Editing Principal branch (section)

=== Complex logarithms ===

One way to view a principal branch is to look specifically at the [[exponential function]], and the [[logarithm]], as it is defined in [[complex analysis]].

The exponential function is single-valued, where {{math|''e<sup>z</sup>''}} is defined as:

:<math>e^z = e^a \cos b + i e^a \sin b</math>
where <math>z = a + i b</math>.

However, the periodic nature of the trigonometric functions involved makes it clear that the logarithm is not so uniquely determined.  One way to see this is to look at the following:

:<math>\operatorname{Re} (\log z) = \log \sqrt{a^2 + b^2}</math>

and

:<math>\operatorname{Im} (\log z) = \operatorname{atan2}(b, a) + 2 \pi k</math>
where {{math|''k''}} is any integer and {{math|[[atan2]]}} continues the values of the {{math|arctan(b/a)}}-function from their principal value range <math>(-\pi/2,\; \pi/2]</math>, corresponding to <math>a > 0</math> into the principal value range of the {{math|arg(z)}}-function <math>(-\pi,\; \pi]</math>, covering all four quadrants in the complex plane.

Any number {{math|log ''z''}} defined by such criteria has the property that {{math|''e''<sup>log ''z''</sup> {{=}} ''z''}}.

In this manner log function is a [[multi-valued function]] (often referred to as a "multifunction" in the context of complex analysis).  A branch cut, usually along the negative real axis, can limit the imaginary part so it lies between {{math|−π}} and {{math|π}}. These are the chosen [[principal value]]s.

This is the principal branch of the log function.  Often it is defined using a capital letter, {{math|Log ''z''}}.