Here is an outline of a proof that is not elementary. By this
I wish to prove that **W** is not contained in any tower of logarithmic,
exponential, or algebraic extensions. I do not have all the details
cold, yet.

Suppose to the contrary that but is not in .

This means that we can express

where the coefficients of **p** and **q** are in **F**, and .
Expand this in the
square-free partial fraction decomposition.

Now suppose that is a transcendental extension. Then when we differentiate each side, we get

and on the right we get

where the indicates differentiation with respect to **x**. If is
a logarithmic extension, will not alter the degrees of the
square-free factorization, nor will they affect the partial fraction expansion.
If is an exponential extension, then we have to check if it is
a monomial (because then it can divide ) which I have not done.
But otherwise it does not affect the expansion. (Actually even if
is a monomial it doesn't matter---the expansion is still in terms of
factors of **q**).

But the square-free decomposition of the **p + q** form above is completely
different, because **p** and **q** have no common factors as polynomials in . Hence cannot be a transcendental extension.

Since the ordering of the extensions is arbitrary, we can conclude that
**W** must be in a purely algebraic extension of .

However, I believe this is impossible also. cannot occur
for **a** algebraic and **r** rational, because is always transcendental
(this is a known result). Therefore is transcendental for every
rational **r**. Therefore cannot be contained in any chain of
algebraic extensions of .

Hence **W** is not elementary.

** Remark.** Just because is not elementary does not mean the
Risch algorithm doesn't apply! With the change of variable
and , so , we can transform any
elementary function of **x** and into an elementary function of
**w**, for which the Risch algorithm applies. Some people say that a
function is Liouvillian if its derivative is a rational function of
**x** and itself; according to this definition is Liouvillian
(but not elementary).

So the moral is that the notion of elementary function is a bit more
slippery than just this `finite tower' of elementary extensions notion,
because if we can change variables to fit into such a tower, then we
have a function that is `morally equivalent' to an elementary function.
Integration of sn, cn, and dn are in such a class as well (put **u = **am,
the Jacobian amplitude function; then sn, and cn and dn
can also be expressed in terms of **u**, and **du = **dn.).

But change of variables is a heuristic...

Thu Nov 23 10:59:42 PST 1995