ''g''<sub>3</sub> = 0 is the [[lemniscatic elliptic function|lemniscatic]] case.

==Differential equation==
With this notation, the ℘ function satisfies the following [[differential equation]]:

:<math> [\wp'(z)]^2 = 4[\wp(z)]^3-g_2\wp(z)-g_3, \, </math>

where dependence on <math>\omega_1</math> and <math>\omega_2</math> is suppressed.

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

:<math>
[\wp'(z)]^2\Big|_{z=0}\sim \frac{4}{z^6}-\frac{24}{z^2}\sum \frac{1}{(m\omega_1+n\omega_2)^4}-80\sum \frac{1}{(m\omega_1+n\omega_2)^6} </math>

while the pole at ''z''&nbsp;=&nbsp;0 of

:<math>
[\wp(z)]^3\Big|_{z=0}\sim \frac{1}{z^6}+\frac{9}{z^2}\sum \frac{1}{(m\omega_1+n\omega_2)^4}+15\sum \frac{1}{(m\omega_1+n\omega_2)^6}.</math>

Comparing these two yields the relation above.

==Integral equation==
The Weierstrass elliptic function can be given as the inverse of an [[elliptic integral]]. Let

:<math>u = \int_y^\infty \frac {ds} {\sqrt{4s^3 - g_2s -g_3}}.</math>

Here, ''g''<sub>2</sub> and ''g''<sub>3</sub> are taken as constants. Then one has

:<math>y=\wp(u).</math>

The above follows directly by integrating the differential equation.

==Modular discriminant==
[[Image:Discriminant real part.jpeg|thumb|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:

:<math> \Delta=g_2^3-27g_3^2. \, </math>

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 <math>\Delta=(2\pi)^{12}\eta^{24}</math> where <math>\eta</math> is the [[Dedekind eta function]].

The presence of [[24 (number)|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

:<math>\Delta \left( \frac {a\tau+b} {c\tau+d}\right) =
\left(c\tau+d\right)^{12} \Delta(\tau) </math>

with {{mvar|&tau;}} being the half-period ratio, and ''a'',''b'',''c'' and ''d'' being integers, with ''ad''&nbsp;&minus;&nbsp;''bc'' = 1.

For the Fourier coefficients of <math>\Delta</math>, see [[Ramanujan tau function]].

==The constants ''e''<sub>1</sub>, ''e''<sub>2</sub> and ''e''<sub>3</sub>==
Consider the [[cubic function|cubic polynomial equation]] 4''t''<sup>3</sup> &minus; ''g''<sub>2</sub>''t'' &minus; ''g''<sub>3</sub> = 0 with roots ''e''<sub>1</sub>, ''e''<sub>2</sub>, and ''e''<sub>3</sub>. Its discriminant is 16 times the modular discriminant Δ = ''g''<sub>2</sub><sup>3</sup> &minus; 27''g''<sub>3</sub><sup>2</sup>. 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

:<math>
e_1+e_2+e_3=0. \,
</math>

The linear and constant coefficients (''g''<sub>2</sub> and ''g''<sub>3</sub>, respectively) are related to the roots by the equations (see [[Elementary symmetric polynomial]]).<ref>Abramowitz and Stegun, p. 629</ref> 
 
:<math>g_2 = -4 \left( e_1 e_2 + e_1 e_3 + e_2 e_3 \right) = 2 \left( e_1^2 + e_2^2 + e_3^2 \right)</math>

:<math>g_3 = 4 e_1 e_2 e_3</math>

The roots ''e''<sub>1</sub>, ''e''<sub>2</sub>, and ''e''<sub>3</sub> of the equation <math>4 x^3 - g_2 x - g_3=0</math> depend on {{mvar|&tau;}} and can be expressed in terms of [[theta function]]s. As before, let,

:<math>a=\theta_{2}(0; e^{\pi i \tau})=\vartheta_{10}(0; \tau)</math>

:<math>b=\theta_{3}(0; e^{\pi i \tau})=\vartheta_{00}(0; \tau)</math>

:<math>c=\theta_{4}(0; e^{\pi i \tau})=\vartheta_{01}(0; \tau)</math>

then

:<math>e_1(\tau) = \tfrac{\pi^2}{3} (b^4+c^4)</math>
:<math>e_2(\tau) = \tfrac{\pi^2}{3} (-a^4-b^4)</math>
:<math>e_3(\tau) = \tfrac{\pi^2}{3} (a^4-c^4)</math>

Since <math>g_2 = 2 \left( e_1^2 + e_2^2 + e_3^2 \right)</math> and <math>g_3 = 4e_1e_2e_3</math>, then these can also be expressed as theta functions. In simplified form,

:<math>g_2(\tau) = \tfrac{2}{3}\pi^4(a^8+b^8+c^8)</math>

:<math>g_3(\tau) = \tfrac{4}{27}\pi^6 \sqrt{\frac{(a^8+b^8+c^8)^3-54(abc)^8}{2}}</math>

:<math>\Delta = g_2^3-27g_3^2 = 16\pi^{12} a^8b^8c^8 = (2\pi)^{12}\eta^{24}(\tau)</math>

Where <math>\eta</math> is the [[Dedekind eta function]]. In the case of real invariants, the sign of '''Δ''' = ''g''<sub>2</sub><sup>3</sup> &minus; 27''g''<sub>3</sub><sup>2</sup> determines the nature of the roots.  If <math>\Delta>0</math>, all three are real and it is conventional to name them so that <math>e_1>e_2>e_3</math>.  If <math>\Delta<0</math>, it is conventional to write <math>e_1=-\alpha+\beta i</math> (where <math>\alpha\geq 0</math>, <math>\beta>0</math>), whence <math>e_3=\overline{e_1}</math>, and <math>e_2</math> is real and non-negative.

The half-periods ω<sub>1</sub>/2 and ω<sub>2</sub>/2 of Weierstrass' elliptic function are related to the roots
:<math>
\wp\left(\frac{\omega_1}{2}\right)=e_1\qquad
\wp\left(\frac{\omega_2}{2}\right)=e_2\qquad
\wp\left(\frac{\omega_3}{2}\right)=e_3
</math>
where <math>\omega_3=-(\omega_1+\omega_2)</math>.  Since the square of the derivative of Weierstrass' elliptic function equals the above cubic polynomial of the function's value, <math>\wp'\left(\frac{\omega_i}{2}\right)^2=\wp'\left(\frac{\omega_i}{2}\right)=0</math> for <math>i=1,2,3</math>. Conversely, if the function's value equals a root of the polynomial, the derivative is zero.

If ''g''<sub>2</sub> and ''g''<sub>3</sub> are real and Δ &gt; 0, the ''e''<sub>''i''</sub> are all real, and <math>\wp()</math> is real on the perimeter of the rectangle with corners 0, ω<sub>3</sub>, ω<sub>1</sub> + ω<sub>3</sub>, and ω<sub>1</sub>.  If the roots are ordered as above (''e''<sub>1</sub> > ''e''<sub>2</sub> &gt; ''e''<sub>3</sub>), then the first half-period is completely real

:<math>
\frac{\omega_1}{2} = \int_{e_1}^\infty \frac{dz}{\sqrt{4z^3 - g_2 z - g_3}}
</math>

whereas the third half-period is completely imaginary

:<math>
\frac{\omega_3}{2} = i \int_{-e_3}^\infty \frac{dz}{\sqrt{4z^3 - g_2 z + g_3}}.
</math>

==Addition theorems==

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

:<math>
\det\begin{bmatrix}
\wp(z) & \wp'(z) & 1\\
\wp(y) & \wp'(y) & 1\\
\wp(z+y) & -\wp'(z+y) & 1
\end{bmatrix}=0</math>

A symmetrical version of the same identity is
:<math>
\det\begin{bmatrix}
\wp(u) & \wp'(u) & 1\\
\wp(v) & \wp'(v) & 1\\
\wp(w) & \wp'(w) & 1
\end{bmatrix}=0 \text{ if } u+v+w=0. </math>

Also
:<math>
\wp(z+y)=\frac 1 4
\left\{
\frac{\wp'(z)-\wp'(y)}{\wp(z)-\wp(y)}
\right\}^2
-\wp(z)-\wp(y)</math>

and the ''duplication formula''
:<math>
\wp(2z)=
\frac 1 4 \left\{
\frac{\wp''(z)}{\wp'(z)}\right\}^2-2\wp(z),</math>
unless 2''z'' is a period.

==The case with 1 a basic half-period==

If <math>\omega_1=1</math>, much of the above theory becomes simpler; it is then conventional to
write <math>\tau</math> for <math>\omega_2</math>. For a fixed {{mvar|&tau;}} in the [[upper half-plane]], so that the imaginary part of {{mvar|&tau;}} is positive, we define the '''Weierstrass ℘ function''' by

:<math>\wp(z;\tau) =\frac{1}{z^2} + \sum_{(m,n) \ne (0,0)}{1 \over (z+m+n\tau)^2} - {1 \over (m+n\tau)^2}.</math>

The sum extends over the [[lattice (group)|lattice]] {{math|{{mset|''n'' + ''m&tau;'' | ''n'', ''m'' ∈ '''Z'''}}}} with the origin omitted.
Here we regard {{mvar|&tau;}} as fixed and ℘ as a function of ''z''; fixing ''z'' and letting {{mvar|&tau;}} vary leads into the area of [[elliptic modular function]]s.

==General theory==
℘ is a [[meromorphic]] function in the complex plane with a double [[pole (complex analysis)|pole]] at each lattice point. It is doubly periodic with periods 1 and {{mvar|&tau;}}; this means that ℘ satisfies

:<math>\wp(z+1) = \wp(z+\tau) = \wp(z).</math>

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

:<math>\wp(cz;c\omega_1,c\omega_2) = \wp(z;\omega_1,\omega_2)/c^2</math>

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

:<math>\wp'^2 = 4\wp^3 - g_2 \wp - g_3</math>

where <math>g_2</math> and <math>g_3</math> depend only on {{mvar|&tau;}}, being [[modular forms]]. The equation

:<math>Y^2 = 4 X^3 - g_2 X - g_3</math>

defines an [[elliptic curve]], and we see that <math>(\wp, \wp')</math> 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

:<math>\mathbb{C}(\wp, \wp'),</math>

so that all such functions are [[rational function]]s 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.

:<math>\wp(z; \tau) = \pi^2 \vartheta^2(0;\tau) \vartheta_{10}^2(0;\tau){\vartheta_{01}^2(z;\tau) \over \vartheta_{11}^2(z;\tau)} + e_2(\tau).</math>

The function ℘ has two zeros ([[Modulo (jargon)|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 integer]]s). An expression was found, by [[Don Zagier|Zagier]] and [[Martin Eichler|Eichler]].<ref>{{cite journal |first=M. |last=Eichler |first2=D. |last2=Zagier |title=On the zeros of the Weierstrass ℘-Function |journal=[[Mathematische Annalen]] |volume=258 |issue=4 |year=1982 |pages=399–407 |doi=10.1007/BF01453974 }}</ref>

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 elliptic functions|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<ref>{{cite book | author = Korn GA, Korn TM | year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw–Hill | location = New York | pages = 721 | lccn = 59014456}}</ref>

:<math>
\wp(z) = e_3 + \frac{e_1 - e_3}{\operatorname{sn}^2 w}
= e_2 + ( e_1 - e_3 ) \frac{\operatorname{dn}^2 w}{\operatorname{sn}^2 w}
= e_1 + ( e_1 - e_3 ) \frac{\operatorname{cn}^2 w}{\operatorname{sn}^2 w}
</math>

where ''e''<sub>1–3</sub> are the three roots described above and where the modulus ''k'' of the Jacobi functions equals

:<math>
k \equiv \sqrt{\frac{e_2 - e_3}{e_1 - e_3}}
</math>

and their argument ''w'' equals

:<math>
w \equiv z \sqrt{e_1 - e_3}.
</math>

== Typography ==
The Weierstrass's elliptic function is usually written with a rather special, lower case script letter &weierp;.{{refn | group=footnote |
This symbol was used already at least in 1890. The first edition of  ''[[A Course of Modern Analysis]]'' by [[E. T. Whittaker]]<!-- The first edition was by Whittaker alone. --> in 1902 also used it.<ref>{{citation | title= The letter &weierp; Name & origin? | author = [https://mathoverflow.net/users/56062/teika-kazura teika kazura] | publisher = [[MathOverflow]] | url= https://mathoverflow.net/q/278130 | date= 2017-08-17 | accessdate=2018-08-30 }}</ref> }}

In computing, the letter &weierp; is available as <code>\wp</code> in [[TeX]]. In [[Unicode]] the code point is {{unichar|2118|script capital p|html=}}, with the more correct alias {{smallcaps|1=weierstrass elliptic function}}.{{refn|group="footnote" |

The [[Unicode Consortium]] has acknowledged two problems with the letter's name: the letter is in fact lowercase, and it is not a "script"  class letter, like {{unichar|1d4c5|mathematical script small p}}, but the letter for Weierstrass's elliptic function.

Unicode added the alias as a correction.<ref>{{cite web | work=Unicode Technical Note #27 | title=Known Anomalies in Unicode Character Names | url=http://unicode.org/notes/tn27/ | version=version 4 | publisher=Unicode, Inc. | date=2017-04-10 | accessdate=2017-07-20 }}</ref><ref>{{cite web | url=https://www.unicode.org/Public/10.0.0/ucd/NameAliases.txt | title=NameAliases-10.0.0.txt | date=2017-05-06 | accessdate=2017-07-20  | publisher=Unicode, Inc.}}</ref>

}} In [[HTML]], it can be escaped as <code>&amp;weierp;</code>.

== Footnotes ==
{{Reflist|group=footnote}}

== References ==
{{Reflist}}

*{{AS ref|18|627}}
* [[Naum Akhiezer|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==
{{commons category|Weierstrass's elliptic functions}}
* {{springer|title=Weierstrass elliptic functions|id=p/w097450}}
* [http://mathworld.wolfram.com/WeierstrassEllipticFunction.html Weierstrass's elliptic functions on Mathworld].
* Chapter 23, [http://dlmf.nist.gov/23 Weierstrass Elliptic and Modular Functions] in DLMF ([[Digital Library of Mathematical Functions]]) by W. P. Reinhardt and P. L. Walker.

[[Category:Modular forms]]
[[Category:Algebraic curves]]
[[Category:Elliptic functions]]