 Application Center - Maplesoft

# Section 1.6 The Topology of Complex Numbers

You can switch back to the summary page by clicking here.

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: : for     .

Example 1.22, Page 40.
If and are two given points, then the straight line segment joining to is  C: for     .

 > 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);     Extra Eample, Page 40.
Find the equation of the line segment with the initial point and the terminal point .

 > 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);     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]); >

End of Section 1.6.