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MathGroup Archive 2004

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RE: question: How to get Mathematica to show more than one value of a complex valued function?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg46334] RE: [mg46286] question: How to get Mathematica to show more than one value of a complex valued function?
  • From: "David Park" <djmp at earthlink.net>
  • Date: Sat, 14 Feb 2004 22:19:44 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Steve,

This is a little tangential to your question. Basically I think you have to
learn which functions are multifunctions and can't depend upon Mathematica
automatically telling you.

Using the Cardano2 package at my web site I made some plots of the Log
function. One of the capabilities of the package is to make complex vector
plots, and animations of complex vector plots. A complex vector plot shows
the value of the complex function as a vectors attached to a grid of points.
Many aspects of complex functions are besy demonstrated by taking a few
points or a single point and moving it around in the complex plane via an
animation. So, for example, you can run a point around a circle and see if
the circle encloses a critical point by whether or not the vector makes a
complete rotation about the moving point.

The complex vector plots also have the capability to plot multifunctions as
an array of vectors. The tutorial notebook, UsingComplexGraphics, that comes
with the package has an example near the end that shows a multifunction
complex vector animation of the function z^(1/3). A point moves in a circle
around the origin. Attached to the point is a trio of vectors that represent
the three values of the function. Each time the point moves around the
circle, the vectors make 1/3 of a rotation about the point. You have to run
around the circle three times to get back to the original situation.

Prompted by your question I made a notebook for the Log function, making a
similar plot using multifunctions. For Log there are an infinite number of
values, but I just plotted 9 of them, enough to go offscale on the plot.
Here, each time one circles the origin, a lower (imaginary) valued vector is
SMOOTHLY transformed into the next higher one. You can see that the original
situation will never repeat. From now on I'm going to think of the Log
function as the barber-pole function!

I'll send the notebook to anyone who wants it. (If you're a new person to me
you will have to go through my challange-response email system.)

David Park
djmp at earthlink.net
http://home.earthlink.net/~djmp/


From: steve_H [mailto:nma124 at hotmail.com]
To: mathgroup at smc.vnet.net

hello;

this is Mathematica 5.0

given a multi-valued complex function, how can I get Mathematica to show
more than one value? or to show the general formula for all solutions?

for example, (-i)^i  is multivalued. The result returned by Mathematica is
Exp[Pi/2], which is the first possible value. But there are infinitly many
others (if my math is correct), such as Exp[-3/2 Pi], Exp[5/2 Pi] etc...
where the general function value is Exp[Pi/2 +-  2 n Pi]  n=0,1,2,3,....

Same as for Log[z] where z is a complex number, which is multi-valued.

or can I at least tell Mathematica to tell me if the result returned is the
ONLY
one possible, (i.e. the complex function is NOT multi-valued)? becuase
right now it is not clear from the result if result returned is the only
value or if there are others.

thanks,
Steve



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