Thompson - Computing for Scientists and Engineers, страница 8

Note that complex conjugation performs an operation on a complex number, albeit a simple one. So it does not return a value in the sense of a C function value.Therefore, the “function” CConjugate is declared to be “void.” The value of the2.2 THE COMPLEX PLANE AND PLANE GEOMETRY27complex conjugate is returned in the argument list of CConjugate as xc and yc,which are dereferenced variables (preceded by a *, which should not be confusedwith complex conjugation, or even with / * and * / used as comment terminators).2. On the other hand, CModulus is declared as “double” because it is a functionwhich returns a value, namely mod (inside CModulus), which is assigned to zmodwithin the main program. Note that CModulus might also be used within an arithmetic statement on the right-hand side of the = sign for zmod.3. The program continues to process complex-number pairs while the input numberpair is nonzero.

If the zero-valued complex number (x = 0 and y = 0 ) is entered,the program exits gracefully by exit (0) rather than with the signal of an ungracefultermination, exit (1).With this background to programming complex conjugation and modulus, plusthe program for arguments in Section 2.6, you are ready to compute with complexvariables.Exercise 2.6Run several complex-number pairs through the program Conjugate & ModulusFunctions. For example, check that the complex conjugate of a complex-conjugate number produces the original number. Also verify that the modulus values of (x, y) and (y, x) are the same.

■2.2THE COMPLEX PLANE AND PLANE GEOMETRYIn the preceding section on algebra and computing with complex numbers we hadseveral hints that there is a strong connection between complex-number pairs (x, y)and the coordinates of points in a plane. This connection, formally called a “mapping,” is reinforced when we consider successive multiplications of z = x + i yby i itself.Exercise 2.7(a) Show that if z is represented by (x, y), then i z is (-y, x), then i2z is(-x,-y), i3z is (y,-x), and i4z regains (x , y).(6) Verify that in plane-polar geometry these coordinates are displaced fromeach other by successive rotations through / 2, as shown in Figure 2.1.

nThe geometric representation of complex numbers shown in Figure 2.1 is variously known as the complex plane, the Argand diagram, or the Gauss plane. Therelations between rotations and multiplication with complex numbers are summarized in the following subsections.28COMPLEX VARIABLESFIGURE 2.1 Rotations of complex numbers by /2 in the complex plane. Note that rotationsdo not change the length of a complex number, as indicated by the dashed circle.Cartesian and plane-polar coordinatesBefore reviewing the connections between the complex plane and plane geometry,let us recall some elementary relations between Cartesian and plane-polar coordinates. If the polar coordinates are r, the (positive) distance from the origin along aline to a point in the plane and the angle (positive in the anticlockwise direction)that this line makes with the x axis, then the Cartesian coordinates are given asin a right-handed coordinate system.

The polar coordinates are indicated in Figure 2.1. Inverting these equations to determine r and which is not as trivial as itmay look, is discussed in the first part of Section 2.6.In the complex plane we may therefore write z as(2.24)so that the modulus, which is also the radius, is given by2.2 THE COMPLEX PLANE AND PLANE GEOMETRY29(2.25)and the polar angle with respect to the x axis is given by(2.26)The principal value of the polar angle is the smallest angle lying betweenandSuch a specification of the principal value allows unique location of a pointin the complex plane. Other choices that limit may also be encountered, forexample, the range 0 to 27.Complex conjugation is readily accomplished by reflecting from to since(2.27)In the language of physics, z, and its complex conjugate are related through a paritysymmetry in a two-dimensional space.With this angular representation of complex variables, we can derive several interesting results.De Moivre’s theorem and its usesA theorem on multiplication of complex numbers in terms of their polar-coordinaterepresentations in the complex plane was enunciated by Abraham De Moivre (16671754).

We derive his theorem as follows. Suppose that we have two complexnumbers. the first as(2.28)and the second as(2.29)Their product can be obtained by using the trigonometric identities for expanding cosines and sines of sums of angles, to obtain(2.30)From this result we see that multiplication of complex numbers involves conventional multiplication of their moduli, the r1r2 part of (2.30), and addition of their angles.30COMPLEX VARIABLESTherefore, multiplication in the complex plane, as well as addition, can readily beshown, as in Figure 2.2.FIGURE 2.2 Combination of complex numbers in the complex plane. The complex numbersand their sum are indicated by the dashed lines, while their product is shown by the solid line.Equation (2.30) can be generalized to the product of n complex numbers, as thefollowing exercise suggests.Exercise 2.8(a) Prove by the method of mathematical induction that for the product of ncomplex numbers, z1,z2, ..., zn, one has in polar-coordinate form(2.31)(b) From this result, setting all the complex numbers equal to each other, provethat for the (positive-integer) nth power of a complex number(2.32)which is called De Moivre’s theorem.

nThis remarkable theorem can also be proved directly by using induction on n.Reciprocation of a complex number is readily performed in polar-coordinateform, and therefore so is division, as you may wish to show.2.3 FUNCTIONS OF COMPLEX VARIABLES31Exercise 2.9(a) Show that for a nonzero complex number, z, its reciprocal in polar-coordinate form is given by(2.33)(b) From the result in (a) show that the quotient of two complex numbers can bewritten in polar-coordinate form as(2.34)where it is assumed that r2 is not zero, that is, z2 is not zero. nThus the polar-coordinate expressions for multiplication and division are much simpler than the Cartesian-coordinate forms, (2.8) and (2.10).Although we emphasized in Exercise 2.2 that complex-number multiplicationand division in Cartesian form are much slower than such operations with real numbers, these operations may be somewhat speedier in polar form, especially if severalnumbers are to be multiplied.

An overhead is imposed by the need to calculate cosines and sines. Note that such advantages and disadvantages also occur when using logarithms to multiply real numbers.2.3 FUNCTIONS OF COMPLEX VARIABLESIn the preceding two sections we reviewed complex numbers from algebraic, computational, and geometric viewpoints. The goal of this section is to summarize howcomplex variables appear in the most common functions, particularly the exponential, the cosine and sine, and the hyperbolic functions. We also introduce the idea oftrajectories of functions in the complex plane.Complex exponentials: Euler’s theoremIn discussing De Moivre’s theorem at the end of the preceding section we noticedthat multiplication of complex numbers may be done by adding their angles, a procedure analogous to multiplying exponentials by adding their exponents, just theprocedure used in multiplication using logarithms.

Therefore, there is probably aconnection between complex numbers in polar-coordinate form and exponentials.This is the subject of Euler’s theorem.A nice way to derive Euler’s theorem is to write(2.35)32COMPLEX VARIABLESthen to note the derivative relation(2.36)But the general solution of an equation of the form(2.37)is given by(2.38)Exercise 2.10Show, by identifying (2.36) and (2.37) with the result in (2.38), that = i.Then choose a special angle, say = 0, to show that = 1.

Thus, you haveproved Euler’s theorem,(2.39)which is a remarkable theorem showing a profound connection between the geometry and algebra of complex variables. nIt is now clear from Euler’s theorem why multiplication of complex numbers involves addition of angles, because the angles are added when they appear in the exponents. Now that we have the formal derivation of Euler’s theorem out of the way,it is time to apply it to interesting functions.Applications of Euler’s theoremThere are several interesting and practical results that follow from Euler’s theoremand the algebra of complex numbers that we reviewed in Section 2.1. The trigonometric and complex-exponential functions can be related by noting that, for real angles,(2.40)which follows by taking complex conjugates on both sides of (2.39).

On combining(2.39) and (2.40) for the cosine we have(2.41)while solving for the sine function gives2.3 FUNCTIONS OF COMPLEX VARIABLES33(2.42)Both formulas are of considerable usefulness for simplifying expressions involvingcomplex exponentials.Exercise 2.11Use the above complex-exponential forms of the cosine and sine functions toprove the familiar trigonometric identity(2.43)the familiar theorem of Pythagoras. nAlthough our derivation of Euler’s theorem does not justify it, since is assumed tobe real in the differentiation (2.36) the theorem holds even for being a complexvariable. Thus the Pythagoras theorem also holds for complex .Some remarkable results, which are also often useful in later chapters, are foundif multiples of /2 are inserted in Euler’s identity, (2.39).