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# Section 2.1 Functions of a Complex Variable

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C02-1.mws

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 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.