In case anyone might be interested, I thought I would mention that
finding the period for a 1-D anharmonic oscillator with a generic power
law spring is no harder than doing the special case that Ed asked about.
If we assume the restoring force is: F(x) = - sgn(x)*k*|x|^(a-1) and the
corresponding potential energy is: V(x) = (1/a)*|x|^a, (where a is a real
constant) then the period T for displacement amplitude A is given by Ed's
formula:
T = N*sqrt(m/k)*f(A)
where the constant N is given by:
N = sqrt(8*[pi]/a)*GAMMA(1/a)/GAMMA(1/a + 1/2) when a > 0, and is
N = sqrt(8*[pi]*|a|)*GAMMA(1/|a| + 1/2)/GAMMA(1/|a|) when a < 0.
In both cases the function f(A) is determined to be f(A) = A^(1 - a/2).