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Help with project needed


I'm about to attempt a project in which so many Mathematica related
things are unclear to me that I will describe the project, rather than
try to ask individual questions.

Suppose that C is a function of alpha, but it also depends on a
parameter epsilon, so I think of it as C[epsilon][alpha] -- but I'm OK
with it if the implementation treats C as a function of two variables.
This C is given. It's complicated, but one could easily obtain a
series for it in epsilon.

Now, I have the following operator that acts on functions of "r", and
"alpha" (d is partial):

L = C''(1-C')/(1+C^2) * d/dr + C'(1-C'') /(1-C')* d/dalpha

For a given function f[r, alpha], I need to take successive orders in
epsilon of the operator and apply it to f.

I realize that for a given function  f, I can form L[f] and the do
Series[L[f], {epsilon, 0, 5}]. However, I'm very interested in seeing
the operator decomposed into a series. I want to be able to say,
here's the epsilon term of the operator, here's the epsilon squared
term, and so forth. Can this be done. Can I decompose L into a series:

L = L0 + L1 + L2 + ....

and then be able to apply Ln to f?

Many thanks in advance,


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