Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONSf(z)543211234 zFigure 3.1 The function f(z) =z 2 − 4z +5.the first term is called a real term. The full solution is the sum of a real termand an imaginary term and is called a complex number. A plot of the functionf(z) =z 2 − 4z + 5 is shown in figure 3.1. It will be seen that the plot does notintersect the z-axis, corresponding to the fact that the equation f(z) =0hasnopurely real solutions.The choice of the symbol z for the quadratic variable was not arbitrary; theconventional representation of a complex number is z, wherez is the sum of areal part x and i times an imaginary part y, i.e.z = x + iy,where i is used to denote the square root of −1. The real part x and the imaginarypart y are usually denoted by Re z and Im z respectively. We note at this pointthat some physical scientists, engineers in particular, use j instead of i. However,for consistency, we will use i throughout this book.In our particular example, √ −4=2 √ −1=2i, and hence the two solutions of(3.1) arez 1,2 =2± 2i =2± i.2Thus, here x = 2 and y = ±1.For compactness a complex number is sometimes written in the formz =(x, y),where the components of z may be thought of as coordinates in an xy-plot. Sucha plot is called an Argand diagram and is a common representation of complexnumbers; an example is shown in figure 3.2.84