COMPLEX ANALYSIS: Maple Worksheets, 2001

COMPLEX ANALYSIS: Maple Worksheets, 2009
(c) John H. Mathews Russell W. Howell
mathews@fullerton.edu howell@westmont.edu

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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/

Check out the new Complex Analysis Projects page.

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 = 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

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

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

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

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

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

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

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

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

`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 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=[11,11],
scaling=constrained,
labels=[`u `,` v`],
view=[-4.1..0.3,-0.1..2.1]);

[Maple Plot]

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=[9,9],
scaling=constrained,
labels=[`u `,`v `],
view=[-0.1..2.5,-0.1..2.5]);

[Maple Plot]

Definition 2.2: Principal n-th root

The function

= = , for ,

is called the principal n-th root function .

End of Section 2.3.