Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site watmath.UUCP Path: utzoo!watmath!csc From: csc@watmath.UUCP (Computer Sci Club) Newsgroups: net.math Subject: Re: References on i ** i, "principal logs" Message-ID: <6964@watmath.UUCP> Date: Tue, 21-Feb-84 12:18:14 EST Article-I.D.: watmath.6964 Posted: Tue Feb 21 12:18:14 1984 Date-Received: Wed, 22-Feb-84 01:59:04 EST References: <205@pucc-i>, <1960@mcnc.UUCP>, <1964@mcnc.UUCP> Organization: U of Waterloo, Ontario Lines: 28 The complex log fuction can be defined in a number of ways. Perhaps the most intuitive is as the inverse of the exponential function (which can be defined as the sum of an infinite series). Hence we define log(z) to be a complex number such that exp(log(z))=z. Now any non zero complex number z can be written as exp(r + it), (r and t real numbers). r is uniquely determined but t isn't. If t is valid so is t+2n(pi) with n an integer. Therefore there are an infinitely many complex numbers g such that exp(g)=z. (if z is non zero). Hence log is multi valued. There is no complex number g such that exp(g)=0, therefore log(0) is undefined. Hence log is a multi valued function defined on the complex plane minus zero. It does not make sense to define log(0)=0 as then exp(log(0))=1. The article which argued that log(0)=0 contained a division by zero which implied i*2n(pi)=i*2n(pi)exp(i*(pi)/2) or as exp(i*pi/2)=i this implies i*2n(pi)=-2n(pi). A contradiction. It is because dividing by zero leads to such contradictions that such division is not defined. One can define a single valued log function by choosing one value of t for each z, usually done by restricting t to some half open interval of length 2(pi). However one cannnot do this in such a way as to have the resulting function continuous on the complex plane (minus zero). Also such equations as log(ab)=log(a) + log(b) and log(exp(z))=z cannot hold for all a,b,z. The usual practice is to use whichever "branch" (ie. choice of an interval for t) that is most convenient for the task at hand. William Hughes