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====Logarithm function==== We have examined the [[logarithm function]] above, i.e., :<math>\log{z} = \ln{|z|} + i\left(\mathrm{arg}\ z\right).</math> Now, {{math|arg ''z''}} is intrinsically multivalued. One often defines the argument of some complex number to be between <math>-\pi</math> (exclusive) and <math>\pi</math> (inclusive), so we take this to be the principal value of the argument, and we write the argument function on this branch {{math|Arg ''z''}} (with the leading capital A). Using {{math|Arg ''z''}} instead of {{math|arg ''z''}}, we obtain the principal value of the logarithm, and we write<ref>{{Cite book |last1=Zill |first1=Dennis |url=https://books.google.com/books?id=YKZqY8PCNo0C |title=A First Course in Complex Analysis with Applications |last2=Shanahan |first2=Patrick |date=2009 |publisher=Jones & Bartlett Learning |isbn=978-0-7637-5772-4 |pages=166 |language=en}}</ref> :<math>\mathrm{pv}\log{z} = \mathrm{Log}\,z = \ln{|z|} + i\left(\mathrm{Arg}\,z\right).</math>
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