This routine produces a plot of the ranges of the branches of the complex multi-valued function f, if this information is known.

One of the major difficulties with multi-valued functions over the complex number field is determining a principal branch, and then assigning the values of the function consistently on the branch cut(s).  This latter assignment is referred to as closure.

branches can use one of two methods to indicate how closure on the branch cut(s) of the function f is defined: color or curve thickening. The default is to use color.  In this case, boundary curves of the same color belong to the same range.  This is usually sufficient to deduce the closure.  In addition, the principal branch of the function is labeled in the same color as its boundary.

For non-color devices, or for devices on which the color variation is insufficient, curve thickening can be used to indicate closure by specifying the option thick.  This will use a default line thickness to widen the boundary curve into the region to which it is attached.

Curve thickening is simulated by drawing parallel curves. If the default values for this mechanism are not adequate for the plotting device, or if there is still insufficient visual information available to deduce closure, the following additional options can be used.  (The presence of any of these options implies that curve thickening will be used.)

thick = nonnegative

gap = positive

skip = nonnegint

An additional option, colors, can be used to change the selection of colors used to display the plots.  The format is:

colors = list

The following functions are currently known to branches:

$\mathrm{branches}\left(\mathrm{arccosh}\right)$

$\mathrm{branches}\left(\arcsin \,\mathrm{thick}=0.1\,\mathrm{gap}=0.02\,\mathrm{skip}=2\right)$

$\mathrm{branches}\left(\mathrm{LambertW}\right)$

$\mathrm{branches}\left(\mathrm{LambertW}\,\mathrm{colors}=\left[\mathrm{black}\,''Purple''\,\mathrm{magenta}\right]\right)$

$\mathrm{branches}\left(\arccos \,\mathrm{show\_defaults}\right)$

thick = .6e-1
gap   = .2e-1
skip  = 0
colors= ["magenta", "blue", "red", "green", "cyan", "yellow"]