Obstacles in Dealing with More Complicated FunctionsLet us now (try to) deal with a slightly complicated example, say .
A contour plot nicely shows the location of the branch cuts.
Numerically we can determine the branch cuts by looking for the lines in the complex plane where , the argument of log, equals a negative real number and where equals a real number of absolute value greater than 1.
Looking at this picture it becomes obvious that even for this relatively simple function it would be a difficult undertaking to mesh the -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.
For algebraic functions, using elimination techniques, it is possible in
principle--although computationally quite expensive--to produce an implicit equation in , , and or and then use
To eliminate , we rewrite the terms containing in a rational form.
Now we have arrived at an implicit equation that can be visualized using
Converted by Mathematica |