Editing Branch point (section)

==Examples==

* 0 is a branch point of the [[square root]] function.  Suppose ''w''&nbsp;=&nbsp;''z''<sup>1/2</sup>, and ''z'' starts at 4 and moves along a [[circle]] of [[radius]] 4 in the [[complex plane]] centered at 0.  The dependent variable ''w'' changes while depending on ''z'' in a continuous manner.  When ''z'' has made one full circle, going from 4 back to 4 again, ''w'' will have made one half-circle, going from the positive square root of 4, i.e., from 2, to the negative square root of 4, i.e., &minus;2.
* 0 is also a branch point of the [[natural logarithm]].  Since ''e''<sup>0</sup> is the same as ''e''<sup>2{{pi}}''i''</sup>, both 0 and 2{{pi}}''i'' are among the multiple values of ln(1).  As ''z'' moves along a circle of radius 1 centered at 0, ''w'' = ln(''z'') goes from 0 to 2{{pi}}''i''.
* In [[trigonometry]], since tan({{pi}}/4) and tan (5{{pi}}/4) are both equal to 1, the two numbers {{pi}}/4 and 5{{pi}}/4 are among the multiple values of arctan(1).  The imaginary units ''i'' and &minus;''i'' are branch points of the arctangent function arctan(''z'') = (1/2''i'')log[(''i''&nbsp;&minus;&nbsp;''z'')/(''i''&nbsp;+&nbsp;''z'')].  This may be seen by observing that the derivative (''d''/''dz'') arctan(''z'') = 1/(1&nbsp;+&nbsp;''z''<sup>2</sup>) has simple [[pole (complex analysis)|poles]] at those two points, since the denominator is zero at those points.
* If the derivative ''ƒ''<nowiki>&nbsp;'</nowiki> of a function ''ƒ ''  has a simple [[pole (complex analysis)|pole]] at a point ''a'', then ''ƒ'' has a logarithmic branch point at ''a''.  The converse is not true, since the function ''ƒ''(''z'') = ''z''<sup>α</sup> for irrational α has a logarithmic branch point, and its derivative is singular without being a pole.