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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.)

[Graphics:../Images/index_gr_37.gif]
[Graphics:../Images/index_gr_38.gif]

[Graphics:../Images/index_gr_39.gif]

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.

[Graphics:../Images/index_gr_41.gif]

[Graphics:../Images/index_gr_42.gif]

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

[Graphics:../Images/index_gr_43.gif]

[Graphics:../Images/index_gr_44.gif]

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.

[Graphics:../Images/index_gr_50.gif]

[Graphics:../Images/index_gr_51.gif]

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.

[Graphics:../Images/index_gr_52.gif]

[Graphics:../Images/index_gr_53.gif]


Converted by Mathematica      May 9, 2000

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