Weierstrass elliptic function

"P-function" redirects here. For the phase-space function representing a quantum state, see Glauber–Sudarshan P representation.

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, i.e., meromorphic functions that are doubly periodic. 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 ${\displaystyle \wp }$-function

Model of Weierstrass ${\displaystyle \wp }$-function

Motivation

A cubic of the form ${\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x-g_{3}\}}$, where ${\displaystyle g_{2},g_{3}\in \mathbb {C} }$ are complex numbers with ${\displaystyle g_{2}^{3}-27g_{3}^{2}\neq 0}$, cannot be rationally parameterized.^[1] Yet one still wants to find a way to parameterize it.

For the quadric ${\displaystyle K=\left\{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}=1\right\}}$; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function: $${\displaystyle \psi$$ :\mathbb {R} /2\pi \mathbb {Z} \to K,\quad t\mapsto (\sin t,\cos t).} Because of the periodicity of the sine and cosine ${\displaystyle \mathbb {R} /2\pi \mathbb {Z} }$ is chosen to be the domain, so the function is bijective.

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

There is another analogy to the trigonometric functions. Consider the integral function $${\displaystyle a(x)=\int _{0}^{x}{\frac {dy}{\sqrt {1-y^{2}}}}.}$$ It can be simplified by substituting ${\displaystyle y=\sin t}$ and ${\displaystyle s=\arcsin x}$: $${\displaystyle a(x)=\int _{0}^{s}dt=s=\arcsin x.}$$ That means ${\displaystyle a^{-1}(x)=\sin x}$. So the sine function is an inverse function of an integral function.^[3]

Elliptic functions are the inverse functions of elliptic integrals. In particular, let: $${\displaystyle u(z)=\int _{z}^{\infty }{\frac {ds}{\sqrt {4s^{3}-g_{2}s-g_{3}}}}.}$$ Then the extension of ${\displaystyle u^{-1}}$ to the complex plane equals the ${\displaystyle \wp }$-function.^[4] This invertibility is used in complex analysis to provide a solution to certain nonlinear differential equations satisfying the Painlevé property, i.e., those equations that admit poles as their only movable singularities.^[5]

Definition

Visualization of the ${\displaystyle \wp }$-function with invariants ${\displaystyle g_{2}=1+i}$ and ${\displaystyle g_{3}=2-3i}$ in which white corresponds to a pole, black to a zero.

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

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

This series converges locally uniformly absolutely in the complex torus ${\displaystyle \mathbb {C} \setminus \Lambda }$.

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

Properties

• ${\displaystyle \wp }$ is a meromorphic function with a pole of order 2 at each period ${\displaystyle \lambda }$ in ${\displaystyle \Lambda }$.
• ${\displaystyle \wp }$ is an even function. That means ${\displaystyle \wp (z)=\wp (-z)}$ for all ${\displaystyle z\in \mathbb {C} \setminus \Lambda }$, which can be seen in the following way:

${\displaystyle {\begin{aligned}\wp (-z)&={\frac {1}{(-z)^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(-z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)\\&={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z+\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)\\&={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)=\wp (z).\end{aligned}}}$

The second last equality holds because ${\displaystyle \{-\lambda$ :\lambda \in \Lambda \}=\Lambda } . Since the sum converges absolutely this rearrangement does not change the limit.

• The derivative of ${\displaystyle \wp }$ is given by:^[6] $${\displaystyle \wp '(z)=-2\sum _{\lambda \in \Lambda }{\frac {1}{(z-\lambda )^{3}}}.}$$
• ${\displaystyle \wp }$ and ${\displaystyle \wp '}$ are doubly periodic with the periods ${\displaystyle \omega _{1}}$ and ${\displaystyle \omega _{2}}$.^[6] This means: $${\displaystyle {\begin{aligned}\wp (z+\omega _{1})&=\wp (z)=\wp (z+\omega _{2}),\ {\textrm {and}}\\[3mu]\wp '(z+\omega _{1})&=\wp '(z)=\wp '(z+\omega _{2}).\end{aligned}}}$$ It follows that ${\displaystyle \wp (z+\lambda )=\wp (z)}$ and ${\displaystyle \wp '(z+\lambda )=\wp '(z)}$ for all ${\displaystyle \lambda \in \Lambda }$.

Laurent expansion

Let ${\displaystyle r:=\min\{{|\lambda }|:0\neq \lambda \in \Lambda \}}$. Then for ${\displaystyle 0<|z|<r}$ the ${\displaystyle \wp }$-function has the following Laurent expansion $${\displaystyle \wp (z)={\frac {1}{z^{2}}}+\sum _{n=1}^{\infty }(2n+1)G_{2n+2}z^{2n}}$$ where $${\displaystyle G_{n}=\sum _{0\neq \lambda \in \Lambda }\lambda ^{-n}}$$ for ${\displaystyle n\geq 3}$ are so called Eisenstein series.^[6]

Differential equation

Set ${\displaystyle g_{2}=60G_{4}}$ and ${\displaystyle g_{3}=140G_{6}}$. Then the ${\displaystyle \wp }$-function satisfies the differential equation^[6] $${\displaystyle \wp '^{2}(z)=4\wp ^{3}(z)-g_{2}\wp (z)-g_{3}.}$$ This relation can be verified by forming a linear combination of powers of ${\displaystyle \wp }$ and ${\displaystyle \wp '}$ to eliminate the pole at ${\displaystyle z=0}$. This yields an entire elliptic function that has to be constant by Liouville's theorem.^[6]

Invariants

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 ${\displaystyle \Lambda }$ they can be viewed as functions in ${\displaystyle \omega _{1}}$ and ${\displaystyle \omega _{2}}$.

The series expansion suggests that g₂ and g₃ are homogeneous functions of degree −4 and −6. That is^[7] $${\displaystyle g_{2}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-4}g_{2}(\omega _{1},\omega _{2})}$$ $${\displaystyle g_{3}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-6}g_{3}(\omega _{1},\omega _{2})}$$ for ${\displaystyle \lambda \neq 0}$.

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

Let ${\displaystyle \tau ={\tfrac {\omega _{2}}{\omega _{1}}}}$. One has:^[8] $${\displaystyle g_{2}(1,\tau )=\omega _{1}^{4}g_{2}(\omega _{1},\omega _{2}),}$$ $${\displaystyle g_{3}(1,\tau )=\omega _{1}^{6}g_{3}(\omega _{1},\omega _{2}).}$$ That means g₂ and g₃ are only scaled by doing this. Set $${\displaystyle g_{2}(\tau ):=g_{2}(1,\tau )}$$ and $${\displaystyle g_{3}(\tau ):=g_{3}(1,\tau ).}$$ As functions of ${\displaystyle \tau \in \mathbb {H} }$ ${\displaystyle g_{2},g_{3}}$ are so called modular forms.

The Fourier series for ${\displaystyle g_{2}}$ and ${\displaystyle g_{3}}$ are given as follows:^[9] $${\displaystyle g_{2}(\tau )={\frac {4}{3}}\pi ^{4}\left[1+240\sum _{k=1}^{\infty }\sigma _{3}(k)q^{2k}\right]}$$ $${\displaystyle g_{3}(\tau )={\frac {8}{27}}\pi ^{6}\left[1-504\sum _{k=1}^{\infty }\sigma _{5}(k)q^{2k}\right]}$$ where $${\displaystyle \sigma _{m}(k):=\sum _{d\mid {k}}d^{m}}$$ is the divisor function and ${\displaystyle q=e^{\pi i\tau }}$ is the nome.

Modular discriminant

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 characteristic polynomial of the differential equation $${\displaystyle \wp '^{2}(z)=4\wp ^{3}(z)-g_{2}\wp (z)-g_{3}}$$ as follows: $${\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}.}$$ The discriminant is a modular form of weight 12. That is, under the action of the modular group, it transforms as $${\displaystyle \Delta \left({\frac {a\tau +b}{c\tau +d}}\right)=\left(c\tau +d\right)^{12}\Delta (\tau )}$$ where ${\displaystyle a,b,d,c\in \mathbb {Z} }$ with ad − bc = 1.^[10]

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

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

The constants e₁, e₂ and e₃

${\displaystyle e_{1}}$, ${\displaystyle e_{2}}$ and ${\displaystyle e_{3}}$ are usually used to denote the values of the ${\displaystyle \wp }$-function at the half-periods. $${\displaystyle e_{1}\equiv \wp \left({\frac {\omega _{1}}{2}}\right)}$$ $${\displaystyle e_{2}\equiv \wp \left({\frac {\omega _{2}}{2}}\right)}$$ $${\displaystyle e_{3}\equiv \wp \left({\frac {\omega _{1}+\omega _{2}}{2}}\right)}$$ They are pairwise distinct and only depend on the lattice ${\displaystyle \Lambda }$ and not on its generators.^[12]

${\displaystyle e_{1}}$, ${\displaystyle e_{2}}$ and ${\displaystyle e_{3}}$ are the roots of the cubic polynomial ${\displaystyle 4\wp (z)^{3}-g_{2}\wp (z)-g_{3}}$ and are related by the equation: $${\displaystyle e_{1}+e_{2}+e_{3}=0.}$$ Because those roots are distinct the discriminant ${\displaystyle \Delta }$ does not vanish on the upper half plane.^[13] Now we can rewrite the differential equation: $${\displaystyle \wp '^{2}(z)=4(\wp (z)-e_{1})(\wp (z)-e_{2})(\wp (z)-e_{3}).}$$ That means the half-periods are zeros of ${\displaystyle \wp '}$.

The invariants ${\displaystyle g_{2}}$ and ${\displaystyle g_{3}}$ can be expressed in terms of these constants in the following way:^[14] $${\displaystyle g_{2}=-4(e_{1}e_{2}+e_{1}e_{3}+e_{2}e_{3})}$$ $${\displaystyle g_{3}=4e_{1}e_{2}e_{3}}$$ ${\displaystyle e_{1}}$, ${\displaystyle e_{2}}$ and ${\displaystyle e_{3}}$ are related to the modular lambda function: $${\displaystyle \lambda (\tau )={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}},\quad \tau ={\frac {\omega _{2}}{\omega _{1}}}.}$$

Relation to Jacobi's 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:^[15] $${\displaystyle \wp (z)=e_{3}+{\frac {e_{1}-e_{3}}{\operatorname {sn} ^{2}w}}=e_{2}+(e_{1}-e_{3}){\frac {\operatorname {dn} ^{2}w}{\operatorname {sn} ^{2}w}}=e_{1}+(e_{1}-e_{3}){\frac {\operatorname {cn} ^{2}w}{\operatorname {sn} ^{2}w}}}$$ where ${\displaystyle e_{1},e_{2}}$ and ${\displaystyle e_{3}}$ are the three roots described above and where the modulus k of the Jacobi functions equals $${\displaystyle k={\sqrt {\frac {e_{2}-e_{3}}{e_{1}-e_{3}}}}}$$ and their argument w equals $${\displaystyle w=z{\sqrt {e_{1}-e_{3}}}.}$$

Relation to Jacobi's theta functions

The function ${\displaystyle \wp (z,\tau )=\wp (z,1,\omega _{2}/\omega _{1})}$ can be represented by Jacobi's theta functions: $${\displaystyle \wp (z,\tau )=\left(\pi \theta _{2}(0,q)\theta _{3}(0,q){\frac {\theta _{4}(\pi z,q)}{\theta _{1}(\pi z,q)}}\right)^{2}-{\frac {\pi ^{2}}{3}}\left(\theta _{2}^{4}(0,q)+\theta _{3}^{4}(0,q)\right)}$$ where ${\displaystyle q=e^{\pi i\tau }}$ is the nome and ${\displaystyle \tau }$ is the period ratio ${\displaystyle (\tau \in \mathbb {H} )}$.^[16] This also provides a very rapid algorithm for computing ${\displaystyle \wp (z,\tau )}$.

Relation to elliptic curves

See also: Elliptic curve § Elliptic curves over the complex numbers

Consider the embedding of the cubic curve in the complex projective plane

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

For this cubic there exists no rational parameterization, if ${\displaystyle \Delta \neq 0}$.^[1] In this case it is also called an elliptic curve. Nevertheless there is a parameterization in homogeneous coordinates that uses the ${\displaystyle \wp }$-function and its derivative ${\displaystyle \wp '}$:^[17]

${\displaystyle \varphi (\wp ,\wp '):\mathbb {C} /\Lambda \to {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} },\quad z\mapsto {\begin{cases}\left[\wp (z):\wp '(z):1\right]&z\notin \Lambda \\\left[0:1:0\right]\quad &z\in \Lambda \end{cases}}}$

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

${\displaystyle \mathbb {C} /\Lambda }$ 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 ${\displaystyle g_{2},g_{3}\in \mathbb {C} }$ with ${\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}\neq 0}$ there exists a lattice ${\displaystyle \mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}}$, such that

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

The statement that elliptic curves over ${\displaystyle \mathbb {Q} }$ can be parameterized over ${\displaystyle \mathbb {Q} }$, 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 theorems

Let ${\displaystyle z,w\in \mathbb {C} }$, so that ${\displaystyle z,w,z+w,z-w\notin \Lambda }$. Then one has:^[19] $${\displaystyle \wp (z+w)={\frac {1}{4}}\left[{\frac {\wp '(z)-\wp '(w)}{\wp (z)-\wp (w)}}\right]^{2}-\wp (z)-\wp (w).}$$

As well as the duplication formula:^[19] $${\displaystyle \wp (2z)={\frac {1}{4}}\left[{\frac {\wp ''(z)}{\wp '(z)}}\right]^{2}-2\wp (z).}$$

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

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

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

This is equivalent to: $${\displaystyle \det \left({\begin{array}{rrr}1&\wp (u+v)&-\wp '(u+v)\\1&\wp (v)&\wp '(v)\\1&\wp (u)&\wp '(u)\\\end{array}}\right)=0,}$$ where ${\displaystyle \wp (u)=a}$, ${\displaystyle \wp (v)=b}$ and ${\displaystyle u,v\notin \Lambda }$.^[21]

Typography

The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘, which was Weierstrass's own notation introduced in his lectures of 1862–1863.^{[footnote 1]} It should not be confused with the normal mathematical script letters P, 𝒫 and 𝓅.

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 ℘.

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	℘, ℘

See also

• Weierstrass functions
• Jacobi elliptic functions
• Lemniscate elliptic functions

Footnotes

1. This symbol was also used in the version of Weierstrass's lectures published by Schwarz in the 1880s. The first edition of A Course of Modern Analysis by E. T. Whittaker in 1902 also used it.^[22]
2. 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]

References

1. 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
2. Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 259, ISBN 978-3-540-32058-6
3. 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{{citation}}: CS1 maint: location missing publisher (link)
4. Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 294, ISBN 978-3-540-32058-6
5. Ablowitz, Mark J.; Fokas, Athanassios S. (2003). Complex Variables: Introduction and Applications. Cambridge University Press. p. 185. doi:10.1017/cbo9780511791246. ISBN 978-0-521-53429-1.
6. 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
7. 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.
8. 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
9. 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.
10. 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.
11. 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) CS1 maint: numeric names: authors list (link)
12. Busam, Rolf (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 270, ISBN 978-3-540-32058-6
13. 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
14. K. Chandrasekharan (1985), Elliptic functions (in German), Berlin: Springer-Verlag, p. 33, ISBN 0-387-15295-4
15. Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw–Hill. p. 721. LCCN 59014456.
16. 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.
17. 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
18. 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
19. Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 286, ISBN 978-3-540-32058-6
20. Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 287, ISBN 978-3-540-32058-6
21. Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 288, ISBN 978-3-540-32058-6
22. teika kazura (2017-08-17), The letter ℘ Name & origin?, MathOverflow, retrieved 2018-08-30
23. "Known Anomalies in Unicode Character Names". Unicode Technical Note #27. version 4. Unicode, Inc. 2017-04-10. Retrieved 2017-07-20.
24. "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 links

• "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