Sketch a Graph

Date: 6/1/96 at 21:56:38
From: Anonymous
Subject: Sketch a graph

z=(3-2i)^1/2 then find z^- . Sketch an appropriate graph to 
illistrate z^- and z if z is in quadrant 1. Show and describe how they 
are related geometrically.

Date: 6/2/96 at 7:37:40
From: Doctor Anthony
Subject: Re: Sketch a graph

Let z = a+ib  then z^2 = a^2-b^2 + 2iab = 3-2i

Equate real and imaginary parts and we have two equations for finding 
a and b.

        a^2-b^2 = 3
            2ab = -2  so ab = -1  and b = -1/a  

Substitute for b in the first equation

        a^2 - 1/a^2 = 3    multiply through by a^2

        a^4 - 1 = 3a^2   so  a^4 - 3a^2 - 1 = 0 

This is a quadratic in a^2, and using the quadratic formula we have

  a^2 = {3 +or-sqrt(9+4)}/2  We must take the positive square root 
since a^2 must be positive.
 
  a^2 = {3+sqrt(13)}/2 = 5.408   and a = 2.32558  b = - 0.43 

So  z = 2.3256 - 0.43i  conjugate z will be 2.3256 + 0.43i

z^- is simply the reflection of z in the x axis.

-Doctor Anthony,  The Math Forum
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