The Mathematica Journal
Departments
Feature Articles
Columns
New Products
New Publications
Classifieds
Calendar
News Bulletins
Mailbox
Letters
Write Us
About the Journal
Staff and Contributors
Submissions
Subscriptions
Advertising
Back Issues
Home
Download this Issue

Obstacles in Dealing with More Complicated Functions

Let us now (try to) deal with a slightly complicated example, say [Graphics:../Images/index_gr_149.gif].

[Graphics:../Images/index_gr_150.gif]
[Graphics:../Images/index_gr_151.gif]

[Graphics:../Images/index_gr_152.gif]

A contour plot nicely shows the location of the branch cuts.

[Graphics:../Images/index_gr_153.gif]

[Graphics:../Images/index_gr_154.gif]

Numerically we can determine the branch cuts by looking for the lines in the complex plane where [Graphics:../Images/index_gr_155.gif], the argument of log, equals a negative real number and where [Graphics:../Images/index_gr_156.gif] equals a real number of absolute value greater than 1.

[Graphics:../Images/index_gr_157.gif]

[Graphics:../Images/index_gr_158.gif]

Looking at this picture it becomes obvious that even for this relatively simple function it would be a difficult undertaking to mesh the [Graphics:../Images/index_gr_159.gif]-plane to exclude the branch cuts. For more complicated functions the problem would be, of course, still more difficult. That is why we will not mesh the plane to exclude the branch cuts, but just avoid the branch points.

Many other "simple"-looking functions will show a similar behavior--the branch cuts will have a quite complicated shape. Here is one more example that requires quite a bit of calculation to describe analytically the nice symmetric shape of the branch cuts.

[Graphics:../Images/index_gr_160.gif]

[Graphics:../Images/index_gr_161.gif]

For algebraic functions, using elimination techniques, it is possible in principle--although computationally quite expensive--to produce an implicit equation in [Graphics:../Images/index_gr_162.gif], [Graphics:../Images/index_gr_163.gif], and [Graphics:../Images/index_gr_164.gif] or [Graphics:../Images/index_gr_165.gif] and then use ContourPlot3D to display the function. However, for transcendental functions this method is rarely applicable because the elimination of [Graphics:../Images/index_gr_166.gif] or[Graphics:../Images/index_gr_167.gif] is algorithmically not feasible, except in very simple examples. Here we use [Graphics:../Images/index_gr_168.gif] to get an implicit equation for [Graphics:../Images/index_gr_169.gif], [Graphics:../Images/index_gr_170.gif], and [Graphics:../Images/index_gr_171.gif].

[Graphics:../Images/index_gr_172.gif]
[Graphics:../Images/index_gr_173.gif]

To eliminate [Graphics:../Images/index_gr_174.gif], we rewrite the terms containing [Graphics:../Images/index_gr_175.gif]in a rational form.

[Graphics:../Images/index_gr_176.gif]
[Graphics:../Images/index_gr_177.gif]
[Graphics:../Images/index_gr_178.gif]
[Graphics:../Images/index_gr_179.gif]

Now we have arrived at an implicit equation that can be visualized using ContourPlot3D.

[Graphics:../Images/index_gr_180.gif]
Show[Graphics3D[{EdgeForm[],Thickness[0.0001],
SurfaceColor[Hue[0.82],Hue[0.22],1.37],
Cases[pic,_Polygon,∞]}]];[Graphics:../Images/index_gr_181.gif]

[Graphics:../Images/index_gr_182.gif]


Converted by Mathematica     

[Article Index] [Prev Page][Next Page]