Transcription of The complex logarithm, exponential and power functions
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The complex logarithm, exponential and power functions
Physics 116 AWinter 2011 The complex logarithm, exponential and power functionsIn these notes, we examine the logarithm, exponential and power functions , wherethe arguments of these functions can be complex numbers. In particular, we areinterested in how their properties differ from the properties of thecorrespondingreal-valued functions . 1. Review of the properties of the argument of a complex numberBefore we begin, I shall review the properties of the argument of anon-zerocomplex numberz, denoted by argz(which is a multi-valued function), and theprincipal valueof the argument, Argz, which is single-valued and conventionallydefined such that: <Argz .(1)Details can be found in the class handout entitled,The argument of a complexnumber.
where the integer Nn is given by: Nn = 1 2 − n 2π Arg z , (16) and [ ] is the greatest integer bracket function introduced in eq. (4). 2. Properties of the real-valued logarithm, exponential and power func-
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