# Power series for complex function

1. Nov 17, 2009

1. The problem statement, all variables and given/known data
Find the power series for the function
f(z) = (1-z)^-m

Hint: Differentiation gives:
f'(z) = m(1-z)^m-1
= m(1-z)^-1.f(z)

or:
zf'(z) + mf(z) = f'(z)

Use the formula for differentiation of power series to determine the coefficients of the power series for f.

2. Relevant equations

3. The attempt at a solution

Hi, here's my attempt so far:

From real analysis,
(1-x)^-1 = Σ(a_n).x^n, where the sum is from n=0 to infinity

Differentiating:f'(x) (1-x)^-2 = Σ(a_n).n.(x^n-1) , from n=0 to infinity

and so on until we get that the j-th derivative

i.e. f^j.(x) = Σa_n(n C j).(x^n-j), where (n C j) is "n choose j" (binomial coefficient.

Thus, substituting z for x and m for j,

(1-x)^-m = Σa_n(n C m).(x^n-m), sum from 0 to infinity.

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I don't think this is correct; it seems too straightforward and I haven't used the hint.
Can someone please point me in the right direction?

Thanks for any help!

2. Nov 17, 2009

### turin

... not to mention that you still have an infinite number of undetermined coefficients.

They give you a first order ODE. Solve it using power series.