--- asimovd@aol.com wrote:
Gene wrote:
<< The proper place to introduce elliptic functions is in a course in complex variables,
I'm moved to mention that there are several opverlapping meanings for
"elliptic functions".
There are the Jacobi elliptic functions cn, sn, dn, so named since they were originally designed to compute the arclength of an ellipse. Cf. < http://mathworld.wolfram.com/JacobiEllipticFunctions.html>.
There are also the more general just plain "elliptic functions", which are any doubly periodic meromorphic (analytic except for poles) functions on C, of which the Jacobi elliptic functions are a special case (as are the Weierstrass elliptic functions). Cf. <http://mathworld.wolfram.com/EllipticFunction.html>. (The general elliptic function can be thought of as a nonconstant homolorphic map from a Riemann surface of genus 1 to one of genus 0 (a torus to a sphere).)
I'm not sure why these more general functions are called "elliptic". (Maybe the ellipse arclength functions just kept getting generalized and the name stuck?)
Yes, the calculation of the arclength of an ellipse is a minor historical footnote in the theory of elliptic functions.
In any case, a whole lot more about the elliptic integrals that underlie elliptic functions can be found at < http://mathworld.wolfram.com/EllipticIntegral.html>.
--Dan
The modern approach to the theory regards the elliptic functions as more fundamental (or, better stated, as conceptually simpler) than the elliptic integrals. By analogy, the circular function sin is simpler than the circular integral asin x = int(1/sqrt(1 - t^2), t=0..x). The amount of algebraic manipulation required is far greater if one tries to derive the elliptic functions from the integrals rather than vice versa. __________________________________ Do you Yahoo!? SBC Yahoo! DSL - Now only $29.95 per month! http://sbc.yahoo.com