The Origin of Elliptic Functions
Michael Somos
27 Nov 1999
The primary problem, as I see it, is to introduce elliptic functions in
a way that is as simple as possible and builds on knowledge of already
known functions and results. There is a variety of known ways to do this,
but, in my opinion, they all fail to provide a satisfactory answer.
The current approach is to emphasize the double periodicity of the
elliptic functions regarded as complex-valued functions. This can be
used abstractly, or else, by using doubly infinite sums or products to
explicitly define elliptic functions. The original approaches were thru
elliptic integrals and inversion of them, or as Gauss found through the
arithmetic-geometric mean algorithm. Other approaches are possible.
The approach I am advocating starts from almost nothing and builds up
the elliptic functions through the study of sequences of numbers that
satisfy certain simple functional equations. Examples of such sequences
would be my Somos sequences, but they are not suitable by themselves
since the functional equations seemingly come from nowhere.
So, there is a need to find some way to come up with the right kind of
functional equations which seem natural in some way. It may be actually
impossible to come up with a completely natural way, but we may find
something that comes close. At least I think so.
How is this going to work? Consider the example of the sine function. It
is defined for all complex numbers using a standard definition such as
the following : sin(z) = ( exp(i*z)-exp(-i*z) )/(2*i) . Instead of this,
we consider the sequence s(n) = sin(n*z) where z is some fixed complex
number. It is possible to define this sequence via the recursion :
s(n) = 2*cos(z)*s(n-1) - s(n-2) and initial values s(0)=0, s(1)=sin(z) .
Now there are a few little things wrong with this simple definition via
recursion. First, where did this recursion formula come from? Certainly
you can't pick a formula at random and expect to get any interesting
sequences from it. Second, the formula includes a parameter, namely the
"2*cos(z)", which is required for it. The other parameter, "sin(z)",
is included among the initial values where it is more appropriate there.
That said, it is still the simplest definition I can think of.
Where is the desired formula going to come from? Certainly it can't come
from a developed theory since that would require building up the theory
which would require an advanced knowledge, almost certainly requiring
prior knowledge of elliptic function themselves. This is unsatisfactory.
So, we are forced to rely on some ad hoc origin for the desired formula.
However, we don't want a deus ex machina. The formula can't spring forth
fully complete from Zeus' brow, so to speak.
That was our dilemma until something clicked just before Thanksgiving in
1999. The specific formulas had been known to me already as a consequence
of prior theories, and the critical detail about the sum of squares being
equal for three sets of numbers was an obvious detail that did not seem
very significant before. However, a shift in point of view made precisely
that detail the key missing starting point. This is often what happens in
mathematics. A theorem turned around becomes a definition.