Weierstrass's Elliptic Functions

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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 they are usually denoted by the symbol ℘. They play an important role in 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.

elliptic functions elliptic curves ℘-function

References

Apostol, Tom M. (1976). Modular functions and Dirichlet series in number theory. New York: Springer-Verlag. pp. 9. ISBN 0-387-90185-X. OCLC 2121639. https://www.worldcat.org/oclc/2121639.
Hulek, Klaus. (2012) (in German), Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen (2., überarb. u. erw. Aufl. 2012 ed.), Wiesbaden: Vieweg+Teubner Verlag, p. 8, ISBN 978-3-8348-2348-9
Rolf Busam (2006) (in German), Funktionentheorie 1 (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 259, ISBN 978-3-540-32058-6
Jeremy Gray (2015) (in German), Real and the complex : a history of analysis in the 19th century, Cham, p. 71, ISBN 978-3-319-23715-2
Rolf Busam (2006) (in German), Funktionentheorie 1 (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 294, ISBN 978-3-540-32058-6
Apostol, Tom M. (1976) (in German), Modular functions and Dirichlet series in number theory, 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. pp. 14. ISBN 0-387-90185-X. OCLC 2121639. https://www.worldcat.org/oclc/2121639.
Apostol, Tom M. (1976) (in German), Modular functions and Dirichlet series in number theory, 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. pp. 20. ISBN 0-387-97127-0. OCLC 20262861. https://www.worldcat.org/oclc/20262861.
Apostol, Tom M. (1976). Modular functions and Dirichlet series in number theory. New York: Springer-Verlag. pp. 50. ISBN 0-387-90185-X. OCLC 2121639. https://www.worldcat.org/oclc/2121639.
Chandrasekharan, K. (Komaravolu), 1920- (1985). Elliptic functions. Berlin: Springer-Verlag. pp. 122. ISBN 0-387-15295-4. OCLC 12053023. https://www.worldcat.org/oclc/12053023.
Rolf Busam (2006) (in German), Funktionentheorie 1 (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 270, ISBN 978-3-540-32058-6
Tom M. Apostol (1976) (in German), Modular functions and Dirichlet series in number theory, New York: Springer-Verlag, p. 13, ISBN 0-387-90185-X
K. Chandrasekharan (1985) (in German), Elliptic functions, Berlin: Springer-Verlag, p. 33, ISBN 0-387-15295-4
Hulek, Klaus. (2012) (in German), Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen (2., überarb. u. erw. Aufl. 2012 ed.), Wiesbaden: Vieweg+Teubner Verlag, p. 12, ISBN 978-3-8348-2348-9
Hulek, Klaus. (2012) (in German), Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen (2., überarb. u. erw. Aufl. 2012 ed.), Wiesbaden: Vieweg+Teubner Verlag, p. 111, ISBN 978-3-8348-2348-9
Rolf Busam (2006) (in German), Funktionentheorie 1 (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 286, ISBN 978-3-540-32058-6
Rolf Busam (2006) (in German), Funktionentheorie 1 (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 287, ISBN 978-3-540-32058-6
Rolf Busam (2006) (in German), Funktionentheorie 1 (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 288, ISBN 978-3-540-32058-6
Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw–Hill. pp. 721.
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.[21]
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 ? MATHEMATICAL SCRIPT SMALL P, but the letter for Weierstrass's elliptic function. Unicode added the alias as a correction.[22][23]

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