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 5 ELEMENTARY FUNCTIONS

Section 5.1 The Complex Exponential Function

How should complex-valued functions such as , , , etc., be defined? Clearly, any responsible definition should satisfy the following criteria:

i. The functions so defined must give the same values as the corresponding functions for real variables when the number is a real number.

ii. As far as possible, the properties of these new functions must correspond with their real counterparts. For example, we would want

to be valid regardless of whether and were real or complex.

These requirements may seem like a tall order to fill. There is a procedure, however, that offers promising results. It is to put the expansion of the real functions , , , etc., as power series into complex form. This will be our stategy for the next few sections.

Recall that the real exponential function can be represented by the power series . Thus it is only natural to define the complex exponential , also written as , in the following way:

Definition 5.1: The complex exponential function

= =

Clearly this definition agrees with that of the real exponential function when is a real number.

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

> with(plots):

Theorem. The exponential function is a solution to the differential equation with the initial condition .

> f:='f': z:='z': Z:='Z':
f := z -> exp(z):
f1 := z -> subs(Z=z, diff(f(Z),Z)):
`f(z) ` = f(z);
`f '(z) ` = f1(z);
`f(0) ` = f(0);

To see its real and imaginary parts, use complex expand.

> x:='x': y:='y': Z:='Z':
Z := x + I*y:
`exp(z) ` = exp(Z);
`exp(z) ` = evalc(exp(Z));

Now verify that the "rules of exponents" hold.

> x:='x': y:='y': z:='z':
w1 := evalc(exp(x[1]+I*y[1])):
w2 := evalc(exp(x[2]+I*y[2])):
w3 := w1*w2: exp(z[1])*exp(z[2]) = w3; ` `;
w3 := evalc(w1*w2): exp(z[1])*exp(z[2]) = w3; ` `;
w3 := expand(w3): exp(z[1])*exp(z[2]) = w3; ` `;
z4 := (x[1]+I*y[1])+(x[2]+I*y[2]):
w4 := exp(z4): exp(z[1]*z[2]) = w4; ` `;
w4 := evalc(exp(z4)): exp(z[1]*z[2]) = w4; ` `;
w4 := expand(w4, trig): exp(z[1]*z[2]) = w4; ` `;
`Does exp(z1 z2) = exp(z1) exp(z2) ?`;
w3 = w4; ` `;
evalb(expand(w3 = w4));

Theorem 5.1 The function is an entire function satisfying:

i. (Using Leibniz notation, .)

ii. exp( ) = exp( )exp( ) (That is, .)

iii. If is a real number, then .

Example 5.1, Page 164. The points for
n =..., -2, -1, 0, 1, 2, ... are mapped onto a single point .

> Z0 := 5/4 + I*11*Pi/6:
for k from 0 to 5 do
exp(Z0+I*2*Pi*k)=evalc(exp(Z0+I*2*Pi*k));
od;

Example 5.2, Page 166. Show that the transformation maps the rectangle
onto a portion of an annular region bounded by two rays.
For illustration we use R = {(x,y): -1 <= x <= 1 and - p /3 <= y <= p /4} .

> f:='f': z:='z':
f := z -> exp(z):
`f(z) ` = f(z);
conformal(f(z), z=-1-I*Pi/3..1+I*Pi/4,
title=`w = exp(z)`,
grid=[8,8],numxy=[50,50],
scaling=constrained,
labels=[`u`,`v `],
tickmarks=[3,5],
view=[0..2.8,-2.4..2.0]);

End of Section 5.1.