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 New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: 629 18. Weierstrass Elliptic and Related Functions Mathematical Properties 18.1. Definitions, Symbolism, Restrictione and Conventions An elliptic function is a single-valued doubly periodic function of a single complex variable which is analytic except at poles and whose only singularities in the finite plane are poles. If u and o’ are a pair of (primitive) half-periods of such a function f(z), then f(z+2Mw+2Nw’) = j ( z ) , M and N being integers. Thus the study of any such function can be reduced to consideration of its behavior in a fundamental period parallelo- gram (FPP). An elliptic function has a finite number of poles (and the same number of zeros) in a FPP; the number of such poles (zeros) (an irreducible set) is the order of the function (poles and zeros are counted according to their multi- plicity). All other poles (zeros) are called con- W n t to the irreducible set. The simplest (non- trivial) elliptic functions are of order two. One may choose as the standard function of order two either a function with two simple poles (Jacobi’s choice) or one double pole (Weierstrass’ choice) in a FPP. Weierstrass’ @-Function. Let o, o‘ denote a pair of complex numbers with Y(w’/o)>O. Then @ ( z ) = @ (210, o’) is an elliptic function of order two with periods 2w, 2w’ and having a double pole at z=O, whose principal part is z - ~ ; @ ( Z ) - Z - ~ is analytic in a neighborhood of the origin, and van- ishes at z=O. Weierstrass’ S-Function f(z) = f(zlw, w’) satisfies the condition f’(z) = - @ ( 2 ) ; further, ((2) has a simple pole at z=O whose principal part is 2-l; T(z)--z-l vanishes at z=O and is analytic in a neighborhood of the origin. { ( z ) is NOT an elliptic €unction, since it is not periodic. However, it is quasi-periodic (see “period” relations), so reduction to FPP is possible. Weierstrass’ u-Function a(z) = u(zIw, o’) satisfies the condition u ’ ( z ) / u ( z ) = ~ ( z ) ; further, u(z) is an entire function which vanishes at the origin. Like {, it is NOT an elliptic function, since it is not periodic. However, it is quasi-periodic (see “period” relations), so reduction to FPP is pos- sible. Invariants ga and ga Let W=2Mw+2Nw’, M and N being integers. Then 18.1.1 g2=602‘W-4 and g3=1402‘W-6 are the INVARIANTS, summation being over all pairs M, N except M= N= 0. Alternate Symbolism Emphasizing Invariants 18.1.2 @ (4 = @ ( 2 ; g2, 93) 18.1.3 @’(z>=@’(z; 92, gal 18.1.4 I ( z ) = l ( z ; g2, 93) .(z> = 4; gat 93) 18.1.5 Fundamental Differential Equation, Discriminant and Related Quantities 18.1.6 @ ”(2)=4 @‘(z>-g2 @ (2)-g3 18.1.7 =4( @ (4 -el> ( P (4 -ed ( @ (4 -4 18.1.8 A=\$4--27&= 16(e2-e3)’(e3-el)2(el-e2)2 g2= -4 (ele2+ele3 +e2e3) = 2 (4 +eZ + 4) 18.1.9 18.1.10 g3=4eleze3=+(e:+ ei +e:) 18.1.11 el + e~ +e3=0 18.1.12 e:+e:+e:=&/8 18.1.13 4e?-g2er-g3=0(i=l, 2, 3) Agreement about Values of Invariants (and Discrim- inant) We shall consider, in this chapter, only real g2 and g3 (this seems to cover most applications)- hence A is real. We shall dichotomize most of what follows (either A>O or A