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:

• f(z) = z^{1/2}. The square root function. How many square roots does a number have, and how are they related to each other? Notice that the phase of the square roots bisect the phase of the original number. Drag the mouse slowly once counterclockwise around the origin. In which direction does the square root function rotate, and by how much? Now select just the principal branch. Note that the value of the principal branch stays in one half-plane. Which one?

  Do you think one could make a branch of the square root function that did not jump at any point in the complex plane?

  Clear the screen, and draw some grid lines on the screen. What curves are drawn on the range? (Hint: they're conic sections). Do perpendicular lines remain perpendicular?

• f(z) = log(z). There are actually an infinite number of values to this function, but we only display three at a time. What happens to the logarithm if you move the mouse toward the origin? Away? (Slowly) counter-clockwise around the origin? Clockwise?

  What happens when you cross the branch cut?

  The values of the principal branch of the logarithm stay inside a certain strip. What is this strip? (You won't be able to reach all of it from this applet due to size restrictions on the domain grid).

Previous applet: Möbius transforms

Next applet: The complex derivative

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