Editing Branch point (section)

== Branch cuts ==
Roughly speaking, branch points are the points where the various sheets of a multiple valued function come together.  The branches of the function are the various sheets of the function.  For example, the function ''w''&nbsp;=&nbsp;''z''<sup>1/2</sup> has two branches: one where the square root comes in with a plus sign, and the other with a minus sign.  A '''branch cut''' is a curve in the complex plane such that it is possible to define a single analytic branch of a multi-valued function on the plane minus that curve.  Branch cuts are usually, but not always, taken between pairs of branch points.

Branch cuts allow one to work with a collection of single-valued functions, "glued" together along the branch cut instead of a multivalued function.  For example, to make the function

:<math>F(z) = \sqrt{z} \sqrt{1-z}\,</math>

single-valued, one makes a branch cut along the interval [0,&nbsp;1] on the real axis, connecting the two branch points of the function. The same idea can be applied to the function {{radic|''z''}}; but in that case one has to perceive that the ''point at infinity'' is the appropriate 'other' branch point to connect to from 0, for example along the whole negative real axis.

The branch cut device may appear arbitrary (and it is); but it is very useful, for example in the theory of special functions. An invariant explanation of the branch phenomenon is developed in [[Riemann surface]] theory (of which it is historically the origin), and more generally in the ramification and [[monodromy]] theory of [[algebraic function]]s and [[differential equation]]s.

=== 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.

=== Continuum of poles ===

One reason that branch cuts are common features of complex analysis is that a branch cut can be thought of as a sum of infinitely many poles arranged along a line in the complex plane with infinitesimal residues. For example,

: <math>
f_a(z) = {1\over z-a}
</math>

is a function with a simple pole at ''z''&nbsp;=&nbsp;''a''.  Integrating over the location of the pole:

: <math>
u(z) = \int_{a=-1}^{a=1} f_a(z) \,da = \int_{a=-1}^{a=1} {1\over z-a} \,da = \log \left({z+1\over z-1}\right)
</math>

defines a function ''u''(''z'') with a cut from &minus;1 to 1. The branch cut can be moved around, since the integration line can be shifted without altering the value of the integral so long as the line does not pass across the point ''z''.