Differentiating a complex power series

1. Nov 17, 2009

1. The problem statement, all variables and given/known data

Say f(z) = Σ(z^n), with sum from 0 to infinity

Then we can say f'(z) = Σn(z^n-1), with sum from 0 to infinity (i)

= Σn(z^n-1), with sum from 1 to infinity (as the zero-th term is 0)

= Σ(n+1)(z^n), with sum from 0 to infinity (ii)

2. Relevant equations

3. The attempt at a solution
Hi,

I am solving an ODE using power series, zf'(z) + af(z) = f'(z), so is it ok for me to sub in eq. (i) for the first f'(z) and eq. (ii) for the second? (This way all the terms will contain a z^n. Or is this cheating? If so, what can I do instead?)

Thanks for any help

2. Nov 17, 2009

Dick

Well, no. You aren't cheating. That seems like the right way to shift the sum. Carry on.

3. Nov 18, 2009

clamtrox

Don't forget the constants in the sum, $$f(z) = \sum a_n z^n$$