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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Figure 3.2: Surface plot <strong>of</strong> cos(z) in the complex plane, using an RGB color wheel to<br />
represent the phase. Red indicates real positive values.<br />
Pro<strong>of</strong>. The uniqueness <strong>of</strong> R follows from the first and the last property.<br />
Let R ∈ [0,∞] be defined by the Cauchy–Hadamard formula (we set 1<br />
= ∞ and 0<br />
1 = 0). ∞<br />
Let r ∈ [0,R), and choose r ′ ∈ (r,R). It follows that<br />
�<br />
n<br />
limsup |an| =<br />
n→∞<br />
1 1<br />
<<br />
R r ′,<br />
so that there exists n0 ∈ N such that n� |an| < 1<br />
r ′ whenever n ≥ n0, i.e.<br />
�<br />
1<br />
|an|<<br />
r ′<br />
�n for all n ≥ n0. For n ≥ n0 and z ∈ Br[z0], we then have<br />
|an(z −z0) n �<br />
r<br />
|≤<br />
r ′<br />
�n .<br />
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