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
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CHAPTER 2 COMPLEX FUNCTIONS

Section 2.1 Functions of a Complex Variable

A complex valued function of the complex variable is a rule that assigns to each complex number in a set one and only one complex number . We write and call the image of under . The set is called the domain of , and the set of all images is called the range of . As we saw in section 1.6, the term domain is also used to indicate a connected open set. When speaking about the domain of a function , however, mathematicians mean only the set of points on which the function is defined. This is a distinction worth noting.

Just as can be expressed by its real and imaginary parts, , we write , where and are the real and imaginary parts of , respectively. This gives us the representation

= = = = .

Since and depend on and , they can be considered to be real valued functions of the real variables and ; that is

and .

Combining these ideas it is customary to write a complex function f in the form

= = .

Definition. A function of the complex variable can be written:

.

Definition. The polar coordinate form of a complex function is:

= .

There are two approaches to defining a complex function in Maple.

Method 1. Make a function of two real variables .

Method 2. Make a function of the complex variable .

Example 2.1, Page 49. Write in the form.

Method 1. Make a function of two real variables .

> f:='f': x:='x': y:='y': z:='z':
f := proc(x,y)
local z,w;
z := x + I*y;
w := expand(z^4);
end:
`f(z) ` = z^4;
`f(x,y) ` = f(x,y); ` `;
`At z = 1 + 2i: `;
`f(1,2) ` = f(1,2);

Method 2. Make a function of .

> F:='F': x:='x': y:='y': z:='z':
F := proc(z)
local w;
w := expand(z^4);
end:
`F(z) ` = F(z);
`F(x + I y) ` = F(x + I*y); ` `;
`At z = 1 + 2i: `;
`F(1 + I 2) ` = F(1 + I*2);

Example 2.2, Page 50. Write in the form.

Method 1. Make a function of two real variables .

> f:='f': x:='x': y:='y':
f := proc(x,y)
local w;
w := (x - I*y)*x + (x + I*y)^2 + y;
end:
`f(x,y) ` = f(x,y);
`f(x,y) ` = evalc(f(x,y)); ` `;
`At z = 1 + 2i: `;
`f(1,2) ` = f(1,2);

Method 2. Make a function of .

> F:='F': x:='x': y:='y': z:='z':
F := proc(z)
local w;
w := conjugate(z)*Re(z) + z^2 + Im(z);
end:
`F(z) ` = F(z);
`F(x + I y) ` = (x-I*y)*x + (x+I*Y)^2 + y; ` `;
`At z = 1 + 2i: `;
`F(1 + I 2) ` = F(1 + I*2);

Example 2.3, Page 50. Express by a formula involving and .

Method 1. Make a function of two real variables .

> f:='f': x:='x': y:='y':
f := proc(x,y)
local w;
w := 4*x^2 + I*4*y^2;
end:
`f(x,y) ` = f(x,y); ` `;
`At z = 1 + 2i: `;
`f(1,2) ` = f(1,2);

Method 2. Make a function of .

> F:='F': w:='w': z:='z': Z:='Z':
w := subs({x=(Z+conjugate(Z))/2, y=(Z-conjugate(Z))/(2*I)},f(x,y)):
F := z -> subs(Z=z, expand(w)):
`f(x,y) ` = f(x,y);
`F(z) ` = F(z); ` `;
`At z = 1 + 2i: `;
`F(1 + I 2) ` = F(1+I*2); ` `;
`F(1 + I 2) ` = evalc(F(1+I*2));

Example 2.5, Page 51. Express in the polar coordinate form.

Method 1. Make a function of two real variables .

> F:='F': x:='x': y:='y': z:='z':
F := proc(z)
local w;
w := z^5 + 4*z^2 - 6;
end:
`F(z) ` = z^5 + 4*z^2 - 6;
`F(x + I y) ` = F(x + I*y);` `;
`At z = 1 + i: `;
`F(1 + I) ` = F(1 + I);

Method 2. Make a function of .

> f:='f': r:='r': t:='t': z:='z':
f := proc(r,t)
local w;
w := subs({z^2=r^2*cos(2*t) + I*r^2*sin(2*t),
z^5=r^5*cos(5*t) + I*r^5*sin(5*t)}, F(z));
end:
`F(z) ` = z^5 + 4*z^2 - 6;
`f(r,t) ` = f(r,t); ` `;
`At z = 1 + i: `;
`f(sqrt(2),Pi/4) ` = f(sqrt(2),Pi/4);

End of Section 2.1.