Editing Square root (section)

===Principal square root of a complex number===
{{Visualisation complex number roots}}
To find a definition for the square root that allows us to consistently choose a single value, called the [[principal value]], we start by observing that any complex number <math>x + i y</math> can be viewed as a point in the plane, <math>(x, y),</math> expressed using [[Cartesian coordinate system|Cartesian coordinates]]. The same point may be reinterpreted using [[polar coordinates]] as the pair <math>(r, \varphi),</math> where <math>r \geq 0</math> is the distance of the point from the origin, and <math>\varphi</math> is the angle that the line from the origin to the point makes with the positive real (<math>x</math>) axis. In complex analysis, the location of this point is conventionally written <math>r e^{i\varphi}.</math> If<math display=block>z = r e^{i \varphi} \text{ with } -\pi < \varphi \leq \pi,</math>
then the {{em|{{visible anchor|principal square root}}}} of <math>z</math> is defined to be the following:<math display=block>\sqrt{z} = \sqrt{r} e^{i \varphi / 2}.</math> 
The principal square root function is thus defined using the non-positive real axis as a [[branch cut]]. If <math>z</math> is a non-negative real number (which happens if and only if <math>\varphi = 0</math>) then the principal square root of <math>z</math> is <math>\sqrt{r} e^{i (0) / 2} = \sqrt{r};</math> in other words, the principal square root of a non-negative real number is just the usual non-negative square root. It is important that <math>-\pi < \varphi \leq \pi</math> because if, for example, <math>z = - 2 i</math> (so <math>\varphi = -\pi/2</math>) then the principal square root is<math display=block>\sqrt{-2 i} = \sqrt{2 e^{i\varphi}} = \sqrt{2} e^{i\varphi/2} = \sqrt{2} e^{i(-\pi/4)} = 1 - i</math>
but using <math>\tilde{\varphi} := \varphi + 2 \pi = 3\pi/2</math> would instead produce the other square root <math>\sqrt{2} e^{i\tilde{\varphi}/2} = \sqrt{2} e^{i(3\pi/4)} = -1 + i = - \sqrt{-2 i}.</math> 

The principal square root function is [[Holomorphic function|holomorphic]] everywhere except on the set of non-positive real numbers (on strictly negative reals it is not even [[Continuous function|continuous]]). The above Taylor series for <math>\sqrt{1 + x}</math> remains valid for complex numbers <math>x</math> with <math>|x| < 1.</math>

The above can also be expressed in terms of [[trigonometric function]]s:<math display=block>\sqrt{r \left(\cos \varphi + i \sin \varphi \right)} = \sqrt{r} \left(\cos \frac{\varphi}{2} + i \sin \frac{\varphi}{2} \right).</math>