Hi! I need some help with the following problem:

Let f(z) be analytic in |z|< 1, and f(0) = 0. Define F(z)=f(z)/z for every z in 0<|z|<1. What value can we give to F(0) for F(z) to be analytic in |z|<1? (Hint: Apply Cauchy's integral formula for derivatives to F(z) in |z|=r < 1. Then show that the resulting function, analytic in |z|< 1, concurs with F in 0<|z|<r. Use partial fractions.)

So, Cauchy's integral formula for derivatives states that

But I'm not sure how to apply it to F(z).

Any help will be appreciated.