"P-function" redirects here. For the phase-space function representing a quantum state, see Glauber–Sudarshan P representation.
"℘" redirects here; the symbol can also be used to denote a power set.
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 -function
Model of Weierstrass -function
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mathematics, the Weierstrassellipticfunctions are ellipticfunctions that take a particularly simple form. They are named for Karl Weierstrass. This class...
mathematics, the Weierstrassfunctions are special functions of a complex variable that are auxiliary to the Weierstrassellipticfunction. They are named...
of an ellipse. Important ellipticfunctions are Jacobi ellipticfunctions and the Weierstrass ℘ {\displaystyle \wp } -function. Further development of...
{\displaystyle \sin } . The Jacobi ellipticfunctions are used more often in practical problems than the Weierstrassellipticfunctions as they do not require notions...
equation is called a Weierstrass equation, and said to be in Weierstrass form, or Weierstrass normal form. The definition of elliptic curve also requires...
quotients of the above four theta functions, and could have been used by him to construct Weierstrass'sellipticfunctions also, since ℘ ( z ; τ ) = − ( log...
Ellipticfunctions: The inverses of elliptic integrals; used to model double-periodic phenomena. Jacobi's ellipticfunctionsWeierstrass'selliptic functions...
theorem of the gamma function and investigated the connection between the gamma function and elliptic integrals. Karl Weierstrass further established the...
by sigma function one can mean one of the following: The sum-of-divisors function σa(n), an arithmetic functionWeierstrass sigma function, related to...
function with just one zero. Ellipticfunction Abel ellipticfunctions Jacobi ellipticfunctionsWeierstrassellipticfunctions Lemniscate elliptic functions...
to define the Weierstrass transform. Gaussian functions arise by composing the exponential function with a concave quadratic function: f ( x ) = exp...
Spirograph (special case of the hypotrochoid) Jacobi's ellipticfunctionsWeierstrass'sellipticfunction Formulae are given as Taylor series or derived from...
Fourier series of the Eisenstein series and the invariants of the Weierstrassellipticfunctions. For k > 0 {\displaystyle k>0} , there is an explicit series...
Y\to \pm iY,\quad X\to -X} in line with the action of i on the Weierstrassellipticfunctions. More generally, consider the lattice Λ, an additive group in...
ratio Jacobi's ellipticfunctionsWeierstrass'sellipticfunctions Theta functionElliptic modular function J-function Modular function Modular form Analytic...
{1}{r^{2}}}\right)}}}}.} This can be expressed in terms of the Weierstrassellipticfunction ℘ {\textstyle \wp } . Unlike in classical mechanics, in Schwarzschild...
Elliptic curve scalar multiplication is the operation of successively adding a point along an elliptic curve to itself repeatedly. It is used in elliptic...
‣ Modular Functions ‣ Chapter 23 WeierstrassElliptic and Modular Functions". dlmf.nist.gov. Retrieved 2023-07-07. A meromorphic function can only have...
fact about elliptic operators, of which the Laplacian is a major example. The uniform limit of a convergent sequence of harmonic functions is still harmonic...
In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve E {\displaystyle E} defined over the field of rational...
Chowla–Selberg formula Ramanujan–Sato series q-series Weierstrass'sellipticfunctions Partition function Kronecker limit formula Affine Lie algebra Siegel...
(℘, ℘) (actually a misnomer, name is corrected into WEIERSTRASSELLIPTICFUNCTION) Antiqua (typeface class) Blackletter Chancery hand Record type...
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