Taylor Series

Question 1

A taylor series

Answer 1

A taylor series (with centre zo) for a function f is a series of the form
SUM (from n=0, to inf) f^(n) (zo) / n! (z-zo)^n

Question 2

A Maclaurin series

Answer 2

A Maclaurin series for a function f is a Taylor series for f with centre 0, that is, a series of the form
SUM (n=0 to inf) f^(n)(0) / n! z^n.

Question 3

Taylor expansion of a holomorphic function for the taylor expansion of its derivative

Answer 3

Suppose that f e H(B(zo, r)), that f'(z) = SUM (n=0, inf) an (z-zo)^n in B(zo, r) and that r>0. Then 
f(z) =f(zo) + SUM (n=0, inf) an/ (n+1) (z-zo)^n+1 
= SUM (n=0, inf) bm (z -zo)^m 
in B(zo, r) where 
bm = { f(zo) when m=0, am-1/m when m>=1}