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

Complex analysis can roughly be thought of as that subject which applies the ideas of calculus to imaginary numbers. But what exactly are imaginary numbers? Usually, students learn about them in high school with introductory remarks from their teachers along the following lines: "We can't take the square root of a negative number. But, let's pretend we can---and since these numbers are really imaginary, it will be convenient notationally to set i = (Sqrt[-1])." Rules are then learned for doing arithmetic with these numbers. The rules make sense. If i = (Sqrt[-1]), it stands to reason that (i^2) = -1. On the other hand, it is not uncommon for students to wonder all along whether they are really doing magic rather than mathematics.

	If you ever felt that way, congratulate yourself! You're in the company of some of the great mathematicians from the sixteenth through the nineteenth centuries. They, too, were perplexed with the notion of roots of negative numbers. The purpose of this section is to highlight some of the episodes in what turns out to be a very colorful history of how imaginary numbers were introduced, investigated, avoided, mocked, and---eventually---accepted by the mathematical community. We intend to show you that, contrary to popular belief, there is really nothing imaginary about "imaginary numbers'' at all. In a metaphysical sense, they are just as real as are "real numbers.''

Our story begins in 1545. In that year the Italian mathematician Girolamo Cardano published Ars Magna (The Great Art), a 40-chapter masterpiece in which he gave for the first time an algebraic solution to the general cubic equation  
			(x^3) + a (x^2) + b x  +  c  =  0. 
His technique involved transforming this equation into what is called a depressed cubic. This is a cubic equation without the (x^2) term, so that it can be written as 
			(x^3) + b x  +  c  =  0.  
	Cardano knew how to handle this type of equation. Its solution had been communicated to him by Niccolo Fontana (who, unfortunately, came to be known as Tartaglia---the stammerer---because of a speaking disorder). The solution was also independently discovered some 30 years earlier by Scipione del Ferro of Bologna. Ferro and Tartaglia showed that one of the solutions to the depressed cubic equation is  
			\(x\  = \ \@\(\(-c\)\/2\  + \ \@\(c\^2\/4 + b\^3\/27\)\)\%3\  + \ \@\(\(-\(c\/2\)\)\  - \ \@\(c\^2\/4 + b\^3\/27\)\)\%3\).  
	This value for x could then be used to factor the depressed cubic into a linear term and a quadratic term, the latter of which could be solved with the quadratic formula. So, by using Tartaglia's work, and a clever transformation technique, Cardano was able to crack what had seemed to be the impossible task of solving the general cubic equation.

Surprisingly, this development played a significant role in helping to establish the legitimacy of imaginary numbers. Roots of negative numbers, of course, had come up earlier in the simplest of quadratic equations such as (x^2) + 1 = 0. The solutions we know today as x = ±(Sqrt[-1]), however, were easy for mathematicians to ignore. In Cardano's time, negative numbers were still being treated with some suspicion, as it was difficult to conceive of any physical reality corresponding with them. Taking square roots of such quantities, therefore, was all the more ludicrous. Cardano himself, although making some attempts to deal with this notion, at one point said that the process of arithmetic that deals with quantities such as (Sqrt[-1]) is, "as refined as it is useless.'' Many mathematicians had this view, but finally there was a breakthrough.

In his 1572 treatise Algebra, Rafael Bombelli showed that roots of negative numbers have great utility indeed. Consider the simple depressed cubic equation (x^3) - 15  x - 4 = 0. Letting b = -15 and c = -4 in the "Ferro-Tartaglia" formula for the depressed cubic, we compute x = ((2 + (Sqrt[-121]))^(1/3)) + ((2 - (Sqrt[-121]))^(1/3)), or in a somewhat different form, x = ((2 + 11 (Sqrt[-1]))^(1/3)) + ((2 - 11 (Sqrt[-1]))^(1/3)). 

	With a flash of insight, Bombeli suspected that if the original depressed cubic equation had real solutions, then the two parts of x in the preceding equation could be written as u + v (Sqrt[-1]) and u - v (Sqrt[-1]) for some real numbers u and v. 
	That is, Bombeli believed u + v (Sqrt[-1]) = ((2 + 11 (Sqrt[-1]))^(1/3)) and u - v (Sqrt[-1]) = ((2 - 11 (Sqrt[-1]))^(1/3)). Indeed, using the well-known identity (((a + b))^3) = (a^3) + 3 (a^2) b + 3 a (b^2) + (b^3), and blindly pretending that roots of negative numbers obey the standard rules of algebra, we can see (letting a = u and b = v (Sqrt[-1])) that 
	
	(((u + v (Sqrt[-1])))^3)= (u^3) + 3  (u^2) (v (Sqrt[-1])) + 
        3 (u) (((v (Sqrt[-1])))^2) + (((v (Sqrt[-1])))^3)  
	            =(u^3) + 3 (u) (((v (Sqrt[-1])))^2) + 
          3  (u^2) (v (Sqrt[-1])) + (((v (Sqrt[-1])))^3)   
	                          =((u^3) - 
              3  u (v^2)) + (3  (u^2) v - (v^3)) (Sqrt[-1])  
	                          =u ((u^2) - 3 (v^2)) + 
          v (3  (u^2) - (v^2)) (Sqrt[-1])  
	                          ="2 + 11\!\(\@\(-121\)\)".

Thus, by equating like parts of the equations Bombeli reasoned that u ((u^2) - 3 (v^2)) = 2 and v (3  (u^2) - (v^2)) = 11. Perhaps with more luck than insight, Bombeli then supposed that u and v were integers. Since 2 is a prime number, its only integer factors are 2 and 1, so the equation u ((u^2) - 3 (v^2)) = 2 led Bombeli to conclude that u = 2 and (u^2) - 3 (v^2) = 1. From this is follows that (v^2) = 1, or v = ±1. Amazingly, u = 2 and v = 1 solve the second equation v (3  (u^2) - (v^2)) = 11, so Bombeli declared the values for u and v to be u=2 and v=1 respectively.

	Bombeli then claimed that if  (((2 + (Sqrt[-1])))^3) = 2 + 11 (Sqrt[-1]), it must be that 2 + (Sqrt[-1]) = ((2 + 11 (Sqrt[-1]))^(1/3)). Likewise, he asserted  2 - (Sqrt[-1]) = ((2 - 11 (Sqrt[-1]))^(1/3)). But then we clearly have 
	
	((2 + 11 (Sqrt[-1]))^(1/3)) + ((2 - 11 (Sqrt[-1]))^(1/3))=(2 + (Sqrt[-1])) + (2 - (Sqrt[-1])) = 4,  
	
and this was a proverbial bombshell. Heretofore, mathematicians could easily scoff at imaginary numbers when they arose as solutions to quadratic equations. With cubic equations, they no longer had this luxury. That x = 4 was a correct solution to the equation (x^3) - 15  x - 4 = 0 was indisputable, as it could be checked easily. However, to arrive at this very real solution, one was forced to detour through the uncharted territory of "imaginary numbers.'' Thus, whatever else one might say about these numbers (which, today, we call complex numbers), their utility could no longer be ignored.

But even this breakthrough did not authenticate complex numbers. After all, a real number could be represented geometrically on the number line. What possible representation could these new numbers have? In 1673 John Wallis made a stab at a geometric picture of complex numbers that comes close to what we use today. He was interested at the time in representing solutions to general quadratic equations, which we shall write as (x^2) + 2  b x  + (c^2)  = 0 so as to make the following discussion more tractable. Using the quadratic formula, the preceding equation has solutions x = -b - (Sqrt[(b^2) - (c^2)]) and x = -b + (Sqrt[(b^2) - (c^2)]).

Wallis imagined these solutions as displacements to the left and right from the point -b. He saw each displacement, 
whose value was (Sqrt[(b^2) - (c^2)]), as the length of the sides of the right triangles.

	Wallis' method has the undesirable consequence that -(Sqrt[-1]) is represented by the same point as is (Sqrt[-1]). Nevertheless, with this interpretation, the stage was set for thinking of complex numbers as "points in the plane.'' By 1800, the great Swiss mathematician Leonhard Euler (pronounced "oiler'') adopted this view concerning the n solutions to the equation (x^n) - 1 = 0. We shall learn shortly that these solutions can be expressed as cos (θ) + (Sqrt[-1]) sin (θ) for various values of  θ;  Euler thought of them as being located at the vertices of a regular polygon in the plane. Euler was also the first to use the symbol i for (Sqrt[-1]). Today, this notation is still the most popular, although some electrical engineers prefer the symbol j instead so that i can be used to represent current.

Perhaps the most influential figure in helping to bring about the acceptance of complex numbers was the brilliant German mathematician Carl Friedrich Gauss, who reinforced their utility by using them in his several proofs of the fundamental theorem of algebra (see Chapter 6). In an 1831 paper, he produced a clear geometric representation of x + i y by identifying it with the point (x, y) in the coordinate plane. He also described how to perform arithmetic operations with these new numbers.

	In would be a mistake, however, to conclude that in 1831 complex numbers were transformed into legitimacy. In that same year the prolific logician Augustus De Morgan commented in his book;  On the Study and Difficulties of Mathematics, "We have shown the symbol  (Sqrt[-a])to be void of meaning, or rather self-contradictory and absurd. Nevertheless, by means of such symbols, a part of algebra is established which is of great utility.''

To be sure, De Morgan had raised some logical problems associated with complex numbers. On the other hand, there were sufficient answers to these problems floating around at the time. Even if De Morgan was unaware of Gauss' paper when he wrote his book, others had done work similar to that of Gauss as early as 1806. De Morgan's remark illustrates that "raw logic'' by itself is at times insufficient to persuade the entire mathematical community to adopt new ideas. Certainly, a firm logical foundation was necessary, but so, too, was the willingness of a sufficient number of respectable mathematicians to modify their ideas concerning certain well-established properties of numbers. For example, with real numbers (Sqrt[ab]) = (Sqrt[a]) (Sqrt[b]) so long as both sides of the equation are defined. As you will soon learn, this is no longer true when working with complex numbers. As time progressed more and more mathematicians came to agree with these new ideas, and by the end of the nineteenth century complex numbers were firmly entrenched. Thus, as it is with any new mathematical or scientific theory, the acceptance of Complex Numbers came through a mixture of sociocultural interactions.

Example in narrative of Section 1.1, Page 2. Consider the simple depressed cubic equation  (x^3)  -  15 x  -  4  =  0, use Mathematica to factor the equation and find the roots.

Solution on Page 2.

Enter the cubic equation.

Clear[b, c, r, x, z]; 
eqn  =  (x^3)  -  15 x  -  4  ==  0; 
solset  =  Solve[eqn,  x ]; 
eqn
MapAll[Factor, eqn]
solset

Enter the solution for the cubic equation.

z  =  ((((-c)/2)  +  (Sqrt[((c^2)/4) + ((b^3)/27)]))^(1/3))  +  ((-(c/2)  -  (Sqrt[((c^2)/4) + ((b^3)/27)]))^(1/3))

Use the Ferro-Tartaglia formula  \(z = \@\(\(-c\)\/2 + \@\(c\^2\/4 + b\^3\/27\)\)\%3 - \@\(c\/2 + \@\(c\^2\/4 + b\^3\/27\)\)\%3\)  and verify that  z  it is a solution.

Clear[z];  
b  =  - 15;  
c  =  - 4;  
Print["Solve  ", 0, " = ", 
        (z^3)  +  b z  +  c];  
Print["Use the `Ferro-Tartaglia formula.`"];  
\(\(Print[\*""0 = \!\(z\^3\) + b z + c""];\)\ \ \)
z  =  ((((-c)/2)  +  (Sqrt[((c^2)/4) + ((b^3)/27)]))^(1/3))  +  ((-(c/2)  -  (Sqrt[((c^2)/4) + ((b^3)/27)]))^(1/3));  
\(\(Print[\*""z = \!\(\@\(\(-c\)\/2 + \\ \@\(c\^2\/4 + b\^3\/27\)\)\%3\) + \!\(\@\(\(-\(c\/2\)\) - \\ \@\(c\^2\/4 + b\^3\/27\)\)\%3\)""];\)\ \ \)
Print["Substitute  b = ", b, "  and  c = ", 
        c,  "  and get"];  
\(\(Print[\*""    \!\(\@\(\(-c\)\/2 + \\ \@\(c\^2\/4 + b\^3\/27\)\)\%3\) = "", \ \@\(\(-c\)\/2\  + \ \@\(c\^2\/4 + b\^3\/27\)\)\%3];\)\ \ \)
\(\(Print[\*""    \!\(\@\(\(-\(c\/2\)\) - \\ \@\(c\^2\/4 + b\^3\/27\)\)\%3\) = "", \ \ \@\(\(-\(c\/2\)\)\  - \ \@\(c\^2\/4 + b\^3\/27\)\)\%3];\)\ \ \)
Print["The details are"];  
\(\(Print[\*""z = ((2 + (Sqrt[-121]))^(1/3))+ ((2 + (Sqrt[-121]))^(1/3))""];\)\ \ \)
\(\(Print[\*""z = ((2 + 11 (Sqrt[-1]))^(1/3))+ ((2 + 11 (Sqrt[-1]))^(1/3))""];\)\ \ \)
\(\(Print[\*""    ((2 + 11 (Sqrt[-1]))^(1/3)) = "", \@\(2 + 11 \@\(-1\)\)\%3];\)\ \ \)
\(\(Print[\*""    ((2 - 11 (Sqrt[-1]))^(1/3)) = "", \@\(2 - 11 \@\(-1\)\)\%3];\)\ \ \)
Print["Add these quantities and obtain"];  
\(\(Print["z = ", \ \@\(2 + 11 \@\(-1\)\)\%3, " + ", \@\(2 - 11 \@\(-1\)\)\%3];\)\ \ \)
\(\(Print["z = ", \@\(2 + 11 \@\(-1\)\)\%3 + \@\(2 - 11 \@\(-1\)\)\%3];\)\ \ \)
Print["z = " ,  
        z,  ", using the `Ferro-Tartaglia formula.`"];

Examples of the cubic equation.  Mathematica can construct the solutions to the general cubic equation. However they are quite formidable.

Solution for the cubic equation.

Enter the depressed cubic equation.

Clear[a, b, c, z]; 
eqn  =  (z^3) + b z  +  c  ==  0; 
solset  =  Solve[eqn,  z ]; 
(Subscript[z, 1]) = 
        (Subscript[solset, [[1, 1, 2]]]); 
(Subscript[z, 2]) = 
        (Subscript[solset, [[2, 1, 2]]]); 
(Subscript[z, 3]) = 
        (Subscript[solset, [[3, 1, 2]]]); 
Print["The depressed cubic equation ", 
        eqn]; 
Print["Mathematica can find the roots", "\n"]; 
\(\(Print[\*""\!\(z\_1\) = "", 
        z\_1];\)\ \)
Print[""]; 
\(\(Print[\*""\!\(z\_2\) = "", 
        z\_2];\)\ \)
Print[""]; 
\(\(Print[\*""\!\(z\_3\) = "", 
        z\_3];\)\ \)

Enter the general cubic equation.

Clear[a, b, c, z]; 
eqn  =  (z^3) + a (z^2) + b z  +  c  ==  0; 
solset  =  Solve[eqn,  z ]; 
(Subscript[z, 1]) = 
        (Subscript[solset, [[1, 1, 2]]]); 
(Subscript[z, 2]) = 
        (Subscript[solset, [[2, 1, 2]]]); 
(Subscript[z, 3]) = 
        (Subscript[solset, [[3, 1, 2]]]); 
Print["The general cubic equation ", 
        eqn]; 
Print["Mathematica can find the roots", "\n"]; 
Print[""]; 
\(\(Print[\*""\!\(z\_1\) = "", 
        z\_1];\)\ \)
Print[""]; 
\(\(Print[\*""\!\(z\_2\) = "", 
        z\_2];\)\ \)
Print[""]; 
\(\(Print[\*""\!\(z\_3\) = "", 
        z\_3];\)\ \)

The formulas for the roots (Subscript[z, 1]),(Subscript[z, 2]),  and  (Subscript[z, 3]) are complicated.  Hopefully this text will promote their understanding.

Library Research Experience for Undergraduates

Project I. Write a report on an historical topic about complex analysis.

1. Thor, Anton and Ivan Pineda and David C. Arney, (1991), ''Extending the Complex Number System,'' Math. and Computer Education, Vol. 25, No. 1, pp. 10-16.

2. Apostol, Tom M., Gulbank D. Chakerian, Geraldine C. Darden and John D. Neff (Editors), (1977), Selected Papers on Precalculus: Chapter 2(c) Complex Numbers, The Math. Asso. of Am.

3. Azarnia, Nozar, (1987), ''A Classroom Note: on Elementary Use of Complex Numbers,'' Math. and Comp. Ed., Vol 21, No. 2, pp. 135-136.

4. Diamond, Louis E., (1957), ''Introduction to Complex Numbers,'' Math. Mag., Vol. 30 pp. 233-249.

5. Green, D. R., (1976), ''The Historical Development of Complex Numbers,'' The Math. Gazette, V. 60, No. 412, pp. 99-107.

6. Kleiner, Israel,(1988), ''Thinking the Unthinkable: The Story of Complex Numbers (with a Moral),'' The Math. Teach., V. 81, No. 7, p583-92.

7. Nievergelt, Yves, (1997), ''History and Uses of Complex Numbers,'' UMAP, Module 743, pp. 1-66, COMAP, Inc. Lexington, MA 02420 USA. COMAP   

8. Truesdell, C., ''Notes on the History of the General Equations of Hydrodynamics,'' Am. Math. M., Vol. 60, No. 7. (Aug. - Sep., 1953), pp. 445-458, Jstor.   

9. Walsh, J. L., ''History of the Riemann Mapping Theorem,'' Am. Math. M., Vol. 80, No. 3. (Mar., 1973), pp. 270-276., Jstor.

Section 1.1  Exercises for The Origin of Complex Numbers.
See textbook page 5.

Section 1.2	The Algebra of Complex Numbers

Section 1.3	The Geometry of Complex Numbers

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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