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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18 CHAPTER 3. POWER SERIES<br />
Figure 3.3: Surface plot <strong>of</strong> sin(z) in the complex plane, using an RGB color wheel to<br />
represent the phase. Red indicates real positive values.<br />
Since r<br />
r ′ < 1, we have �∞ � ∞<br />
n=0<br />
�<br />
r<br />
r ′<br />
�n < ∞. The Weierstraß M-test thus yields that<br />
n=0 an(z −z0) n converges absolutely and uniformly on Br[z0].<br />
Since every z ∈ BR(z0) is contained in Br[z0] for some r ∈ [0,R), it follows that<br />
� ∞<br />
n=0 an(z −z0) n converges absolutely for each such z.<br />
Let z /∈ BR[z0], i.e. |z −z0|> R, so that<br />
1 1<br />
<<br />
|z −z0| R<br />
and thus, for infinitely many n ∈ N,<br />
or, equivalently,<br />
= limsup<br />
n→∞<br />
1<br />
|z −z0| < n� |an|<br />
1 < |an(z −z0) n |.<br />
�<br />
n<br />
|an|