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Section 1.6 The Topology of Complex Numbers

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C01-6.mws

COMPLEX ANALYSIS: Maple Worksheets,  2001
(c) John H. Mathews          Russell W. Howell
mathews@fullerton.edu     howell@westmont.edu

Complimentary software to accompany the textbook:

COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9
Jones and Bartlett Publishers, Inc.,      40  Tall  Pine  Drive,      Sudbury,  MA  01776
Tele.  (800) 832-0034;      FAX:  (508)  443-8000,      E-mail:  mkt@jbpub.com,      http://www.jbpub.com/


CHAPTER 1  COMPLEX NUMBERS

Section 1.6  The Topology of Complex Numbers

In this section we investigate some basic ideas concerning sets of points in the plane.
The first concept is that of a curve.

Definition:  Curve

 

A curve in the complex plane is:  

    C :  z(t) = x(t)+i*y(t)    for   a `` <= `` t `` <= `` b .    


Example 1.22, Page 40.   If   z[0] = x[0]+i*y[0]   and   z[1] = x[1]+i*y[1] are two given points, then the straight line segment joining  z[0]  to  z[1]  is  C:  z(t) = x[0]+(x[1]-x[0])*t+i(y[0]+(y[1]-y[0])*t)    for   0 `` <= `` t `` <= `` 1 .     

> t:='t':x0:='x0':x1:='x1':y0:='y0':y1:='y1':z:='z':
z := t -> x0 + (x1-x0)*t + I*(y0 + (y1-y0)*t):
`Equation of a line segment:`;
`z(t) ` = z(t), `   for  0 <= t <= 1`; ` `;
`Initial  point    z(0) ` = z(0);
`Terminal point    z(1) ` = z(1);

`Equation of a line segment:`

`z(t) ` = x0+(x1-x0)*t+I*(y0+(y1-y0)*t), ` for 0 <= t <= 1`

` `

`Initial point z(0) ` = x0+I*y0

`Terminal point z(1) ` = x1+I*y1


Extra Eample, Page 40.  Find the equation of the line segment with the initial point  z[0] = -3+2*i  and the terminal point  z[1] = 1+i .

> t:='t':x0:='x0':x1:='x1':y0:='y0':y1:='y1':z:='z':
z0 := - 3 + 2*I:
z1 := 1 + I:
x0 := Re(z0): y0 := Im(z0): x1 := Re(z1): y1 := Im(z1):
z := t -> x0 + (x1-x0)*t + I*(y0 + (y1-y0)*t):
`Equation of a line segment:`;
`z(t) ` = z0 + (z1 - z0)*t, `   for  0 <= t <= 1`; ` `;
`Initial  point    z(0) ` = z(0);
`Terminal point    z(1) ` = z(1);

`Equation of a line segment:`

`z(t) ` = (-3+2*I)+(4-I)*t, ` for 0 <= t <= 1`

` `

`Initial point z(0) ` = -3+2*I

`Terminal point z(1) ` = 1+I

The graph for this line segment can is drawn with the plot subroutine.

> plot([evalf(Re(z(t))),evalf(Im(z(t))), t=0..1],
title=`Line segment between z0 and z1.`,
scaling=constrained, color=red,
labels=[`  x`,`  y`],
view=[-3.5..1.5,-1.0..3.50]);

[Plot]

>

End of Section 1.6.