Editing Branch point (section)

=== Complex logarithm ===
[[File:Riemann surface log.svg|thumb|right|A plot of the multi-valued imaginary part of the complex logarithm function, which shows the branches. As a complex number ''z'' goes around the origin, the imaginary part of the logarithm goes up or down. This makes the origin a ''branch point'' of the function.]]
{{Main|Complex logarithm|Principal branch}}
The typical example of a branch cut is the complex logarithm.  If a [[complex number]] is represented in polar form ''z''&nbsp;=&nbsp;''r''e<sup>i''θ''</sup>, then the logarithm of ''z'' is
:<math>\ln z = \ln r + i\theta.\,</math>
However, there is an obvious ambiguity in defining the angle ''θ'': adding to ''θ'' any integer multiple of 2{{pi}} will yield another possible angle.  A branch of the logarithm is a continuous function ''L''(''z'') giving a logarithm of ''z'' for all ''z'' in a connected open set in the complex plane.  In particular, a branch of the logarithm exists in the complement of any ray from the origin to infinity: a ''branch cut''.  A common choice of branch cut is the negative real axis, although the choice is largely a matter of convenience.

The logarithm has a jump discontinuity of 2{{pi}}i when crossing the branch cut.  The logarithm can be made continuous by gluing together [[Countable set|countably]] many copies, called ''sheets'', of the complex plane along the branch cut.  On each sheet, the value of the log differs from its principal value by a multiple of 2{{pi}}i. These surfaces are glued to each other along the branch cut in the unique way to make the logarithm continuous. Each time the variable goes around the origin, the logarithm moves to a different branch.