The Mathematica Journal
Feature Articles
New Products
New Publications
News Bulletins
Write Us
About the Journal
Staff and Contributors
Back Issues
Download this Issue

Why Use Multivalued Functions?

If we have a 1:1 function [Graphics:../Images/index_gr_17.gif], say [Graphics:../Images/index_gr_18.gif] for instance, then the inverse function is also 1:1, here [Graphics:../Images/index_gr_19.gif]. If a function is not 1:1, but rather many:1, then the inverse is no longer unique. It is now 1:many and we have various [Graphics:../Images/index_gr_20.gif] ([Graphics:../Images/index_gr_21.gif]) for (almost) each [Graphics:../Images/index_gr_22.gif]. (Here and in the following we will always use [Graphics:../Images/index_gr_23.gif] as the independent variable and [Graphics:../Images/index_gr_24.gif] as the dependent variable.) One of the simplest examples of a many:1 function is the exponential function [Graphics:../Images/index_gr_26.gif]. We have the periodicity property [Graphics:../Images/index_gr_27.gif]. If we restrict [Graphics:../Images/index_gr_28.gif] to vary over the domain [Graphics:../Images/index_gr_29.gif], then [Graphics:../Images/index_gr_30.gif] takes every complex number, with the exception of  0, exactly once.  We can invert the relation [Graphics:../Images/index_gr_31.gif] and obtain [Graphics:../Images/index_gr_32.gif]. The following parametric plot shows [Graphics:../Images/index_gr_33.gif] parametrized in the form [Graphics:../Images/index_gr_34.gif] for points from the strip [Graphics:../Images/index_gr_35.gif]. One clearly sees that the whole [Graphics:../Images/index_gr_36.gif]-plane is covered once and a discontinuity forms along the negative imaginary axis. (Using another shape of the strip, we would arrive at another line or curve of discontinuity.)



Here the real part of [Graphics:../Images/index_gr_40.gif] is shown. Because the real part is the same on all sheets we do not see a discontinuity here.



Here is the difference between the imaginary and the real part.



The resulting function will not be continuous along the negative real axis, but rather it has a discontinuity of size [Graphics:../Images/index_gr_45.gif]. Dividing the complex [Graphics:../Images/index_gr_46.gif]-plane into strips of the form [Graphics:../Images/index_gr_47.gif], we get infinitely many copies of complex planes. These copies can now be "glued together" along the negative real axis to form one Riemann surface of the function [Graphics:../Images/index_gr_48.gif]. This allows moving "smoothly" from one sheet to another. Here is [Graphics:../Images/index_gr_49.gif] for three such strips.



Using the concept of splitting the complex plane into subparts (which are all congruent here), all of the above mentioned multivalued functions can be constructed from their single-valued inverses. The names of the multivalued elementary transcendental functions Arc* strongly resemble their inverse functions. Here are some sheets of the Riemann surface of ArcSin.



Converted by Mathematica      May 9, 2000

[Trott's Corner Index] [Prev Page][Next Page]