Multi-valued complex maps

This applet is similar to Applet 2, but displays functions which are multi-valued rather than single-valued. Thus, dragging the mouse on the left grid will draw several curves on the right-hand side, rather than just one.

One of the values has been designated as the "principal" value of the function; this value is marked by a red ball. By making such a choice, we can restrict the multi-valued function to a single-valued function; this function is called a branch of the original function. Change "Show all branches" to "Principal branch only" to see this single-valued function.

Except for the rather silly multiple valued function f(z) = +/- z, all the branches here have a discontinuity at the negative real axis: if you move the mouse across this line, the principal branch will suddenly switch from one value of the multiple-valued function to another. The negative real line is called a branch cut for the principal branch. Branch cuts are a necessary evil that occur whenever one tries to prune a multiple-valued function into a single valued function.

Note: If you drag the mouse around too rapidly, the applet may not draw curves completely.

Notes on selected functions:

Previous applet: Möbius transforms

Next applet: The complex derivative

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