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Freeman, New York, 1991.Wolfram, S., Mathematica: A System for Doing Mathematics by Computer,Addison-Wesley, Redwood City, California, second edition, 1991.Wylie, C. R., and L. C. Barrett, Advanced Engineering Mathematics, McGrawHill, New York, fifth edition, 1982.References on learning and using CDarnell, P. A., and P. E.

Margolis, C, a Software Engineering Approach, SpringerVerlag, New York, 1991.Eliason, A. L., C, a Practical Learning Guide, Macmillan, New York, 1988.Gehani, N., C: An Advanced Introduction, Computer Science Press, Rockville,Maryland, 1985.Harbison, S. P., and G. L Steele, Jr., C: A Reference Manual, Prentice Hall,Englewood Cliffs, New Jersey, third edition, 199 1.Kernighan, B. W., and D.

M. Ritchie, The C Programming Language, PrenticeHall, Englewood Cliffs, New Jersey, second edition, 1988.Kerrigan, J. F., From Fortran to C, Windcrest Books, Blue Ridge Summit,Pennsylvania, 199 1.Koenig, A., C Traps and Pitfalls, Addison-Wesley, Reading, Massachusetts, 1989.Miildner, T., and P. W. Steele, C us a Second Language, Addison-Wesley,Reading, Massachusetts, 1988.Plauger, P.

J., The Standard C Library, Prentice Hall, Englewood Cliffs, NewJersey, 1992.Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, NumericalRecipes in C, Cambridge University Press, New York, 1988.Schildt, H., Teach Yourself C, Osborne McGraw-Hill, Berkeley, California, 1990.Shammas, N., Introducing C to Pascal Programmers, Wiley, New York, 1988.16Previous HomeChapter 2A REVIEW OF COMPLEX VARIABLESThe purpose of this chapter is to review your understanding of complex numbersand complex variables, and to summarize results that are used extensively in subsequent chapters. Complex variables are treated elegantly and completely in manymathematics texts, but a treatment in which the aim is to develop intuition in the scientific applications of complex numbers may have much more modest goals, and isbest done from a geometrical perspective, rather than from an analytic or algebraicviewpoint.We start with the algebra and arithmetic of complex numbers (in Section 2.1)including a simple program, then turn in Section 2.2 to the complex-plane representation because of its similarities to plane-polar coordinates and to planar vectors.The simplest (and most common) functions of complex variables-complex exponentials and hyperbolic functions- are reviewed in Section 2.3.

Phase angles, vibrations, and complex-number representation of waves, which are all of great interest to scientists and engineers, are summarized in Section 2.4 before we take a diversion in Section 2.5 to discuss the interpretation of complex numbers. Project 2,which includes program Cartesian & Polar Coordinate Interconversionfor converting between plane-polar and Cartesian coordinates, concludes the text ofthis chapter.

This program serves to emphasize the ambiguities in calculating polarcoordinates from Cartesian coordinates, and it will be useful in the later programming applications. References on complex numbers complete the chapter.The discussion of complex variables is limited to the above topics, and does notdevelop extensively or with any rigor the topics of analytic functions in the complexplane, their differentiation, or their integration. Although several derivations later inthis book, especially those involving integrals, would be simplified if the methods ofcontour integration were used, the methods of derivation used here are usually directand do not require the extra formalism of contours and residues.

Readers who haveexperience with functions of a complex variable will often be able to substitute theirown methods of proof, which may be more direct than those provided here.17Next18COMPLEX VARIABLESMany of the examples and exercises in this chapter anticipate steps in our laterdevelopments that use complex variables, especially the material on Fourier expansions (Chapters 9 and 10). Since we always have a reference to the later material,you may wish to look ahead to see how the complex variables are used.2.1ALGEBRA AND COMPUTING WITH COMPLEX NUMBERSIn this section we summarize the algebraic properties of complex numbers, their properties for numerical computation, the relation between complex numbers and planegeometry, and the special operations on complex numbers — complex conjugation,modulus, and argument.When you have reviewed this section, you will have the conceptual and technicalskills for the more extensive developments of complex numbers that are presented inthe remainder of this chapter.

In particular, Project 2 — the program for convertingbetween coordinates (Section 2.6) -requires most of the ideas from this section.If you are experienced with complex variables, you may try the project before working this section. If you have difficulties with the mathematics in the project (as distinguished from the programming involved), return and rework this section.The algebra of complex numbersWe indicate a complex number, z, symbolically by(2.1)in which x and y are understood to be both real numbers. The sign + in this formula does not mean arithmetic addition, although it has many rules that are similar tothose for addition.

You will have already encountered yet another meaning of + asa sign used in the addition of vectors, which is also distinct from arithmetic addition.In (2.1), the symbol i has the property that(2.2)with a unique value being assumed for i itself. In engineering contexts it is moreusual to find the symbol i replaced by the symbol j, thus avoiding possible confusion when complex numbers are used to describe currents (i) in electrical circuits, asin our Section 8.1.

We will use i, recalling its origins in the initial letter of the historical term “imaginary.”Complex numbers may also be thought of as pairs of numbers, in which the order in the pair is significant. Thus we might write(2.3)analogously to vectors in a plane. Just as the coordinates (x, y) and (y,x) are usually distinct, so are the analogous complex numbers. The notation in (2.3) avoids2.1 ALGEBRA AND COMPUTING WITH COMPLEX NUMBERS19ambiguities in using the + sign and in the necessity of inventing a symbol satisfying (2.2). Further, many of the rules for manipulating complex numbers have astrong similarity to those for vectors in a plane if the notation (2.3) is used. Although we will write our results for complex-number algebra and arithmetic in thenotation (2.1), you are invited to try the number-pair notation in Exercise 2.1 (c).This notation is also used in some programming languages that allow complexarithmetic operations.The rules for manipulating complex numbers must be consistent with those forpurely real numbers ( y = 0) and for purely imaginary numbers (x = 0).

In the following, let z = x + iy generically, and let z1 = x1+ iy1, z2 = x2 + i y2 represent two particular complex numbers. Then the following properties hold:(2.4)Negation of a complex number is defined by(2.5)which is often written casually as(2.6)A complex number is zero only if both its real and imaginary parts are zero, which isconsistent with zero being the only solution of the equation z = -z.Addition or subtraction of two complex numbers is accomplished by(2.7)Multiplication of complex numbers is performed by(2.8)Reciprocal of a complex number is defined by(2.9)which has the property that z (1/z) = 1, as for the arithmetic of real numbers.Division of one complex number into another is based on the reciprocal of the divisor, and is therefore undefined if the divisor is zero:(2.10)20COMPLEX VARIABLESIn order to check your comprehension of these rules for complex arithmetic, trythe following exercise.Exercise 2.1(a) Verify that the rules (2.7) through (2.10) are consistent with those for realarithmetic by checking them for y1 = y2 = 0.(b) Check the consistency of (2.7) through (2.10) for purely imaginary numbersby setting x1= x2 = 0 and noting the condition on i, (2.2).(c) Use the notation for complex numbers as ordered-number pairs, as indicatedby (2.3), to write down the preceding complex-arithmetic rules, (2.4) through(2.10).