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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Chapter 3<br />
Power Series<br />
Definition. A(complex)power seriesisaninfiniteseries<strong>of</strong>theform �∞ n<br />
n=0an(z−z0) withz,z0,a0,a1,a2,... ∈ C. Thepointz0 iscalledthepoint <strong>of</strong> expansionfortheseries.<br />
Examples.<br />
1. For m ∈ N, we have<br />
m�<br />
n=0<br />
z n = 1−zm+1<br />
1−z<br />
if z �= 1. For |z|< 1, we obtain (letting m → ∞)<br />
2. For z ∈ C, define<br />
Let z �= 0, and note that<br />
∞�<br />
n=0<br />
z n = 1<br />
1−z .<br />
exp(z) :=<br />
�<br />
�<br />
�<br />
z<br />
�<br />
n+1<br />
��<br />
����<br />
�<br />
�<br />
z<br />
(n+1)! �<br />
n<br />
�<br />
�<br />
�<br />
n! �<br />
∞�<br />
n=0<br />
z n<br />
n! .<br />
= |z|<br />
n+1<br />
→ 0<br />
as n → ∞. As the ratio test holds for series with summands in C as well as for<br />
series over R, we conclude that exp(z) converges absolutely.<br />
Let z,w ∈ C, and note that the Cauchy product formula for series over R also<br />
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