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# Section 2.3 The Mappings w = z^n and w = z^`1/n`

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C02-3.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 2   COMPLEX FUNCTIONS

Section 2.3  The Mappings and The mapping or can be expressed in polar coordinates by the function .

The mapping can be expressed in polar coordinates
by the function = .

Load Maple's  "eliminate" and "conformal mapping" procedures.
Make sure this is done only ONCE during a Maple  session.

Warning, the name changecoords has been redefined

Definition 2.1:  Principal Square Root

The function = = ,  for ,

is called the principal square root function.

Example 2.12, Page 63.  The transformation maps lines onto lines or parabolas.
(a)
Find the image of the vertical line .

 > x:='x':y:='y':u:='u':v:='v':U:='U':V:='V': eqns1 := {u = x^2 - y^2, v = 2*x*y}: eqns1; `Substitute  x=a  in the previous equations.`; eqns2 := subs(x=a, eqns1): eqns2; `Eliminate  y  in the previous equations.`; eqns3 := eliminate(eqns2, y): eqns3; `Solve for  u  in the previous equations.`; solset := [solve(eqns3, u)]: `u ` = solset; u1 := v -> expand(solset): `u ` = u1(v);        Hence, the image of the vertical line is a parabola.

(b)  Find the image of the vertical line .

 > x:='x':y:='y':u:='u':v:='v':U:='U':V:='V': eqns1 := {u = x^2 - y^2, v = 2*x*y}: eqns1; `Substitute  y=b  in the previous equations.`; eqns2 := subs(y=b, eqns1): eqns2; `Eliminate  x  in the previous equations.`; eqns3 := eliminate(eqns2, x): eqns3; `Solve for  u  in the previous equations.`; solset := [solve(eqns3, u)]: `u ` = solset; u2 := v -> expand(solset): `u ` = u2(v);        Hence, the image of the vertical line is a parabola.

 > f:='f': z:='z': f := z -> z^2: `f(z) ` = f(z); conformal(f(z), z=0..0.5+2*I,  title=`w = z^2`,  grid=[11,11],numxy=[50,50],  scaling=constrained,  labels=[`u     `,`  v`],  view=[-4.1..0.3,-0.1..2.1]);  Example 2.13, Page 65.   The transformation maps  lines  onto  lines  or hyperbolas.

 > f:='f': z:='z': f := z -> z^(1/2): `f(z) ` = f(z); conformal(f(z), z=-4..4+4*I,  title=`w = z^(1/2)`,  grid=[9,9],numxy=[50,50],  scaling=constrained,  labels=[`u `,`v   `],  view=[-0.1..2.5,-0.1..2.5]);  >

Definition 2.2:  Principal n-th root

The function = = ,  for ,

is called the principal n-th root function.

End of Section 2.3.