MATH253 Chapter Notes - Chapter 1-24: Ellipse, Geometric Progression, Joule

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

Document Summary

Reproduced with kind permission from james lewis"s informal lecture notes. We set ez := exeiy = ex(cos y + i sin y). Note that ez1ez2 = ez1+z2 directly from the de nition. Also |ez| = ex > 0, hence ez (cid:54)= 0 for all z c. [note that ez+2n i = eze2n i = ez for n = 0, 1, 2, . , hence ez is not a 1 1 function. This will imply that the inverse function , namely the complex log function (de ned below), will be multivalued. ] 2 = ie. (cid:18) 1 + i (cid:19) 2 (1 + i). e e(2+ i)/4 = e1/2e i/4 = The complex log function log z, is characterized by the property that z = elog z. If we write z = ew, then solving for w in terms of z will give us w = log z.