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Re: How to get the real and imaginary parts of a power

• To: mathgroup at smc.vnet.net
• Subject: [mg80036] Re: [mg79998] How to get the real and imaginary parts of a power
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Sat, 11 Aug 2007 02:06:29 -0400 (EDT)
• Organization: Mathematics & Statistics, Univ. of Mass./Amherst
• References: <200708100552.BAA10970@smc.vnet.net>
• Reply-to: murray at math.umass.edu

To avoid my getting thoroughly confused, let me change your variable u, 
presumably intended to be complex, to z = u + I v where u and v are real.

Then will this do? The key seems to be to use Normal to get rid of the 
SeriesData wrapper first. And in any case you won't see the real part 
unless you somehow specify the u+I v form of the complex z, as otherwise 
ComplexExpand will assume that z is real.

  s = Series[Cosh[(x + I y) z], {z, 0, 2}];
  ComplexExpand[Re@Normal[s /. z -> u + I v]]
1 + (u^2*x^2)/2 - (v^2*x^2)/2 - 2*u*v*x*y - (u^2*y^2)/2 + (v^2*y^2)/2

Gordon Smith wrote:
> Suppose s = Series[Cosh[(x + I y)u, {u,0,2}]. How can I get the real part 1 + 1/2(x^2 - y^2) u^2 + O(u^3) and the imaginary part x y u^2 + O(u^3) ? I thought ComplexExpand[Re[s]] should give me the real part of s, but it just gives me s unchanged. (Mathematica newbie here!)
> 

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305

• References:
  • How to get the real and imaginary parts of a power series?
    • From: Gordon Smith <gsmithsf@hotmail.com>