Principal value: Difference between revisions

Revision as of 18:54, 5 August 2013

Revision as of 17:01, 21 March 2014

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Now, arg ''z'' is intrinsically multivalued. One often defines the argument of some complex number to be between -π (exclusive) and π (inclusive), so we take this to be the principal value of the argument, and we write the argument function on this branch Arg ''z'' (with the leading capital A). Using Arg ''z'' instead of arg ''z'', we obtain the principal value of the logarithm, and we write

:<math>\mathrm{pv}\ \log{z} = \mathrm{Log}\ z = \ln{|z|} + i\left(\mathrm{Arg}\ z\right).</math>

====Exponential function====

So far we have only considered the logarithm function. What about [[Exponential function|exponents]]?

Consider <math>z ^\alpha\,</math> with <math> \alpha \in \mathbb{C}</math> . One usually defines ''z''^α to be ''e''^{α log ''z''}. Yet ''e''^{α log ''z''} is multiple-valued since we are using log as opposed to Log. Using Log we obtain the principal value of ''z''^α, i.e.,

: <math>\mathrm{pv}\ z^\alpha = e^{\alpha \mathrm{Log}\ z}.</math>

====Square root====