Notes:
-
To change |z| drag the green circle on the vertical black line
on the right.
-
The blue curve seen on the left is the image of the |z| = constant
circle on the w plane, where w = P(z).
-
To enable the entire infinite
w-plane to fit in the limited space available,
we have compressed it to a circle in which the distance ρ of
the point representing w from the centre is obtained
using the transformation ρ = (|w|/a) / √[(1+(w/a)²].
Here we have used a = 10.
-
To change the coefficients of the polynomial P, click on the word
Coefficients on the lower left of the canvas.
-
The coefficients are entered constant term first.
-
A comma is used to separate the
real and imaginary parts and a semicolon is used to separate
the individual coefficients.
Text book:
The demonstration presented here is based on the
outline of the proof of the Fundamental Theorem of Algebra
given in the following text book.
A Survey of Modern Algebra
Garrett Birkhoff and Saunders Mac Lane
3rd Edition
Macmillan Co.
1965
The following Octave transcript shows the magnitudes of the roots of
the default polynomial.
octave:1> abs(roots([1+0.2*i, -2-3*i, 1+2*i, 1-0.5*i]))
ans =
3.20490
0.88211
0.38780
octave:2>