Transcription of SECTION 3 - University of Manitoba
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SECTION 3 - University of Manitoba
SECTION Complex Logarithm FunctionThe real logarithm function lnxis defined as the inverse of the exponential function y=lnxis the unique solution of the equationx=ey. This works becauseexis aone-to-one function; ifx16=x2, thenex16=ex2. This is not the case forez; we haveseen thatezis 2 i-periodic so that all complex numbers of the formz+2n iaremapped byw=ezonto the same complex number asz. To define the logarithmfunction, logz, as the inverse ofezis clearly going to lead to difficulties, and thesedifficulties are much like those encountered when finding the inverse function ofsinxin real-variable calculus. Let us proceed. We callwa logarithm ofz, and writew= logz,ifz=ew. To findwwe letw=u+vibe the Cartesian form forwandz=re ibe the exponential form forz.
The last two results must be approached with care. Because the logarithm function is multiple-valued, each equation must be interpreted as saying that given values for the logarithm terms, there is a value of k for which the equation holds. It is also possible to write these equations in the forms log(z1z2)=logz1 +logz2, (3.24a) log z1 z2 ...
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