Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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Problem 3.1. Show in Theorem 3.2 that the power series for f ′ and f have the same<br />
radius <strong>of</strong> convergence.<br />
Examples.<br />
1. exp ′ (z) = exp(z).<br />
2. sin ′ (z) = cos(z).<br />
3. cos ′ (z) = −sin(z).<br />
Corollary 3.2.1 (Higher Derivatives <strong>of</strong> Power Series). Let � ∞<br />
n=0 an(z − z0) n be a<br />
complex power series with radius <strong>of</strong> convergence R. Then<br />
f: BR(z0) → C, z ↦→<br />
∞�<br />
an(z −z0) n<br />
n=0<br />
is infinitely <strong>of</strong>ten complex differentiable on BR(z0) with<br />
f (k) (z) =<br />
∞�<br />
n(n−1)···(n−k +1)an(z −z0) n−k .<br />
n=k<br />
for z ∈ BR(z0) and k ∈ N. In particular, when z = z0 we see that<br />
holds for each n ∈ N0.<br />
an = 1<br />
n! f(n) (z0)<br />
Corollary 3.2.2 (Integration <strong>of</strong> Power Series). Let � ∞<br />
n=0 an(z −z0) n be a complex<br />
power series with radius <strong>of</strong> convergence R. Then<br />
F: BR(z0) → C, z ↦→<br />
is complex differentiable on BR(z0) with<br />
for z ∈ BR(z0).<br />
F ′ (z) =<br />
∞�<br />
n=0<br />
∞�<br />
an(z −z0) n<br />
n=0<br />
an n+1<br />
(z −z0)<br />
n+1<br />
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