Differentiating a complex power series

[Pyroadept]

Homework Statement

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)

Homework Equations

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

 

[Dick]

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

 

[clamtrox]

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