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Weierstrass elliptic function

Summary

In mathematics, the Weierstrass 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 ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script p. They play an important role in the theory of elliptic functions. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice.

Symbol for Weierstrass $\wp$ -function

Model of Weierstrass

\wp

-function

DefinitionEdit

Visualization of the

\wp

-function with invariants

g_{2}=1+i

and

g_{3}=2-3i

in which white corresponds to a pole, black to a zero.

Let $\omega _{1},\omega _{2}\in \mathbb {C}$ be two complex numbers that are linearly independent over $\mathbb {R}$ and let $\Lambda :=\mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}:=\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}$ be the lattice generated by those numbers. Then the $\wp$ -function is defined as follows:

\wp (z,\omega _{1},\omega _{2}):=\wp (z,\Lambda ):={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right).

This series converges locally uniformly absolutely in $\mathbb {C} \setminus \Lambda$ . Oftentimes instead of $\wp (z,\omega _{1},\omega _{2})$ only $\wp (z)$ is written.

The Weierstrass $\wp$ -function is constructed exactly in such a way that it has a pole of the order two at each lattice point.

Because the sum ${\textstyle \sum _{\lambda \in \Lambda }{\frac {1}{(z-\lambda )^{2}}}}$ alone would not converge it is necessary to add the term ${\textstyle -{\frac {1}{\lambda ^{2}}}}$ .^[1]

It is common to use $1$ and $\tau$ in the upper half-plane ${H}:=\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}$ as generators of the lattice. Dividing by ${\textstyle \omega _{1}}$ maps the lattice $\mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}$ isomorphically onto the lattice $\mathbb {Z} +\mathbb {Z} \tau$ with ${\textstyle \tau ={\tfrac {\omega _{2}}{\omega _{1}}}}$ . Because $-\tau$ can be substituted for $\tau$ , without loss of generality we can assume $\tau \in \mathbb {H}$ , and then define $\wp (z,\tau ):=\wp (z,1,\tau )$ .

MotivationEdit

A cubic of the form $C_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x+g_{3}\}$ , where $g_{2},g_{3}\in \mathbb {C}$ are complex numbers with $g_{2}^{3}-27g_{3}^{2}\neq 0$ , can not be rationally parameterized.^[2] Yet one still wants to find a way to parameterize it.

For the quadric $K=\left\{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}=1\right\}$ , the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function:

\psi :\mathbb {R} /2\pi \mathbb {Z} \to K,\quad t\mapsto (\sin t,\cos t).

Because of the periodicity of the sine and cosine $\mathbb {R} /2\pi \mathbb {Z}$ is chosen to be the domain, so the function is bijective.

In a similar way one can get a parameterization of $C_{g_{2},g_{3}}^{\mathbb {C} }$ by means of the doubly periodic $\wp$ -function (see in the section "Relation to elliptic curves"). This parameterization has the domain $\mathbb {C} /\Lambda$ , which is topologically equivalent to a torus.^[3]

There is another analogy to the trigonometric functions. Consider the integral function

a(x)=\int _{0}^{x}{\frac {dy}{\sqrt {1-y^{2}}}}.

It can be simplified by substituting $y=\sin t$ and $s=\arcsin x$ :

a(x)=\int _{0}^{s}dt=s=\arcsin x.

That means $a^{-1}(x)=\sin x$ . So the sine function is an inverse function of an integral function.^[4]

Elliptic functions are also inverse functions of integral functions, namely of elliptic integrals. In particular the $\wp$ -function is obtained in the following way:

Let

u(z)=-\int _{z}^{\infty }{\frac {ds}{\sqrt {4s^{3}-g_{2}s-g_{3}}}}.

Then $u^{-1}$ can be extended to the complex plane and this extension equals the $\wp$ -function.^[5]

PropertiesEdit

℘ is an even function. That means $\wp (z)=\wp (-z)$ for all $z\in \mathbb {C} \setminus \Lambda$ , which can be seen in the following way: ${\begin{aligned}\wp (-z)&={\frac {1}{(-z)^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(-z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)\\[4pt]&={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z+\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)\\[4pt]&={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)=\wp (z).\end{aligned}}$ The second last equality holds because $\{-\lambda :\lambda \in \Lambda \}=\Lambda$ . Since the sum converges absolutely this rearrangement does not change the limit.
℘ is meromorphic and its derivative is^[6] $\wp '(z)=-2\sum _{\lambda \in \Lambda }{\frac {1}{(z-\lambda )^{3}}}.$
$\wp$ and $\wp '$ are doubly periodic with the periods $\omega _{1}$ and $\omega _{2}$ .^[6] This means: ${\begin{aligned}\wp (z+\omega _{1})&=\wp (z)=\wp (z+\omega _{2}),\ {\textrm {and}}\\[3mu]\wp '(z+\omega _{1})&=\wp '(z)=\wp '(z+\omega _{2}).\end{aligned}}$

It follows that $\wp (z+\lambda )=\wp (z)$ and $\wp '(z+\lambda )=\wp '(z)$ for all $\lambda \in \Lambda$ . Functions which are meromorphic and doubly periodic are also called elliptic functions.

Laurent expansionEdit

Let $r:=\min\{{|\lambda }|:0\neq \lambda \in \Lambda \}$ . Then for $0<|z|<r$ the $\wp$ -function has the following Laurent expansion

\wp (z)={\frac {1}{z^{2}}}+\sum _{n=1}^{\infty }(2n+1)G_{2n+2}z^{2n}

where

G_{n}=\sum _{0\neq \lambda \in \Lambda }\lambda ^{-n}

for

n\geq 3

are so called Eisenstein series.^[6]

Differential equationEdit

Set $g_{2}=60G_{4}$ and $g_{3}=140G_{6}$ . Then the $\wp$ -function satisfies the differential equation^[6]

\wp '^{2}(z)=4\wp ^{3}(z)-g_{2}\wp (z)-g_{3}.

This relation can be verified by forming a linear combination of powers of $\wp$ and $\wp '$ to eliminate the pole at $z=0$ . This yields an entire elliptic function that has to be constant by Liouville's theorem.^[6]

InvariantsEdit

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

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

The coefficients of the above differential equation g₂ and g₃ are known as the invariants. Because they depend on the lattice $\Lambda$ they can be viewed as functions in $\omega _{1}$ and $\omega _{2}$ .

The series expansion suggests that g₂ and g₃ are homogeneous functions of degree −4 and −6. That is^[7]

g_{2}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-4}g_{2}(\omega _{1},\omega _{2})

g_{3}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-6}g_{3}(\omega _{1},\omega _{2})

for

\lambda \neq 0

If $\omega _{1}$ and $\omega _{2}$ are chosen in such a way that $\operatorname {Im} \left({\tfrac {\omega _{2}}{\omega _{1}}}\right)>0$ , g₂ and g₃ can be interpreted as functions on the upper half-plane $\mathbb {H} :=\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}$ .

Let $\tau ={\tfrac {\omega _{2}}{\omega _{1}}}$ . One has:^[8]

g_{2}(1,\tau )=\omega _{1}^{4}g_{2}(\omega _{1},\omega _{2}),

g_{3}(1,\tau )=\omega _{1}^{6}g_{3}(\omega _{1},\omega _{2}).

That means g₂ and g₃ are only scaled by doing this. Set

g_{2}(\tau ):=g_{2}(1,\tau )

and

g_{3}(\tau ):=g_{3}(1,\tau ).

As functions of $\tau \in \mathbb {H}$ $g_{2},g_{3}$ are so called modular forms.

The Fourier series for $g_{2}$ and $g_{3}$ are given as follows:^[9]

g_{2}(\tau )={\frac {4}{3}}\pi ^{4}\left[1+240\sum _{k=1}^{\infty }\sigma _{3}(k)q^{2k}\right]

g_{3}(\tau )={\frac {8}{27}}\pi ^{6}\left[1-504\sum _{k=1}^{\infty }\sigma _{5}(k)q^{2k}\right]

where

\sigma _{a}(k):=\sum _{d\mid {k}}d^{\alpha }

is the divisor function and

q=e^{\pi i\tau }

is the nome.

Modular discriminantEdit

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

The modular discriminant Δ is defined as the discriminant of the polynomial at right-hand side of the above differential equation:

\Delta =g_{2}^{3}-27g_{3}^{2}.

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

\Delta \left({\frac {a\tau +b}{c\tau +d}}\right)=\left(c\tau +d\right)^{12}\Delta (\tau )

where

a,b,d,c\in \mathbb {Z}

with ad − bc = 1.^[10]

Note that $\Delta =(2\pi )^{12}\eta ^{24}$ where $\eta$ is the Dedekind eta function.^[11]

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

The constants e₁, e₂ and e₃Edit

$e_{1}$ , $e_{2}$ and $e_{3}$ are usually used to denote the values of the $\wp$ -function at the half-periods.

e_{1}\equiv \wp \left({\frac {\omega _{1}}{2}}\right)

e_{2}\equiv \wp \left({\frac {\omega _{2}}{2}}\right)

e_{3}\equiv \wp \left({\frac {\omega _{1}+\omega _{2}}{2}}\right)

They are pairwise distinct and only depend on the lattice $\Lambda$ and not on its generators.^[12]

$e_{1}$ , $e_{2}$ and $e_{3}$ are the roots of the cubic polynomial $4\wp (z)^{3}-g_{2}\wp (z)-g_{3}$ and are related by the equation:

e_{1}+e_{2}+e_{3}=0.

Because those roots are distinct the discriminant $\Delta$ does not vanish on the upper half plane.^[13] Now we can rewrite the differential equation:

\wp '^{2}(z)=4(\wp (z)-e_{1})(\wp (z)-e_{2})(\wp (z)-e_{3}).

That means the half-periods are zeros of $\wp '$ .

The invariants $g_{2}$ and $g_{3}$ can be expressed in terms of these constants in the following way:^[14]

g_{2}=-4(e_{1}e_{2}+e_{1}e_{3}+e_{2}e_{3})

g_{3}=4e_{1}e_{2}e_{3}

$e_{1}$ , $e_{2}$ and $e_{3}$ are related to the modular lambda function:

\lambda (\tau )={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}},\quad \tau ={\frac {\omega _{2}}{\omega _{1}}}.

Relation to Jacobi's elliptic functionsEdit

For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of Jacobi's elliptic functions.

The basic relations are:^[15]

\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}}

where

e_{1},e_{2}

and

e_{3}

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

k={\sqrt {\frac {e_{2}-e_{3}}{e_{1}-e_{3}}}}

and their argument w equals

w=z{\sqrt {e_{1}-e_{3}}}.

Relation to Jacobi's theta functionsEdit

The function $\wp (z,\tau )=\wp (z,1,\omega _{2}/\omega _{1})$ can be represented by Jacobi's theta functions:

\wp (z,\tau )=\left(\pi \theta _{2}(0,q)\theta _{3}(0,q){\frac {\theta _{4}(\pi z,q)}{\theta _{1}(\pi z,q)}}\right)^{2}-{\frac {\pi ^{2}}{3}}\left(\theta _{2}^{4}(0,q)+\theta _{3}^{4}(0,q)\right)

where

q=e^{\pi i\tau }

is the nome and

\tau

is the period ratio

(\tau \in \mathbb {H} )

.^[16] This also provides a very rapid algorithm for computing

\wp (z,\tau )

Relation to elliptic curvesEdit

Consider the projective cubic curve

{\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x+g_{3}\}\cup \{\infty \}\subset \mathbb {P} _{\mathbb {C} }^{2}.

For this cubic, also called Weierstrass cubic, there exists no rational parameterization, if $\Delta \neq 0$ .^[2] In this case it is also called an elliptic curve. Nevertheless there is a parameterization that uses the $\wp$ -function and its derivative $\wp '$ :^[17]

\varphi :\mathbb {C} /\Lambda \to {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} },\quad {\bar {z}}\mapsto {\begin{cases}(\wp (z),\wp '(z),1)&{\bar {z}}\neq 0\\\infty \quad &{\bar {z}}=0\end{cases}}

Now the map $\varphi$ is bijective and parameterizes the elliptic curve ${\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }$ .

$\mathbb {C} /\Lambda$ is an abelian group and a topological space, equipped with the quotient topology.

It can be shown that every Weierstrass cubic is given in such a way. That is to say that for every pair $g_{2},g_{3}\in \mathbb {C}$ with $\Delta =g_{2}^{3}-27g_{3}^{2}\neq 0$ there exists a lattice $\mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}$ , such that

$g_{2}=g_{2}(\omega _{1},\omega _{2})$ and $g_{3}=g_{3}(\omega _{1},\omega _{2})$ .^[18]

The statement that elliptic curves over $\mathbb {Q}$ can be parameterized over $\mathbb {Q}$ , is known as the modularity theorem. This is an important theorem in number theory. It was part of Andrew Wiles' proof (1995) of Fermat's Last Theorem.

Addition theoremsEdit

Let $z,w\in \mathbb {C}$ , so that $z,w,z+w,z-w\notin \Lambda$ . Then one has:^[19]

\wp (z+w)={\frac {1}{4}}\left[{\frac {\wp '(z)-\wp '(w)}{\wp (z)-\wp (w)}}\right]^{2}-\wp (z)-\wp (w).

As well as the duplication formula:^[19]

\wp (2z)={\frac {1}{4}}\left[{\frac {\wp ''(z)}{\wp '(z)}}\right]^{2}-2\wp (z).

These formulas also have a geometric interpretation, if one looks at the elliptic curve ${\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }$ together with the mapping ${\varphi }:\mathbb {C} /\Lambda \to {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }$ as in the previous section.

The group structure of $(\mathbb {C} /\Lambda ,+)$ translates to the curve ${\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }$ and can be geometrically interpreted there:

The sum of three pairwise different points $a,b,c\in {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }$ is zero if and only if they lie on the same line in $\mathbb {P} _{\mathbb {C} }^{2}$ .^[20]

This is equivalent to:

\det \left({\begin{array}{rrr}1&\wp (u+v)&-\wp '(u+v)\\1&\wp (v)&\wp '(v)\\1&\wp (u)&\wp '(u)\\\end{array}}\right)=0,

where

\wp (u)=a

\wp (v)=b

and

u,v\notin \Lambda

.^[21]

TypographyEdit

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 ℘ SCRIPT CAPITAL P (&weierp;, &wp;), with the more correct alias weierstrass elliptic function.^{[footnote 2]} In HTML, it can be escaped as &weierp;.

Character information
Preview	℘
Unicode name	SCRIPT CAPITAL P / WEIERSTRASS ELLIPTIC FUNCTION
Encodings	decimal	hex
Unicode	8472	U+2118
UTF-8	226 132 152	E2 84 98
Numeric character reference	℘	℘
Named character reference	&weierp;, &wp;

FootnotesEdit

^ This symbol 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.^[22]
^ 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 U+1D4C5 𝓅 MATHEMATICAL SCRIPT SMALL P, but the letter for Weierstrass's elliptic function. Unicode added the alias as a correction.^[23]^[24]

ReferencesEdit

^ Apostol, Tom M. (1976). Modular functions and Dirichlet series in number theory. New York: Springer-Verlag. p. 9. ISBN 0-387-90185-X. OCLC 2121639.
^ ^a ^b Hulek, Klaus. (2012), Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen (in German) (2., überarb. u. erw. Aufl. 2012 ed.), Wiesbaden: Vieweg+Teubner Verlag, p. 8, ISBN 978-3-8348-2348-9
^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 259, ISBN 978-3-540-32058-6
^ Jeremy Gray (2015), Real and the complex: a history of analysis in the 19th century (in German), Cham, p. 71, ISBN 978-3-319-23715-2
^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 294, ISBN 978-3-540-32058-6
^ ^a ^b ^c ^d ^e Apostol, Tom M. (1976), Modular functions and Dirichlet series in number theory (in German), New York: Springer-Verlag, p. 11, ISBN 0-387-90185-X
^ Apostol, Tom M. (1976). Modular functions and Dirichlet series in number theory. New York: Springer-Verlag. p. 14. ISBN 0-387-90185-X. OCLC 2121639.
^ Apostol, Tom M. (1976), Modular functions and Dirichlet series in number theory (in German), New York: Springer-Verlag, p. 14, ISBN 0-387-90185-X
^ Apostol, Tom M. (1990). Modular functions and Dirichlet series in number theory (2nd ed.). New York: Springer-Verlag. p. 20. ISBN 0-387-97127-0. OCLC 20262861.
^ Apostol, Tom M. (1976). Modular functions and Dirichlet series in number theory. New York: Springer-Verlag. p. 50. ISBN 0-387-90185-X. OCLC 2121639.
^ Chandrasekharan, K. (Komaravolu), 1920- (1985). Elliptic functions. Berlin: Springer-Verlag. p. 122. ISBN 0-387-15295-4. OCLC 12053023.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Busam, Rolf (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 270, ISBN 978-3-540-32058-6
^ Apostol, Tom M. (1976), Modular functions and Dirichlet series in number theory (in German), New York: Springer-Verlag, p. 13, ISBN 0-387-90185-X
^ K. Chandrasekharan (1985), Elliptic functions (in German), Berlin: Springer-Verlag, p. 33, ISBN 0-387-15295-4
^ Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw–Hill. p. 721. LCCN 59014456.
^ Reinhardt, W. P.; Walker, P. L. (2010), "Weierstrass Elliptic and Modular Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
^ Hulek, Klaus. (2012), Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen (in German) (2., überarb. u. erw. Aufl. 2012 ed.), Wiesbaden: Vieweg+Teubner Verlag, p. 12, ISBN 978-3-8348-2348-9
^ Hulek, Klaus. (2012), Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen (in German) (2., überarb. u. erw. Aufl. 2012 ed.), Wiesbaden: Vieweg+Teubner Verlag, p. 111, ISBN 978-3-8348-2348-9
^ ^a ^b Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 286, ISBN 978-3-540-32058-6
^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 287, ISBN 978-3-540-32058-6
^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 288, ISBN 978-3-540-32058-6
^ teika kazura (2017-08-17), The letter ℘ Name & origin?, MathOverflow, retrieved 2018-08-30
^ "Known Anomalies in Unicode Character Names". Unicode Technical Note #27. version 4. Unicode, Inc. 2017-04-10. Retrieved 2017-07-20.
^ "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. Vol. 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 linksEdit

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

"Weierstrass elliptic functions", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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.
Weierstrass P function and its derivative implemented in C by David Dumas