MATH253 Chapter Notes - Chapter 1-24: Parametric Equation, Constant Function, Winding Number

• School: University of Alberta
• Department: Mathematics
• Course: MATH253
• Professor: Souvik Goswami

Document Summary

From james lewis"s lecture notes, with permission from the author. Assume given f (z) analytic on 0 < |z p| < r, F (z) = bn(z p) n + an(z p)n. (i) f (z) is said to have a removeable singularity at p, if bn = 0 for all n 1 n=1 n=0. Then (i) p is removeable limz p f (z) exists. (ii) p is a pole limz p f (z) = . (iii) p is an essential singularity limz p f (z) does not exist. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) |f (z)| (z p)m + + b1. Part (iii) is a consequence of the picard theorem and the proof is omitted. Thus the fact will follow from showing that if p is a pole, then limz p f (z) = (z p) + analytic part g(z), (where bm (cid:54)= 0), . 1 thus f (z) = bm (cid:12)(cid:12) (z p)m 1 + + (cid:18) |bm 1|