Editing Principal value (section)

==Motivation==
Consider the [[complex logarithm]] function {{math|log ''z''}}. It is defined as the [[complex number]] {{mvar|w}} such that
:<math>e^w = z.</math>
Now, for example, say we wish to find {{math|log i}}. This means we want to solve
:<math>e^w = i</math> 
for <math>w</math>. The value <math>i\pi/2</math> is a solution.

However, there are other solutions, which is evidenced by considering the position of {{mvar|i}} in the [[complex plane]] and in particular its [[argument (complex analysis)|argument]] <math>\arg i</math>. We can rotate counterclockwise <math>\pi/2</math> radians from 1 to reach {{mvar|i}} initially, but if we rotate further another <math>2\pi</math> we reach {{mvar|i}} again. So, we can conclude that <math>i(\pi/2 + 2\pi)</math> is ''also'' a solution for {{math|log ''i''}}. It becomes clear that we can add any multiple of <math>2\pi</math> to our initial solution to obtain all values for {{math|log ''i''}}.

But this has a consequence that may be surprising in comparison of real valued functions: {{math|log ''i''}} does not have one definite value. For {{math|log ''z''}}, we have
:<math>\log{z} =  \ln{|z|} + i\left(\mathrm{arg}\ z \right)
= \ln{|z|} + i\left(\mathrm{Arg}\ z+2\pi k\right)</math>
for an [[integer]] {{mvar|k}}, where {{math|Arg ''z''}} is the (principal) argument of {{mvar|z}} defined to lie in the [[interval (mathematics)|interval]] <math>(-\pi,\ \pi]</math>. Each value of {{mvar|k}} determines what is known as a ''[[branch (mathematical analysis)|branch]]'' (or ''sheet''), a single-valued component of the multiple-valued log function.  When the focus is on a single branch, sometimes a [[branch cut]] is used; in this case removing the non-positive real numbers from the domain of the function and eliminating <math>\pi</math> as a possible value for {{math|Arg ''z''}}. With this branch cut, the single-branch function is [[continuous function|continuous]] and [[analytic function|analytic]] everywhere in its domain.

The branch corresponding to {{math|1=''k'' = 0}} is known as the ''principal branch'', and along this branch, the values the function takes are known as the ''principal values''.