ELLIPTIC FUNCTIONS. It is a familiar result of ele mentary integral calculus that, if R is a linear or quadratic func tion of a variable x, AIR, 'RJR and, more generally, f (x,,l R), where f is any rational function, can be integrated by means of elementary functions. In the next higher case, when R is a cubic or quartic with no repeated factor, no such integration is possible. It was gradually recognized by mathematicians of the 18th cen tury that such integrals were essentially new functions requiring special investigation. Particular cases of f Rdx and of f
,/R) dx occurred respectively in the problem of finding the length of an arc of an ellipse, and in that of the motion of a pendu lum, and on account of the former problem the name elliptic was given to the new functions. Some isolated results connecting two or more arcs of ellipses or of other curves were obtained in the first half of the 18th century, and an important formula connect ing two integrals was discovered in 1761 by Leonhard Euler (1707-83), a formula equivalent to one of what we now call addi tion theorems (sec. 3) . An algebraic transformation of one elliptic integral into another, which was afterwards found to be of great importance for numerical calculation, was given in 1775 by John Landen (1719-90), in a geometrical form.
The first systematic treatment of the new functions is due to Adrien Marie Legendre (1752-1833), whose main results were embodied in the great Traite des fonc tions elliptiques et des integrales Euleriennes (1825-28). In par ticular Legendre showed that any integral of the form f f (x.yR) dx, where, as before, R is a cubic or quartic and f is a rational function, can be reduced to the sum of an elementary function and of constant multiples of integrals of the three standard forms where 6,(0) =1/(1-
, and k, n are constants. These in tegrals are called Legendre's standard (or normal) elliptic in tegrals of the first, second and third kinds, respectively; and, if the integration is from o to 4), are denoted by F(k, 4)), E(k, 4)), II(n, k, 4)) ; 4) is called the amplitude, k the modulus, and n the parameter. If the original integral is real, k can be taken to be a positive number less than 1, and n to be real; if the restriction to reality is removed, k, n may be any real or complex numbers, subject to the exceptions that if
is o or 1 the integrals are elementary functions, and that if n is o the third integral reduces to the first. The integrals may be expressed in the equiva lent algebraic forms where x = sin 4) and
(1-
(1 -
;
may be called Legendre's normal form of the fundamental quartic.
The integrals in their algebraic forms have the functionally important properties that the first is finite for all real or complex values of x (including infinity), the second is simply-infinite (i.e. has a pole of order I) for
o , and the third is logarithmi cally-infinite for
= -1/n. If 4)= 7r/2, we have the corresponding complete integrals, sometimes denoted by
but the two former are more commonly called K and E respectively. If k is replaced by k' - sal(' -
the corresponding complete integrals are denoted by K' and E'. It should be noticed that F(k,
and E(k, 4) are functions of two independent arguments k and 4), II(n, k, 4)) of three arguments; but k, n usually (though not invariably) remain constants throughout any particular investi gation. Legendre gave extensive numerical tables of the integrals of the first and second kinds, which are the basis of nearly all modern tables (see bibliography).
Other standard forms of the integrals have been used by other writers; it is enough to give the important form of the integral of the first kind (where
and g3 are constants), first systematically used at a much later date by Karl Theodor Wilhelm Weierstrass
See sec. 6.
Bef ore Legendre's Traite was finished, the subject was revolutionized by Niels Henrik Abel (1802-29), working in close connection with Carl Gustav Jacob Jacobi (1804-51). In 1825 Abel inverted the re lation between the integral of the first kind F(k,4) and its am plitude 4), by treating 4) as a function of the integral, instead of following Legendre in treating the integral as a function of 4). In Jacobi's notation, if u=F(k,4)) then = amu, sin 4 = sin amu, cos 4 = cos amu, ,l (1-
_ Aamu.
The three latter functions so obtained, now more commonly written in the abbreviated forms sn u, cn u, dn u, or, if it is desirable to express the modulus k explicitly, sn(u,k) cn(u,k), dn(u, k), are called Jacobi's elliptic functions, as distinguished from Legendre's elliptic integrals, which, however, Legendre him self called fonctions elliptiques. Abel immediately recognized as a fundamental property the existence, in the case of each function, of two independent periods, which are respectively real and pure-imaginary in the standard case when o <
<1. In the particular degenerate case
= o, sn u, cn u become the trigono metrical functions sin u, cos u, with the real period 2 7r and dn u becomes I ; when
1, sn u becomes the hyperbolic function tank u, with the imaginary period i7r, and each of cn u, dn u be comes sec u, with the period 2i7r. It is now known that many of the fundamental results of Abel and Jacobi had been antici pated, but not published, by Carl Friedrich Gauss (1777-1855).
It is by no means evident that the elliptic functions thus pro visionally defined are definite functions of the argument u, still less that they are one-valued functions for complex as well as real values of u, as may be illustrated by the fact that in the next higher case, when the quartic is replaced by a quintic, the process of inversion breaks down completely. A satisfactory proof that the elliptic functions, as defined by inversion of an integral, are one-valued analytic functions only became possible after further development of the theory of functions of a com plex variable. From one point of view the essential simplifica tion due to inversion is that elliptic integrals, which are infinitely many-valued functions, are replaced by one-valued functions ; as in trigonometry it is obviously much simpler to work with the one-valued function x= sin 0 rather than with the many-valued inverse function 0= arc sin x.
3. Elementary Properties of Jacobi's Elliptic Functions. Jacobi's book Fundamenta nova theoriae functionum ellipticarum (1829) contains, among other results, the main properties of the new functions and of some allied functions. The derivatives of sn
cn u, dn a follow at once from the definitions. The very important formulae called addition-theorems express sn (u+v), etc. as rational functions of sn u, cn u, dn u and sn v, cn v, dn v, and are analogous to the familiar trigonometrical formulae for sin (A+B), etc. If u is increased by the complete integral K, it follows from the addition theorem that sn u is converted into a simple function of cn u, dn u, and if u is increased by 2K, sn u merely changes sign, so that sn u is periodic in 4K; if K' is the complete integral corresponding to the complementary modulus k', iK' has similar properties and in particular sn u is periodic in 2iK'; similarly for cn u, dn u. The transformations employed show that the three functions are infinite of order i (i.e., have simple poles) when u = iK', and, more generally, if u differs from iK' by any integral combination of 2K, 2iK' of the form 2mK+2niK' where m and n are integers. A very large number of formulae of interest follow immediately from the addition theorems and the periodicity.
in terms of the new variable u, the integral of the first kind. Modifying Legendre's choice of functions, Jacobi replaced E(k, 4)) by E(u, k) = .
du, and introduced an allied function 0 z(u) =E(u) — (E/K)u, where E is Legendre's complete integral of the second kind; Legendre's II(n, k, 0) is replaced by 4. Doubly-periodic Functions.—Jacobi's sn u, cn u, dn u are simple cases of a more general class of functions having the same or similar fundamental properties. We can take any two num bers 2w,
as periods, subject to the one restriction that r = w'/w is not real; then any one-valued analytical function f(u), which is doubly periodic in 2w,
so that f(u+2w) =f(u+2w') =f(u), and has no singularities other than poles for any finite value of u, may be called an elliptic function. It is convenient to use the geometrical representation of complex numbers (q.v.) by points in a plane (the Argand diagram) ; any parallelogram with vertices at a, a+ 2W, a+ 2w+ 2w', a+
(a arbitrary), is called a parallelogram of the periods; if we know the properties of f(u) throughout any one parallelogram, then from the periodicity we know its properties everywhere, so that it is enough to study one parallelogram.
were established by Joseph Liouville (1809-82), mainly by the method of contour-integration ; in particular (i.) an elliptic function f(u) which has no pole in a parallelo gram of the periods is a constant ; (ii.) the sum of the residues of f(u) at the poles in a parallelo gram is zero; (iii.) f(u) has at least two poles, or a multiple pole, in a parallelo gram ; (iv.) the number of poles in a parallelogram is equal to the num ber of zeros and is also equal to the number of points at which the function has any assigned value; (v.) the sum of the values of u at the poles in a parallelogram is equal to the sum of the values of a at the zeros or differs from it by some period.
are multiple, or if any of them lie on the boundary of the parallel ogram, and in the trivial case when f(u) is merely a constant). Liouville's theory gives simple proofs of important properties of the elliptic functions as well as of many elementary identities ; e.g., to prove the identity of two functions f(u), (a), it is enough to prove (I) that the periods are the same, (2) that in any parallelogram they have the same poles and zeros (of the same order), so that by (i.) their quotient is a constant and (3) that the quotient is i for some particular value of u.
In the Fundamenta Nova Jacobi introduced four allied functions 0(u), 9(u+K), H(u), H(u+K), each of which is holomorphic, i.e., is a one-valued analytic func tion with no singularity except at infinity; and showed that the quotients of H(u), H(u+K), e(u+K) by 0(u) are constant multiples of sn u, cn u, do u. He also showed that the integral of the third kind can be expressed in terms of these new functions.
these functions as a basis for the whole theory. Taking one period arbitrarily to be it and the other to be irr where r is an arbitrary complex number a-1--0, subject only to the condition that 13 is positive, we define four functions
(r=o, i, 2, 3) as series of sines or cosines of multiples of a variable x, the coeffi cients being of the form ± 2pn', where p = el' and n is an integer or half an odd integer. From the condition imposed on TIIPI < I the series converge for all values of x, and, except when IP is near i, with great rapidity. The addition of a half-period to x converts any 0 into another 0 multiplied by a simple factor, and the addition of a period merely multiplies any 0 by a simple factor, in such a way that the quotient of any two O's is doubly periodic.
Jacobi obtains a remarkable system of theta-product formulae, which are, in the first instance, linear relations between products of four 9's of eight variables, four of which are independent. By specializing the variables, a great number of interesting and im portant relations between 6's of 4, 3, 2, I or o independent vari ables are obtained. Certain of these formulae yield algebraical addition theorems for quotients of two O's, and differentiation follows by a simple limiting process ; and finally there is con structed, in the form of a quotient of O's, a function which satisfies the condition which defines sn u regarded as the inverse of the elliptic integral of the first kind. The process is entirely independent of any properties of the elliptic functions them selves, and is thus an independent method of solving the inver sion problem (sec. 2), and incidentally gives most of the funda mental properties of Jacobi's elliptic functions.
0 's are also expressible as singly-infinite products of trig onometrical factors, in forms from which the zeros are at once evident ; sn u, cn u, dn u and various simple combinations of them, as well as Z(u) are expressible as trigonometrical series, which unlike the
have only a limited range of convergence; and the constants k, k', K, etc. are expressible in a variety of forms as infinite products involving p , as ratios of series in p, or as series in p.
By inverting the integral of the first kind in the Weierstrassian form (sec. i ), we obtain an elliptic function p (u) which, unlike the Jacobian func tions, has a single pole of order 2 in any parallelogram. The theory is however more simply developed if we define p(u) not by in version, but as a double-series, which converges everywhere except at the poles, has two periods chosen arbitrarily subject to the condition that their ratio is not real, and has a double pole in each parallelogram. An easy application of Liouville's method gives the formula p'(u) = V{
, where
g3 are defined as series involving only the periods, and a solution of the inversion problem follows. Closely connected with p(u) are the functions ?'(u), defined as a doubly-infinite series with simple poles, and v(u) defined as a doubly-infinite product with no singularities. The three functions are connected by the equations
= — p(u), a'(u)/a(u) Neither nor a is periodic, but they undergo simple changes when u is increased by a period;
which is closely allied to Jacobi's z, can be taken as the integral of the second kind, and the in tegral of the third kind is simply expressible in terms of o (u).
the addition-theorem for p(u), formulae for the addition of half-periods, and to many other results, the development being in general much more symmetrical and elegant than for the Jacobian functions. With o(u) are associated three other func tions
(r= I, 2, 3), these four functions being closely related to the four 9's. The Weierstrassian and Jacobian functions are connected by equations of the form p(u) = A +
Q(u) =
el(cu), constants; and the properties of either set of functions can be derived from the others.
Any elliptic function f(u) can be expressed in any one of three standard forms: (i.) as a product of a's, divided by another product of a's, (ii.) as a linear combination of rs and their derivatives, (iii.) as a rational function of p(u) and p'(u).
From the second form the integral of f(u) is at once expressible in terms of o and 7. The Theory of Transformation, which originated with Abel and Jacobi as the problem of algebraic transformation of one elliptic integral into another, has developed into a very extensive subject, intimately connected on one side with algebraic equa tions and groups (q.v.), on another with the theory of numbers (see NUMBERS, THEORY OF). It lies outside our present scope in this article.
As elliptic integrals or functions are re quired for the integration of the square root of a cubic or quartic they necessarily occur in many mathematical problems. Among these problems are: In differential equations: the integration of Lame's equation, and of Briot and Bouquet's equation f (y, dy/dx) =o; In geometry: the arc of an ellipse, hyperbola, lemniscate; the parametric representation of the coordinates of a point on a non-singular, plane, cubic curve, or more generally on any curve of genus I ; Poncelet's polygons inscribed in one conic, circum scribed to another; the surface of an oblique cone and of an ellipsoid; geodesics on a quadric of revolution and umbilical geodesics on a general quadric ; In physics: the elastica, the motion of a projectile under a re sistance varying as the cube of the velocity, the simple and spherical pendulums, the motion of a rigid body under no forces, the symmetrical top; the potential of an ellipsoid ; various po tential problems in which occurs a rectangular boundary, or various other rectilinear boundaries.
M. Legendre's Traite des f cnctions elliptiques et Bibliography.-A. M. Legendre's Traite des f cnctions elliptiques et des integrates Euleriennes (1825-28) ; C. G. J. Jacobi, Fundamenta nova theoriae functionum ellipticarum (1829) ; Weierstrass's lectures, first published in Mathematische Werke, vols. v., vi. (1915) .
Text-books and treatises: A. G. Greenhill, The Applications of Ellip tic Functions (1892) ; A. C. Dixon, Elementary Properties of the Ellip tic Functions (1894) ; A. Cayley, Elliptic Functions (1895) ; H. Hancock, Theory of Elliptic Functions (191o) ; L. V. King, Direct Numerical Calculation of Elliptic Functions and Integrals (1924) .
J. Tannery and J. Molk, Elements de la theorie des fonctions ellip tiques (1893-1902) ; P. Appell and E. Lacour, Principes de la theorie des fonctions elliptiques (1922) .
A. Enneper and F. Muller, Elliptische Funktionen (1890) ; H. A. Schwarz Formeln and Lehrsdtze zum Gebrauch der elliptischen Funk tionen
; F. Klein and R. Fricke, Theorie der elliptischen Modulf unktionen (1890-92) ; H. Weber, Lehrbuch der Algebra (1908) ; A. Hurwitz and R. Courant, Allgemeine Funktionentheorie and elliptische Funktionen (1925) ; R. Fricke, Die elliptischen Funk tionen (vol. i., 1916, vol.
1923, vol. iii. in preparation) .
L. Bianchi, Funzioni di variabile complessa e delle funzioni ellittiche (1898-99).
See also R. Fricke in Encyklopddie der mathematischen Wissen schaften (vol. ii., 1901-21) ; Jordan, Cours d'Analyse (1913) ; Goursat, Cours d'Analyse (1925) ; Whittaker and Watson, Modern Analysis (1927) ; Pascal, Repertorium der hoheren Analysis (1927).
Numerical tables: Tables of F(k,c4) and E (k,4) to 9, 1o, 12 or 14 places of decimals, in Legendre's Traite, vol. ii. ; tables of the same functions to 5 places in J. B. Dale, Five Figure Tables of Mathematical Functions; J. Bertrand, Calcul integral; L. Levy, Precis elementaire de la theorie des fonctions elliptiques; K. Bohlin, Bih. Svenska Ak., vol. 25 (1900) ; tables to 4 places in Jahnke and Emde, Funktionen ta f eln, and in Houel Recueil de formules et de tables numeriques. Tables facilitating the calculation of K, E to 8 places are given by G. Witt in Astronomische Nachrichten (vol. 165) .
Tables of log p to 5 places are given by Jacobi in Crelle's Journal vol. 26 (reprinted in Opuscula Mathematica, vol. i., and in Gesammelte Werke vol. i.), by Bertrand and by Levy; to 4 places by Bohlin, Jahnke and Emde, and by Houel. Tables of the 6's as well as of F (k, 4) , E (k, 4) to io places, are given by R. L. Hippisley, Smith sonian Miscellaneous Collections (vol. 74, no. 1, 5922), and of the 9's and their logarithmic derivatives to 4 places by Jahnke and Emde and by Houel. Tables, mostly to 5 places, of p (u)
(u), Q (ii) for the special case g2=0, ga=l, were calculated by A. G. Hadcock for a paper by Greenhill in Proceedings of Royal Artillery Institution, (1889) and reprinted to 4 places by Jahnke and Emde.
(A. BER.)
