Re: minimize with complex numbers

*To*: mathgroup at smc.vnet.net*Subject*: [mg91273] Re: minimize with complex numbers*From*: Benjamin Hell <hellben at gmx.de>*Date*: Wed, 13 Aug 2008 04:38:47 -0400 (EDT)

Well, minimize doesn't work with complex arguments. And that is in no way a restriction, because seen from a mathematical standpoint searching for a minimum in a complex set doesn't make any sense without further specification. I could go into detail about that, but this may be confusing for you, so let me just mention the main idea: To explain what is a minimum you need to have a smaller than ("<") relation. So one has to define that for a set of complex numbers. Consider the following example of two complex numbers z1=1+2i and z2=2+0.5i. Which one would you say is greater than the other one? Hard to say that here, right? So what usually appears in applications (and that's something you might want to take in account) is that you compare complex numbers by their absolute value respectively their norm. Usually one takes the euclidean norm |.|, which is defined by |z|=sqrt(a^2+b^2) where z=a+ib. So the norm delivers a real number (a norm always does) and your are in your first case of comparing real numbers. Hope this helps understanding the problem.