Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONSIm zr 1 r 2 e i(θ 1+θ 2 )r 2 e iθ 2r 1 e iθ 1Re zFigure 3.8 The multiplication of two complex numbers. In this case r 1 andr 2 are both greater than unity.The algebra of the polar representation is different from that of the real andimaginary component representation, though, of course, the results are identical.Some operations prove much easier in the polar representation, others much morecomplicated. The best representation for a particular problem must be determinedby the manipulation required.3.3.1 Multiplication and division in polar formMultiplication and division in polar form are particularly simple. The product ofz 1 = r 1 e iθ1 and z 2 = r 2 e iθ2 is given byz 1 z 2 = r 1 e iθ1 r 2 e iθ2= r 1 r 2 e i(θ1+θ2) . (3.25)The relations |z 1 z 2 | = |z 1 ||z 2 | and arg(z 1 z 2 ) = arg z 1 +argz 2 follow immediately.An example of the multiplication of two complex numbers is shown in figure 3.8.Division is equally simple in polar form; the quotient of z 1 and z 2 is given byz 1= r 1e iθ1z 2 r 2 e = r 1e i(θ1−θ2) . (3.26)iθ2 r 2The relations |z 1 /z 2 | = |z 1 |/|z 2 | and arg(z 1 /z 2 ) = arg z 1 − arg z 2 are again94