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 5.2 Branches of the Complex Logarithm Function

In Section 5.1 we showed that if is a nonzero complex number the equation has infinitely many solutions. Because the function is a many-to-one function, its inverse (the logarithm) is necessarily multivalued.

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

> with(plots):

Warning, the name changecoords has been redefined

Definition 5.2: Multivalued logarithm

For , we define the multivalued function as the inverse of the exponential function; that is,

if and only if .

Definition 5.3: Principal value of the logarithm

For , we define the single-valued function , the principal value of the logarithm, by

.

The domain for the function is the set of all nonzero complex numbers in the -plane, and its range is the horizontal strip <

in the -plane. We stress again that is a single-valued function and corresponds to setting in the above definition. As we saw in Chapter 2, the function is discontinuous at each point along the negative -axis, hence so is the function . In fact, because any branch of the multi-valued function is discontinuous along some ray, a corresponding branch of the logarithm will have a discontinuity along that same ray.

Example for Page 170.

> r:='r': t:='t': u:='u': U:='U':
v:='v': V:='V': w:='w': x:='x': y:='y':
w := log(x + I*y):
`Log(z) ` = w;
w := evalc(log(x + I*y)):
`Log(z) ` = w;

> u := proc(x,y) ln(x^2+y^2)/2 end:
v := proc(x,y) arctan(y,x) end:
`Log(z) ` = u(x,y) + I*v(x,y);
U := proc(r,t) ln(r) end:
V := proc(r,t) t end:
`z ` = r*exp(I*t);
`Log(z) ` = U(r,t) + I*V(r,t);

To plot the function we need to solve
for as a function of , that is: and use a "cylinderplot."

> cylinderplot(exp(u) ,t=0..2*Pi,u=-2..1,
title=`u(x,y) = ln|z| = ln(r)`);

Example 5.3, Page 170. Find and .

> w:='w': z:='z':
z := 1+I:
`z ` = z;
w := log(z):
`Log(z) ` = w;
w := evalc(log(z)):
`Log(z) ` = w;

The best we can do for the multivalued is to add .

> w:='w': z:='z':
z := 1+I:
`z ` = z;
w := log(z) + I*2*Pi*n:
`log(z) ` = w;
w := evalc(log(z)) + I*2*Pi*n:
`log(z) ` = w;

The transformation maps the punctured -plane slit along the ray
, , one-to-one and onto the horizontal strip < < in the -plane.

The following graph is the image of the disk slit along the ray , .

> f:='f': z:='z':
f := z -> log(z):
`f(z) ` = f(z);
conformal(f(Re(z)*exp(I*Im(z))), z=0.01-I*3.14..100+I*3.14,
title=`w = log(z)`,
grid=[13,13],numxy=[13,13],
scaling=constrained,
labels=[` u`,`v `],
tickmarks=[5,7],
view=[-4.7..4.7,-3.2..3.2]);

Example 5.4, Page 170. (a) Find .

> w:='w': z:='z':
z := -exp(1):
`z ` = z;
w := log(z):
`Log(z) ` = w;
w := evalc(log(z)):
`Log(z) ` = w;
w := ln(abs(z)) + I*argument(z):
`ln|z| + I Arg(z) ` = w;

(b) Find .

> w:='w': z:='z':
z := -1:
`z ` = z;
w := log(z):
`Log(z) ` = w;
w := ln(abs(z)) + I*argument(z):
`ln|z| + I Arg(z) ` = w;

Example 5.5, Page 171. Verify that .

> z:='z':
z1 := -sqrt(3) + I: z[1] = z1;
z2 := - 1 + I*sqrt(3): z[2] = z2;
w1 := log(z1) + log(z2):
log(z1) = evalc(log(z1));
log(z2) = evalc(log(z2));
Log(z[1]) + Log(z[2]) = w1;
w1 :=evalc(w1):
Log(z[1]) + Log(z[2]) = w1; ` `;
z3 := z1*z2:
w3 := log(z3):
Log(z[1]*z[2]) = w3;
z4 := evalc(z3):
w4 := log(z4):
Log(z[1]*z[2]) = w4;
w4 := evalc(w4):
Log(z[1]*z[2]) = w4; ` `;
`Does Log(z1) + Log(z2) = Log(z1 z2) ?`;
w1 = w4;
evalb(w1 = w4);

Theorem 5.2 (A fact concerning principal logarithm)

The identity Log( ) = Log( ) + Log( ) holds true if and only if < Arg( ) + Arg( .

Theorem 5.3 (Facts about the multivalued logarithm)

Let and be nonzero complex numbers. The multivalued function obeys the familiar properties of logarithms:

i. log( ) = log( ) + log( ) ,

ii. Log( ) = Log( ) - Log( ) , and

iii. Log( ) = - Log( ).

Page 172, Properties of .

> z1 := x1 + I*y1:
z2 := x2 + I*y2:
w1 := log(z1):
w2 := log(z2):
v3 := v1 + v2:
`Log(z1) + Log(z2) ` = w3;
w3 := evalc(w1 + w2):
`Log(z1) + Log(z2) ` = w3; ` `;

> z3 := z1*z2:
z4 := evalc(z1*z2):
w4 := log(z3):
`Log(z1 z2) ` = w4;
w4 := log(z4):
`Log(z1 z2) ` = w4;
w4 := evalc(w4):
`Log(z1 z2) ` = w4;
w4 := simplify(w4):
`Log(z1 z2) ` = w4; ` `;
`Does Log(z1 z2) = Log(z1) + Log(z1) ?`;
w4 = w3;

Therefore we will have provided that .
Those pesky arctan's must be taken seriously !

End of Section 5.2.