Sixty years ago, in an undergraduate course in complex variables, I saw that the laws of logarithms and exponents on the real line fail in the complex plane. The book I learned from provided a complicated and difficult to remember set of adjustments. Unfortunately, that same approach was built into my 2001 Advanced Engineering Mathematics text, and was, until a timely change, the approach taken in the ebook version available through Maplesoft.
Recently, I decided to digest the 1996 paper The Unwinding Number (Corless and Jeffries, SIGSAM Bulletin 116, 26 July 1996), a paper I had in my to-do stack for a long, long time. It turns out that the "unwinding number" approach to complex logarithms and exponents is far simpler than the tableau I learned as an undergraduate.
In this video, I'll sketch and give examples of both the 60-year old approach and the one based on the unwinding number, with the conclusion that the sooner the latter is adopted, the better.
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