Gene wrote:

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The proper place to introduce elliptic functions is in a course in complex variables,
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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?)

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