Application Center - Maplesoft
  Submit your work
Application Center Applications Section 2.3 The Mappings w = z^n and w = z^`1/n` Preview

App Preview:

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

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

Learn about Maple
Download Application

 

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  w = z^n  and  w = z^`1/n`

    The mapping  w = z^2  or  w = x^2-y^2+i*2*x*y

can be expressed in polar coordinates by the function  f(z) = r^2*exp(i*2*theta) .

    The mapping  w = sqrt(z)  can be expressed in polar coordinates
by the function  f(z) = f(r*exp(i*theta)) = sqrt(r)*exp(i*theta/2) .  

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

> readlib(eliminate):
with(plots):

Warning, the name changecoords has been redefined

Definition 2.1:  Principal Square Root  

The function

    g(w) = w^`1/2` = abs(w)*exp(i*`Arg(z)/2`) ,  for  w <> 0 ,       

is called the principal square root function.

Example 2.12, Page 63.  The transformation  w = z^2  maps lines onto lines or parabolas.
(a)  Find the image of the vertical line  x = a .

> 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[2][1], u)]:
`u ` = solset[1];
u1 := v -> expand(solset[1]):
`u ` = u1(v);

{u = x^2-y^2, v = 2*x*y}

`Substitute x=a in the previous equations.`

{u = a^2-y^2, v = 2*a*y}

`Eliminate y in the previous equations.`

[{y = 1/2*v/a}, {4*u*a^2-4*a^4+v^2}]

`Solve for u in the previous equations.`

`u ` = -1/4*(-4*a^4+v^2)/a^2

`u ` = a^2-1/4*v^2/a^2

Hence, the image of the vertical line  x = a  is a parabola.

(b)  Find the image of the vertical line  y = b .

> 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[2][1], u)]:
`u ` = solset[1];
u2 := v -> expand(solset[1]):
`u ` = u2(v);

{u = x^2-y^2, v = 2*x*y}

`Substitute y=b in the previous equations.`

{u = x^2-b^2, v = 2*x*b}

`Eliminate x in the previous equations.`

[{x = 1/2*v/b}, {4*u*b^2-v^2+4*b^4}]

`Solve for u in the previous equations.`

`u ` = 1/4*(v^2-4*b^4)/b^2

`u ` = 1/4*v^2/b^2-b^2

Hence, the image of the vertical line  y = b  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]);

`f(z) ` = z^2

[Plot]

Example 2.13, Page 65.   The transformation   w = sqrt(z)  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]);

`f(z) ` = z^(1/2)

[Plot]

>

Definition 2.2:  Principal n-th root  

The function

    g(w) = w^`1/n` = abs(w)*exp(i*`Arg(z)/n`) ,  for  w <> 0 ,       

is called the principal n-th root function.

End of Section 2.3.