COMPLEX ANALYSIS: Mathematica 4.1 Notebooks
(c) John H. Mathews, and
Russell W. Howell, 2002
Complimentary software to accompany our textbook:

COMPLEX ANALYSIS: for Mathematics and Engineering, 
4th Edition,  2001,  ISBN: 0-7637-1425-9
Jones & Bartlett Publishers, Inc.  
40 Tall Pine Drive, Sudbury, MA 01776  
Tele.  (800) 832-0034, FAX:  (508)  443-8000  
E-mail:  mkt@jbpub.com  
Internet:  http://www.jbpub.com/  
This free software is compliments of the authors.
John H. Mathews,   mathews@fullerton.edu 
Russell W. Howell,  howell@westmont.edu

CHAPTER 1  COMPLEX NUMBERS

Needs["Graphics`Arrow`"]; 
Needs["Graphics`Colors`"]; 
a = b = c = e = P = r = t = θ = v = V = w = x = y = z = Z = 0; 
Clear[a, b, bdd, c, disk, dot, e, eqn, ext, int, k, list, P, r, r1, 
        r2, set, solset, t, θ, θ1, θ2, v, V, values, w, 
        x, x0, x1, x2, x3, xx, y, y0, y1, y2, y3, yy, z, z1, z2, z3, z4, 
        Z];

Section 1.1	The Origin of Complex Numbers

Section 1.2	The Algebra of Complex Numbers

Section 1.3  The Geometry of Complex Numbers

Since the complex numbers are ordered pairs of real numbers, there is a one-to-one correspondence between them and points in the plane. In this section we shall see what effect algebraic operations on complex numbers have on their geometric representations.
	
	The number z = x+iy = (x,y) can be represented by a position vector in the xy-plane whose tail is at the origin and whose head is at the point (x,y). When the xy-plane is used for displaying complex numbers, it is called the Complex Plane, or more simply, the z-plane. Recall that Re(z)=x and Im(z)=y. Geometrically, Re(z) is the projection of z=(x, y) onto the x axis, and Im(z) is the projection of z onto the y axis. It makes sense, then, that the x axis is also called the real axis, and the y axis is called the imaginary axis.

	Addition of complex numbers is analogous to addition of vectors in the plane. As we saw in Section 1.2, the sum of (Subscript[z, 1]) = (Subscript[x, 1]) + i (Subscript[y, 1]) = ((Subscript[x, 1]), (Subscript[y, 1])) and (Subscript[z, 2]) = (Subscript[x, 2]) + i (Subscript[y, 2]) = ((Subscript[x, 2]), (Subscript[y, 2]))is ((Subscript[x, 1]) + (Subscript[x, 2]), (Subscript[y, 1]) + (Subscript[y, 2])). Hence, (Subscript[z, 1])  + (Subscript[z, 2]) can be obtained vectorially by using the parallelogram law. 

	The difference (Subscript[z, 1])  - (Subscript[z, 2]) can be represented by the displacement vector from the point (Subscript[z, 2]) = ((Subscript[x, 2]), (Subscript[y, 2])) to the point (Subscript[z, 1]) = ((Subscript[x, 1]), (Subscript[y, 1])).

Definition 1.8, (Modulus ) or Absolute Value,  Page 17.  The modulus, or absolute value, of the complex number z=x+iy is a nonnegative real number denoted by |z|  and is given by the equation  
			|z| = (Sqrt[(x^2) + (y^2)]).  

	The number |z| is the distance between the origin and the point z=(x,y). The only complex number with modulus zero is the number 0. The number z=4+3i has modulus 5. The numbers |Re(z)|, |Im(z)|, and |z| are the lengths of the sides of the right triangle. 
	
	The inequality |(Subscript[z, 1])|<|(Subscript[z, 2])| means that the point (Subscript[z, 1]) is closer to the origin than the point (Subscript[z, 2]) and it follows that  
			\(\(\(|\)\(x\)\(|\)\) = \(\(|\)\(Re \((z)\)\)\(|\)\(\(≤\)\(\(|\)\(z\)\(|\)\)\)\)\) and  \(\(\(|\)\(y\)\(|\)\) = \(\(|\)\(Im \((z)\)\)\(|\)\(\(≤\)\(\(|\)\(z\)\(|\)\)\)\)\).

Since the difference (Subscript[z, 1])  - (Subscript[z, 2]) can represent the displacement vector from (Subscript[z, 2]) to (Subscript[z, 1]), it is evident that the distance between (Subscript[z, 1]) and (Subscript[z, 2]) is 
given by |(Subscript[z, 1])  - (Subscript[z, 2])|. This can be obtained by using identities of Section 1.2 and definition |z| to obtain the familiar formula  

			dist((Subscript[z, 1]), (Subscript[z, 2]))|(Subscript[z, 2]) - (Subscript[z, 1])|=(Sqrt[((((Subscript[x, 1]) - (Subscript[x, 2])))^2) + ((((Subscript[y, 1]) - (Subscript[y, 2])))^2)]).  

	If z= (x,y) = x+iy, then -z = (-x,-y) = -x - iy is the reflection of z through the origin, and \(\(\(z\)\(\ \)\)\&_ = \(\((x, \(-y\))\) = x - i\ y\)\) is the reflection of z through the x axis.

	There is a very important algebraic relationship that can be used in establishing properties of the absolute value that have geometric applications. Its proof is rather straightforward, and it is given in the exercises
	
			\(\(\(|\)\(z\)\( | \^2\)\) = z\ \(\(z\)\(\ \)\)\&_\).  

	A beautiful and important application of the above equation  is its use in establishing the triangle inequality, which states that the sum of the lengths of two sides of a triangle is greater than or equal to the length of the third side.

Theorem 1.2, (Triangle Inequality), Page 15.   If  (Subscript[z, 1]) and (Subscript[z, 2]) are arbitrary complex numbers, then  

			\(\(\(|\)\(z\_1 + 
          z\_2\)\(|\)\(\(≤\)\(\(|\)\(z\_1\)\(|\)\(+\(\(|\)\(z\_2\)\(|\)\)\)\)\)\)\).

Example  1.5,  Page 19.  Use the values  (Subscript[z, 1]) = 7 + i  and  (Subscript[z, 2]) = 
          3 + 5  i  and verify the Triangle Inequality  \(\(\(|\)\(z\_1 + 
          z\_2\)\(|\)\(\(≤\)\(\(|\)\(z\_1\)\(|\)\(+\(\(|\)\(z\_2\)\(|\)\)\)\)\)\)\)  for  (Subscript[z, 1])  and   (Subscript[z, 2]).

Solution 1.5.

Compute \(\(\(|\)\(z\_1 + 
          z\_2\)\(|\)\(\ \ \)\(and\)\(\ \ \)\(|\)\(z\_1\)\(|\)\(+\(\(|\)\(z\_2\)\(|\)\)\)\)\).

Clear[z]; 
(Subscript[z, 1]) = 7 + I; 
(Subscript[z, 2]) = 3 + 5  I; 
(Subscript[w, 1]) = (Subscript[z, 1]) + (Subscript[z, 2]); 
(Subscript[V, 1])  =  
        Graphics[{Red, Arrow[{0,  0}, {Re[(Subscript[z, 1])], Im[(Subscript[z, 1])]}]}]; 
(Subscript[V, 2])  =  
        Graphics[{Green, 
            Arrow[{Re[(Subscript[z, 1])], Im[(Subscript[z, 1])]}, {Re[(Subscript[w, 1])], 
                Im[(Subscript[w, 1])]}]}]; 
(Subscript[V, 3])  =  
        Graphics[{Blue, 
            Arrow[{0,  0}, {Re[(Subscript[w, 1])], Im[(Subscript[w, 1])]}]}]; 
Show[(Subscript[V, 1]), (Subscript[V, 2]), (Subscript[V, 3]), 
        PlotRange -> {{0, 10}, {0, 
              6}}, 
Ticks -> {Range[0, 10, 1], 
            Range[0, 6, 1]}, 
AspectRatio -> (6/10), 
        Axes -> True, 
        AxesLabel -> {"x", "y"}]; 
\(\(Print[\*""          \!\(z\_1\) = ""\ , z\_1];\)\ \)
\(\(Print[\*""          \!\(z\_2\) = ""\ , z\_2];\)\ \)
\(\(Print[\*""     \!\(z\_1\) + \!\(z\_2\) = ""\ , 
        z\_1 + z\_2];\)\ \)
Print[""]; 
\(\(Print[\*""   |\!\(z\_1\) + \!\(z\_2\)| = "", 
        Abs[z\_1 + z\_2]];\)\ \)
\(\(Print[\*"" |\!\(z\_1\)| + |\!\(z\_2\)| = "", 
        Abs[z\_1] + Abs[z\_2]];\)\ \)
\(\(Print[\*""Is |(Subscript[z, 1]) + (Subscript[z, 2])| ≤  |(Subscript[z, 1])| + |(Subscript[z, 2])| ?  "", 
Abs[z\_1 + z\_2] ≤ 
          Abs[z\_1] + Abs[z\_2]];\)\ \)
Print[""]; 
\(\(Print[\*""   |\!\(z\_1\) + \!\(z\_2\)| = "", 
        N[Abs[z\_1 + z\_2]]];\)\ \)
\(\(Print[\*"" |\!\(z\_1\)| + |\!\(z\_2\)| = "", 
        N[Abs[z\_1] + Abs[z\_2]]];\)\ \)
\(\(Print[\*""Is |(Subscript[z, 1]) + (Subscript[z, 2])| ≤  |(Subscript[z, 1])| + |(Subscript[z, 2])| ?  "", 
Abs[z\_1 + z\_2] ≤ 
          Abs[z\_1] + Abs[z\_2]];\)\ \)

Other important identities can also be established by means of the triangle inequality. Note that 
	
				|(Subscript[z, 1])| = |((Subscript[z, 1]) + 
                (Subscript[z, 2])) + (-(Subscript[z, 2]))|  
				            ≤|(Subscript[z, 1]) + 
            (Subscript[z, 2])|+|-(Subscript[z, 2])|  
				            ≤|(Subscript[z, 1]) + 
            (Subscript[z, 2])|+|(Subscript[z, 2])|.  

Subtracting |(Subscript[z, 2])| from the left and right sides of this string of inequalities gives an important relationship that will be used in determining lower bounds of sums of complex numbers: 

			\(\(\(|\)\(z\_1 + 
          z\_2\)\(|\)\(\(≥\)\(\(|\)\(z\_1\)\(|\)\(-\(\(|\)\(z\_2\)\(|\)\)\)\)\)\)\).

Using the equation \(\(\(|\)\(z\)\( | \^2\)\) = z\ \(\(z\)\(\ \)\)\&_\) and the commutative and associative laws it follows that  

			\(\(\(|\)\(\(z\_1\) 
            z\_2\)\( | \^2\)\) = \((\(z\_1\) 
              z\_2)\)\ \(\((\(z\_1\) z\_2)\)\(\ \)\)\&_\)  
			\(\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\(=\)\(\((z\_1\ \(\(z\_1\)\(\ \)\)\&_)\) \((z\_2\ \(\(z\_2\)\(\ \)\)\&_)\)\)\)\)\)  
			\(\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\(=\)\(\(|\)\(z\_1\)\( | \^2\)\(|\)\(z\_2\)\( | \^2\)\)\)\)\)  

Taking square roots of the terms on the left and right establishes another important identity  

			|(Subscript[z, 1]) (Subscript[z, 2])| = |(Subscript[z, 1]) || (Subscript[z, 2])|.  

	As an exercise, we ask you to show

			\(\(\(|\)\(z\_1\/z\_2\)\(|\)\) = \(\(|\)\(z\_1\)\(|\)\)\/\(\(|\)\(z\_2\)\(|\)\)\), provided   (Subscript[z, 2]) ≠ 0.

Properties of the Absolute Value, Page 22.  

The quotient obeys  \(\(\(|\)\(z\_1\/z\_2\)\(|\)\) = \(\(|\)\(z\_1\)\(|\)\)\/\(\(|\)\(z\_2\)\(|\)\)\).

Example  1.6,  Page 20.  Use the values  (Subscript[z, 1]) = 1 + 2  i  and  (Subscript[z, 2]) = 
          3 + 2  i  and verify that |(Subscript[z, 1]) (Subscript[z, 2])| = |(Subscript[z, 1]) || (Subscript[z, 2])|.

Solution 1.6.

Compute |(Subscript[z, 1]) (Subscript[z, 2])|  and  |(Subscript[z, 1]) || 
          (Subscript[z, 2])|.

Clear[z]; 
(Subscript[z, 1])  =  1  +  2  I; 
(Subscript[z, 2])  =  3  +  2  I; 
(Subscript[z, 3])  =  (Subscript[z, 1]) (Subscript[z, 2]); 
(Subscript[V, 1])  =  
        Graphics[{Red, Arrow[{0,  0}, {Re[(Subscript[z, 1])], Im[(Subscript[z, 1])]}]}]; 
(Subscript[V, 2])  =  
        Graphics[{Green, Arrow[{0, 0}, {Re[(Subscript[z, 2])], Im[(Subscript[z, 2])]}]}]; 
(Subscript[V, 3])  =  
        Graphics[{Blue, 
            Arrow[{0,  0}, {Re[(Subscript[z, 3])], Im[(Subscript[z, 3])]}]}]; 
Show[(Subscript[V, 1]), (Subscript[V, 2]), (Subscript[V, 3]), PlotRange -> {{-1, 3}, {0, 8}}, 
        Ticks -> {Range[-1, 3, 1], Range[0, 8, 1]}, 
        AspectRatio -> 2, Axes -> True, 
        AxesLabel -> {"x", "y"}]; 
\(\(Print[\*""\!\(z\_1\) = ""\ , \ z\_1];\)\ \)
\(\(Print[\*""\!\(z\_2\) = ""\ , \ z\_2];\)\ \)
\(\(Print[\*""\!\(z\_1\) \!\(z\_2\) = ""\ , \ z\_3];\)\ \)
\(\(Print[\*""|\!\(z\_1\)| = ""\ , \ Abs[z\_1]];\)\ \)
\(\(Print[\*""|\!\(z\_2\)| = ""\ , \ 
        Abs[z\_2]];\)\ \)
Print[""]; 
\(\(Print[\*"" |\!\(z\_1\) \!\(z\_2\)| = ""\ , \ 
        Abs[z\_1\ z\_2]];\)\ \)
\(\(Print[\*""|\!\(z\_1\)||\!\(z\_2\)| = "", \ Abs[z\_1], 
        Abs[z\_2]];\)\ \)
\(\(Print[\*""Is |(Subscript[z, 1]) (Subscript[z, 2])| = |(Subscript[z, 1])||(Subscript[z, 2])| ?  "", 
\ \ Abs[z\_1\ z\_2]\  == \ 
          Abs[z\_1]\ Abs[z\_2]];\)\ \)
Print[""]; 
\(\(Print[\*"" |\!\(z\_1\) \!\(z\_2\)| = "", \ 
        N[Abs[z\_1\ z\_2]]];\)\ \)
\(\(Print[\*""|\!\(z\_1\)||\!\(z\_2\)| = "", \ 
        N[Abs[z\_1] Abs[z\_2]]];\)\ \)
\(\(Print[\*""Is |(Subscript[z, 1]) (Subscript[z, 2])| = |(Subscript[z, 1])||(Subscript[z, 2])| ?  "", 
\ \ Abs[z\_1\ z\_2]\  == \ 
          Abs[z\_1]\ Abs[z\_2]];\)\ \)

Section 1.3  Exercises for The Geometry of Complex Numbers.
See textbook page 20.

Section 1.4	The Geometry of Complex Numbers, Continued

Section 1.5	The Algebra of Complex Numbers, Revisited

Section 1.6	The Topology of Complex Numbers

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