Fonctions elliptiques weierstrass biography

Class of mathematical functions

"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 is also referred to as ℘-functions and they are usually denoted by the symbol&#;℘, a uniquely fancy scriptp. 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.

Motivation

A cubic of the form , where are complex numbers with , cannot be rationally parameterized. Yet one still wants to find a way to parameterize it.

For the quadric; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function: Because of the periodicity of the sine and cosine is chosen to be the domain, so the function is bijective.

In a similar way one can get a parameterization of by means of the doubly periodic -function (see in the section "Relation to elliptic curves"). This parameterization has the domain , which is topologically equivalent to a torus.

There is another analogy to the trigonometric functions. Consider the integral function It can be simplified by substituting and : That means . So the sine function is an inverse function of an integral function.

Elliptic functions are the inverse functions of elliptic integrals. In particular, let: Then the extension of to the complex plane equals the -function. This invertibility is used in complex analysis to provide a solution to certain nonlinear differential equatio

Class of periodic mathematical functions

In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are in turn named elliptic because they first were encountered for the calculation of the arc length of an ellipse.

Important elliptic functions are Jacobi elliptic functions and the Weierstrass -function.

Further development of this theory led to hyperelliptic functions and modular forms.

Definition

A meromorphic function is called an elliptic function, if there are two -linear independentcomplex numbers such that

and .

So elliptic functions have two periods and are therefore doubly periodic functions.

Period lattice and fundamental domain

If is an elliptic function with periods it also holds that

for every linear combination with .

The abelian group

is called the period lattice.

The parallelogram generated by and

is a fundamental domain of acting on .

Geometrically the complex plane is tiled with parallelograms. Everything that happens in one fundamental domain repeats in all the others. For that reason we can view elliptic function as functions with the quotient group as their domain. This quotient group, called an elliptic curve, can be visualised as a parallelogram where opposite sides are identified, which topologically is a torus.

Liouville's theorems

The following three theorems are known as Liouville's theorems ().

1st theorem

A holomorphic elliptic function is constant.

This is the original form of Liouville's theorem and can be derived from it. A holomorphic elliptic function is bounded since it takes on all of its values on the fundamental domain which is compact. So it is constant by Liouville's theorem.

2nd theorem

Every elliptic function has

Functions on which K. Weierstrass based his general theory of elliptic functions (cf. Elliptic function), exposed in in his lectures at the University of Berlin [1], [2]. As distinct from the earlier structure of the theory of elliptic functions developed by A. Legendre, N.H. Abel and C.G. Jacobi, which was based on elliptic functions of the second order with two simple poles in the period parallelogram, a Weierstrass elliptic function has one second-order pole in the period parallelogram. From the theoretical point of view the theory of Weierstrass is simpler, since the function $ \wp (z) $ , on which it is based, and its derivative serve as elliptic functions which generate the algebraic field of elliptic functions with given primitive periods.

The Weierstrass $ \wp $-function $ \wp (z) $ ( $ \wp $ is Weierstrass' notation) for given primitive periods $ 2 \omega _{1} ,\ 2 \omega _{3} $ , $ \mathop{\rm Im}\nolimits ( \omega _{3} / \omega _{1} ) > 0 $ , is defined as the series $$ \tag{1} \wp (z) = \wp (z; \ 2 \omega _{1} ,\ 2 \omega _{3} ) = $$ $$ = \frac{1}{z ^{2}} + \mathop{ {\sum'}} _ {m _{1} , m _{3} = - \infty } ^ {+ \infty} \left [ \frac{1}{(z-2 \Omega _ {m _{1} , m _{3}} ) ^{2} } - \frac{1}{(2 \Omega _ {m _{1} ,m _{3}} ) ^{2}} \right ] = $$ $$ = \frac{1}{z ^{2}} + c _{2} z ^{2} + c _{4} z ^{4} + \dots , $$ where $ \Omega _ {m _{1} , m _{3}} = m _{1} \omega _{1} +m _{3} \omega _{3} $ , and $ m _{1} ,\ m _{3} $ run through all integers except $ m _{1} = m _{3} = 0 $ . The function $ \wp (z) $ is an even elliptic function of order 2, with a unique second-order pole with zero residue in each period parallelogram. Its derivative $ \wp ^ \prime (z) $ is an odd elliptic function of order 3 with the same primitive periods; $ \wp ^ \prime (z) $ has simple zeros at points congruent with $ \omega _{1} ,\ \omega _{2} = \omega _{1} + \omega _{3} ,\ \omega _{3} $ . The most important property of the function $ \wp (z) $ is that any

(b. Ostenfelde, Westphalia, Germany, 31 October ; d. Berlin, Germany, 19 February ), Mathematics.

Weierstrass was the first child of Wilhelm Weierstrass, secretary to the mayor of Ostenfelde, and Theodora Vonderforst, who were married five months before his birth. The family name first appeared in Mettmann, a small town between Düsseldorf and Elberfeld; since the sixteenth century they had been artisans and small merchants. Weierstrass’ father, an intelligent, educated man with knowledge of the arts and sciences, could have held higher posts than he actually did; little is known about his mother’s family. Weierstrass had a brother and two sisters, none of whom ever married. When Weierstrass was eight his father entered the Prussian taxation service; and as a result of his frequent transfers, young Karl attended several primary schools. In , at the age of fourteen, he was accepted at the Catholic Gymnasium in Paderborn, where his father was assistant and subsequently treasurer at the main customs office.

A distinguished student at the Gymnasium, Weierstrass received several prizes before graduating. Unlike many mathematicians, he had no musical talent; nor did he ever acquire an interest in the theater, painting, or sculpture. He did, however, value lyric poetry and occasionally wrote verses himself. In , a year after his mother’s death, Weierstrass’ father remarried. At the age of fifteen Weierstrass reportedly worked as a bookkeeper for a merchant’s wife–both to utilize his abilities and to ease the strain of his family’s financial situation. A reader of Crelle’s Journal für die reine und angewandte Mathematik while in his teens, he also gave his brother Peter mathematical coaching that does not seem to have proved helpful: Weierstrass’ proofs were generally “knocking,” his brother later admitted.

After leaving the Gymnasium in , Weierstrass complied with his father’s wish that he study public finance and administra