Control Systems
Control Systems
Control Systems
Analytic functions
Hart Smith
Department of Mathematics
University of Washington, Seattle
∞
X
0
Consequence: if f (z) = bk (z − z0 )k on DR (z0 ), then
k =0
∞
X bk −1
f (z) = f (z0 ) + (z − z0 )k , z ∈ DR (z0 )
k
k =1
Expansion of z −1 on |z − 1| < 1:
∞
1 1 X
= = (−1)k (z − 1)k
z 1 + (z − 1)
k =0
1
Expansion of on |z| < 1:
1 + z2
∞
1 X
= (−1)k z 2k
1 + z2
k =0
• If D|z0 | (z0 ) extends across the cut line, the series expansion
does not agree with that branch across the cut line.
Theorem: Cauchy’s estimates
If f (z) is analytic on an open set containing Dr (z0 ), then
|f (k ) (z0 )| M
≤ k, M = max |f (w)|
k! r |w−z0 |=r
∞
X
Remark. This proves that ak (z − z0 )k converges on Dr (z0 );
k =0
we already knew this from the proof that it equals f (z) there.