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
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CHAPTER 5 ELEMENTARY FUNCTIONS

Section 5.4 Trigonometric and Hyperbolic Functions

Given the success we had in using power series to define the complex exponential, we have reason to believe this approach will be fruitful for other elementary functions as well. The power series expansions for the real-valued sine and cosine functions are

and .

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

> with(plots):

Thus it is natural to make the following definitions for the complex sine and cosine.

Definition 5.5, Page 182. The series for is: .

> f:='f': F:='F': p:='p': P:='P': z:='z': Z:='Z':
f := z -> cos(z):
F := series(f(Z), Z=0, 12):
P := convert(F, polynom):
p := z -> subs(Z=z, P):
`f(z) ` = f(z);
`f(Z) ` = F;
`p(z) ` = p(z);

The general term for the series for is:

> a:='a': n:='n': S:='S': z:='z':
a := n -> (-1)^n*z^(2*n)/(2*n)!:
`f(z) = cos(z)`;
`An ` = a(n);
`The sum of five terms:`;
S5 := sum(a(n), n=0..4):
`S5(z) ` = S5;
`The sum of infinitely many terms:`;
S := sum(a(n), n=0..infinity):
`S(z) = `, Sum(a(n), n=0..infinity) = S;

Definition 5.5, Page 182. The series for is: .

> g:='g': G:='G': q:='q': Q:='Q': z:='z': Z:='Z':
g := z -> sin(z):
G := series(g(Z), Z=0, 11):
Q := convert(G, polynom):
q := z -> subs(Z=z, Q):
`g(z) ` = g(z);
`g(Z) ` = G;
`q(z) ` = q(z);

The general term for the series for is:

> b:='b': n:='n': S:='S': z:='z':
b := n -> (-1)^n*z^(2*n+1)/(2*n+1)!:
`f(z) = sin(z)`;
`Bn ` = b(n);
`The sum of five terms:`;
S5 := sum(b(n), n=0..4):
`S5(z) ` = S5;
`The sum of infinitely many terms:`;
S := sum(b(n), n=0..infinity):
`S(z) = `, Sum(b(n), n=0..infinity) = S;

Theorem 5.4 The functions and are entire functions, with the properties

and .

Properties of the trigonometric functions. The derivative of is .

> `f(z) ` = f(z);
`f '(z) ` = diff(f(z), z);
`p '(z) ` = diff(p(z), z);
`-q'(z) ` = -q(z);
`-g(z) ` = -g(z);
`f '(z) = -g(z)`;

Remark. In order to demonstrate that the derivative of is ,
it is necessary to adjust the length of the series.

> g:='g': G:='G': q:='q': Q:='Q': z:='z':
g := z -> sin(z):
G := series(g(Z), Z=0, 11):
Q := convert(G, polynom):
q := z -> subs(Z=z, Q):
f:='f': F:='F': p:='p': P:='P': z:='z':
f := z -> cos(z):
F := series(f(Z), Z=0, 10):
P := convert(F, polynom):
p := z -> subs(Z=z, P):
`g(z) ` = g(z);
`g '(z) ` = diff(g(z), z);
`q '(z) ` = diff(q(z), z);
`p(z) ` = p(z);
`f(z) ` = f(z);
`g '(z) = f(z)`;

To establish additional properties, it will be useful to express and in the cartesian form . (Additionally, the applications in Chapters 9 and 10 will use these formulas.) We begin by observing that

and

which are then used to obtain

= ( )

= ( )

= ( )

from which we conclude that

= .

In an similar fashion it can be shown that

= .

Figure 5.7, Page 187. Graph the transformation .

> f:='f': z:='z':
f := z -> sin(z):
`f(z) ` = f(z);
conformal(f(z), z=-Pi/2-I*2..Pi/2+I*2,
title=`w = sin(z)`,
grid=[9,9],numxy=[100,100],
scaling=constrained,
labels=[` u`,`v `],
view=[-4..4,-4..4]);

Example for Page 186. Verify that , if and only if for an integer.

> x:='x': y:='y':
cos(x + I*y) = `0 `;
evalc(cos(x + I*y)) = `0 `;
`Solve the equations:`;
eqns := {cos(x)*cosh(y) = 0, -sin(x)*sinh(y) = 0}: eqns;
solset := solve(eqns ,{x,y}): solset;

Remark. It is assumed that both and are real numbers. Hence, of the four solutions, the only valid
solutions are , and , . The other solutions are obtained by adding to this result.

> cos(-pi/2) = cos(-Pi/2);
cos(pi/2) = cos(Pi/2);
cos(pi/2+pi) = cos(Pi/2+Pi);
cos(pi/2+2*pi) = cos(Pi/2+2*Pi);
cos(pi/2+3*pi) = cos(Pi/2+3*Pi);

Or showing that the system of equations is satisfied.

> eqns;
eval(subs({x=Pi/2, y=0},eqns));
eval(subs({x=Pi/2+Pi, y=0},eqns));
eval(subs({x=Pi/2+2*Pi,y=0},eqns));
eval(subs({x=Pi/2+3*Pi,y=0},eqns));

Example for Page 185. Establish the trigonometric identity for complex numbers:
.

> x:='x': y:='y': z:='z':
Z1 := x[1] + I*y[1]:
Z2 := x[2] + I*y[2]:
eq1 := cos(Z1+Z2):
cos(z[1] + z[2]) = eq1;
eq1 := evalc(eq1):
cos(z[1] + z[2]) = eq1;
eq1 := expand(evalc(eq1), trig):
cos(z[1] + z[2]) = eq1; ` `;
eq2 := cos(Z1)*cos(Z2) - sin(Z1)*sin(Z2):
cos(z[1])*cos(z[2]) - sin(z[1])*sin(z[2]) = eq2;
eq2 := evalc(eq2):
cos(z[1])*cos(z[2]) - sin(z[1])*sin(z[2]) = eq2;
eq2 := expand(evalc(eq2)):
cos(z[1])*cos(z[2]) - sin(z[1])*sin(z[2]) = eq2; ` `;
`Does cos(z1 + z1) = cos(z1)cos(z2) - sin(z1)sin(z2) ?`;
evalb(eq1 = eq2);

Therefore, .

Properties of and on Page 187.

> `cos(x + I y) ` = evalc(cos(x + I*y)); ` ` ;
`sin(x + I y) ` = evalc(sin(x + I*y)); ` ` ;
`tan(x + I y) ` = evalc(tan(x + I*y)); ` ` ;
`|sin(x + I y)| ` = evalc(abs(sin(x + I*y)));
`|sin(x + I y)| ` = simplify(evalc(abs(sin(x + I*y))));

>

End of Section 5.4.