Yes, my understanding of the history of elliptic functions is also what Gareth said. A Weierstrass "elliptic function" on the complex plane C is a meromorphic function that is doubly periodic (with periods required to be two complex numbers that are independent over R). The quotient of the complex plane by the period lattice of an elliptic function is topologically a torus, considered to be an "elliptic curve" in complex algebraic geometry. It can be shown to be biholomorphically equivalent to the locus in C^2 of an equation of the form (*) y^2 = x^3 + ax + b as long as the locus is non-singular. This is equivalent to knowing that the discriminant of the cubic is nonzero (equivalently, 4a^3 + 27b^2 != 0). This leads to all kinds of deep and unsolved questions about elliptic curves especially when they are defined by (*) or a similar equation over *finite* fields instead of C. They are typically considered to be *projective* "curves" defined by y^2 z = x^3 + ax z^2 + bz^3 in the complex projective plane CP^2. (Over C, these are once again tori.) ----------- On the other hand, *Jacobi* elliptic functions are closely related to the rotations of a solid object in a force-free 3-space about its center of gravity. This situation is what gives rise to the paradoxical motion of a general rectangular solid when one attempts to rotate it about its middle-size axis. (As has been said here before, you *can* try this at home: put a rubber band about a paperback book to keep it from opening, and then try to rotate it in mid-air about its left-right axis.) --Dan On 2013-02-12, at 1:39 PM, Gareth McCaughan wrote:

On 12/02/2013 21:08, Henry Baker wrote:

...

Are elliptical regions (those conic sections with 2 foci) in the complex plane particularly interesting in the study of "elliptic functions" ??

I don't know enough about elliptic functions to have any insights here.

Nor do I, but I do know (quite likely you do too) that the reason why the functions are called elliptic functions has nothing to do with elliptical regions in the complex plane; it's because some particular ones are useful for calculating the circumference of an ellipse.

(More precisely, "elliptic integrals" were first so called for that reason, and "elliptic functions" first arose as inverse functions of elliptic integrals.)

-- g

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