Funzioni ellittiche di Jacobi

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In matematica, le funzioni ellittiche di Jacobi costituiscono una famiglia di funzioni ellittiche basilari che sono state introdotte dal matematico tedesco Carl Gustav Jakob Jacobi intorno al 1830. Esse e le funzioni teta (queste con ruoli ausiliari) hanno importanza storica e presentano molte caratteristiche che contribuiscono a far emergere una importante struttura; inoltre hanno diretta rilevanza per talune applicazioni, ad esempio per le equazioni del pendolo. Esse inoltre presentano utili analogie con le funzioni trigonometriche, come rivelato dalla scelta della notazione sn per una funzione associabile alla funzione sin. Oggi sappiamo che le funzioni ellittiche di Jacobi non sono gli strumenti più semplici per lo sviluppo di una teoria generale, come si vede anche nell'attuale articolo: strumenti migliori sono le funzioni ellittiche di Weierstrass. Le funzioni di Jacobi presentano comunque vari motivi di interesse.

Indice

1 Introduzione
2 Notazione
3 Definizione come funzioni inverse degli integrali ellittici
4 Definizione che si serve delle funzioni teta
5 Funzioni di importanza minore
6 Teeoremi di addizione
7 Relazioni tra i quadrati delle funzioni
8 Sviluppo in termini del nome
9 Bibliografia

[modifica] Introduzione

Ci sono dodici funzioni ellittiche Jacobiane. Ognuna di queste corrisponde a una freccia tracciata da un angolo a un altro di uno stesso rettangolo. Gli angoli del rettangolo vengono chiamati, per convenzione, s, c, d, n. The rectangle is understood to be lying on the complex plane, so that s is at the origin, c is at the point K on the real axis, d is at the point K +iK' and n is at point iK' on the imaginary axis. The numbers K and K' are called the quarter periods. The twelve Jacobian elliptic functions are then pq, where p and q are one of the letters s,c,d,n.

The Jacobian elliptic functions are then the unique doubly-periodic, meromorphic functions satisfying the following three properties:

There is a simple zero at the corner p, and a simple pole at the corner q.
The step from p to q is equal to half the period of the function pq u; that is, the function pq u is periodic in the direction pq, with the period being twice the distance from p to q. Also, pq u is also periodic in the other two directions as well, with a period such that the distance from p to one of the other corners is a quarter period.
If the function pq u is expanded in terms of u at one of the corners, the leading term in the expansion has a coefficient of 1. In other words, the leading term of the expansion of pq u at the corner p is u; the leading term of the expansion at the corner q is 1/u, and the leading term of an expansion at the other two corners is 1.

The Jacobian elliptic functions are then the unique elliptic functions that satisfy the above properties.

More generally, there is no need to impose a rectangle; a parallelogram will do. However, if K and iK' are kept on the real and imaginary axis, respectively, then the Jacobi elliptic functions pq u will be real functions when u is real.

[modifica] Notazione

The elliptic functions can be given in a variety of notations, which can make the subject un-necessarily confusing. Elliptic functions are functions of two variables. The first variable might be given in terms of the amplitude φ, or more commonly, in terms of u given below. The second variable might be given in terms of the parameter m, or as the elliptic modulus k, where $k 2 = m$ , or in terms of the modular angle $α$ , where $m = sin 2 α$ . A more extensive review and definition of these alternatives, their complements, and the associated notation schemes are given in the articles on elliptic integrals and quarter period.

[modifica] Definizione come funzioni inverse degli integrali ellittici

The above definition, in terms of the unique meromorphic functions satisfying certain properties, is quite abstract. There is a simpler, but completely equivalent definition, giving the elliptic functions as inverses of the incomplete elliptic integral of the first kind. This is perhaps the easiest definition to understand. Let

$u=\int_0^\phi \frac{d\theta} {\sqrt {1-m \sin^2 \theta}}$

Then the elliptic function sn u is given by

$\operatorname {sn}\; u = \sin \phi$

and cn u is given by

$\operatorname {cn}\; u = \cos \phi$

and

$\operatorname {dn}\; u = \sqrt {1-m\sin^2 \phi}$ .

Here, the angle $φ$ is called the amplitude. On occasion, $\operatorname {dn}\; u = \Delta(u)$ is called the delta amplitude. In the above, the value m is a free parameter, usually taken to be real, $0\leq m \leq 1$ , and so the elliptic functions can be thought of as being given by two variables, the amplitude $φ$ and the parameter m.

The remaining nine elliptic functions are easily built from the above three, and are given in a section below.

Note that when $φ = π / 2$ , that u then equals the quarter period K.

[modifica] Definizione che si serve delle funzioni teta

Equivalently, Jacobi's elliptic functions can be defined in terms of his theta functions. If we abbreviate $\vartheta(0;\tau)$ as $\vartheta$ , and $\vartheta_{01}(0;\tau), \vartheta_{10}(0;\tau), \vartheta_{11}(0;\tau)$ respectively as $\vartheta_{01}, \vartheta_{10}, \vartheta_{11}$ (the theta constants) then the elliptic modulus k is $k=({\vartheta_{10} \over \vartheta})^2$ . If we set $u = \pi \vartheta^2 z$ , we have

$\mbox{sn}(u; k) = -{\vartheta \vartheta_{11}(z;\tau) \over \vartheta_{10} \vartheta_{01}(z;\tau)}$

$\mbox{cn}(u; k) = {\vartheta_{01} \vartheta_{10}(z;\tau) \over \vartheta_{10} \vartheta_{01}(z;\tau)}$

$\mbox{dn}(u; k) = {\vartheta_{01} \vartheta(z;\tau) \over \vartheta \vartheta_{01}(z;\tau)}$

Since the Jacobi functions are defined in terms of the elliptic modulus $k (τ)$ , we need to invert this and find τ in terms of k. We start from $k' = \sqrt{1-k^2}$ , the complementary modulus. As a function of τ it is

$k'(\tau) = ({\vartheta_{01} \over \vartheta})^2$

Let us first define

$\ell = {1 \over 2} {1-\sqrt{k'} \over 1+\sqrt{k'}} = {1 \over 2} {\vartheta - \vartheta_{01} \over \vartheta + \vartheta_{01}}$

Then define the nome q as $q = exp(π i τ)$ and expand $\ell$ as a power series in the nome q, we obtain

$\ell = {q+q^9+q^{25} \cdots \over 1+2q^4+2q^{16} \cdots}$

Reversion of series now gives

$q = \ell+2\ell^5+15\ell^9+150\ell^{13}+1707\ell^{17}+20910\ell^{21}+268616\ell^{25}+\cdots$

Since we may reduce to the case where the imaginary part of τ is greater than or equal to $\sqrt{3}/2$ , we can assume the absolute value of q is less than or equal to $\exp(-\pi \sqrt{3}/2)$ ; for values this small the above series converges very rapidly and easily allows us to find the appropriate value for q.

[modifica] Funzioni di importanza minore

It is conventional to denote the reciprocals of the three functions above by reversing the order of the two letters of the function name:

$\operatorname{ns}(u)=1/\operatorname{sn}(u)$

$\operatorname{nc}(u)=1/\operatorname{cn}(u)$

$\operatorname{nd}(u)=1/\operatorname{dn}(u)$

The ratios of the three primary functions are denoted by the first letter of the numerator followed by the first letter of the denominator:

$\operatorname{sc}(u)=\operatorname{sn}(u)/\operatorname{cn(u)}$

$\operatorname{sd}(u)=\operatorname{sn}(u)/\operatorname{dn(u)}$

$\operatorname{dc}(u)=\operatorname{dn}(u)/\operatorname{cn(u)}$

$\operatorname{ds}(u)=\operatorname{dn}(u)/\operatorname{sn(u)}$

$\operatorname{cs}(u)=\operatorname{cn}(u)/\operatorname{sn(u)}$

$\operatorname{cd}(u)=\operatorname{cn}(u)/\operatorname{dn(u)}$

More compactly, we can write

$\operatorname{pq}(u)=\frac{\operatorname{pr}(u)}{\operatorname{qr(u)}}$

where p, q, and r are any of the letters s, c, d, n, with the understanding that ss=cc=dd=nn=1.

[modifica] Teeoremi di addizione

The functions satisfy the two algebraic relations

$\operatorname{cn}^2 + \operatorname{sn}^2 = 1$

$\operatorname{dn}^2 + k^2 \operatorname{sn}^2 = 1$

From this we see that (cn, sn, dn) parametrizes an elliptic curve which is the intersection of the two quadrics defined by the above two equations. We now may define a group law for points on this curve by the addition formulas for the Jacobi functions

$\operatorname{cn}(x+y) = {\operatorname{cn}(x)\;\operatorname{cn}(y) - \operatorname{sn}(x)\;\operatorname{sn}(y)\;\operatorname{dn}(x)\;\operatorname{dn}(y) \over {1 - k^2 \;\operatorname{sn}^2 (x) \;\operatorname{sn}^2 (y)}}$

$\operatorname{sn}(x+y) = {\operatorname{sn}(x)\;\operatorname{cn}(y)\;\operatorname{dn}(y) + \operatorname{sn}(y)\;\operatorname{cn}(x)\;\operatorname{dn}(x) \over {1 - k^2 \;\operatorname{sn}^2 (x)\; \operatorname{sn}^2 (y)}}$

$\operatorname{dn}(x+y) = {\operatorname{dn}(x)\;\operatorname{dn}(y) - k^2 \;\operatorname{sn}(x)\;\operatorname{sn}(y)\;\operatorname{cn}(x)\;\operatorname{cn}(y) \over {1 - k^2 \;\operatorname{sn}^2 (x)\; \operatorname{sn}^2 (y)}}$

[modifica] Relazioni tra i quadrati delle funzioni

$-\operatorname{dn}^2(u)+m_1= -m\;\operatorname{cn}^2(u) = m\;\operatorname{sn}^2(u)-m$

$-m_1\;\operatorname{nd}^2(u)+m_1= -mm_1\;\operatorname{sd}^2(u) = m\;\operatorname{cd}^2(u)-m$

$m_1\;\operatorname{sc}^2(u)+m_1= m_1\;\operatorname{nc}^2(u) = \operatorname{dc}^2(u)-m$

$\operatorname{cs}^2(u)+m_1=\operatorname{ds}^2(u)=\operatorname{ns}^2(u)-m$

where $m + m 1 = 1$ and $m = k 2$ .

Additional relations between squares can be obtained by noting that $\operatorname{pq}^2 \cdot \operatorname{qp}^2 = 1$ and that $\operatorname{pq}=\operatorname{pr}/\operatorname{qr}$ where p,q,r are any of the letters s,c,d,n and ss=cc=dd=nn=1.

[modifica] Sviluppo in termini del nome

Let the nome be $q = exp( - π K' / K)$ and let the argument be $v = π u / (2 K)$ . Then the functions have expansions as Lambert series

$\operatorname{sn}(u)=\frac{2\pi}{K\sqrt{m}} \sum_{n=0}^\infty \frac{q^{n+1/2}}{1-q^{2n+1}} \sin (2n+1)v$

and

$\operatorname{cn}(u)=\frac{2\pi}{K\sqrt{m}} \sum_{n=0}^\infty \frac{q^{n+1/2}}{1+q^{2n+1}} \cos (2n+1)v$

and

$\operatorname{dn}(u)=\frac{\pi}{2K} + \frac{2\pi}{K} \sum_{n=0}^\infty \frac{q^{n}}{1+q^{2n}} \cos 2nv$

[modifica] Bibliografia

Milton Abramowitz, Irene A. Stegun, eds. (1972): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover. (Vedi Chapter 16.)
Naum Illyich Akhiezer (1990): Elements of the Theory of Elliptic Functions, AMS Translations of Mathematical Monographs Volume 79, AMS, Rhode Island, ISBN 0-8218-4532-2. Traduzione in inglese del testo in russo pubblicato a Mosca nel 1970.
Shanje Zhang, Janming Jin (1996): Computation of Special Functions, J.Wiley. (Vedi Chapter 18)

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Categorie: Da tradurre - matematica | Da tradurre dalla lingua inglese | Da tradurre febbraio 2006 | Funzioni speciali