Mathematics of Physics and Engineering

The Complex Plane 183

of solving the polynomial equations, but these topics fall outside the scope
of our discussions. For more on the history of complex numbers, see the
book An Imaginary Tale: The Story of \/—I by P. J. Nahin, 1998.

4.1.2 The Complex Plane

The field C can be represented geometrically as a set of points in the plane,

with the real part along the horizontal axis, and the imaginary part along

the vertical axis. This representation identifies a complex number z = x+iy

with either the point (x,y) or the vector r — xi + yj in the Euclidean

space R^2 ; see Figure 4.1.1. The upper (lower) half-plane contains complex

numbers with positive (negative) imaginary part. Similarly, the right or

left half-plane refers to complex numbers with positive or negative real

parts, respectively.

z — x + iy xi + yj

Fig. 4.1.1 Complex plane and K^2

In polar coordinates,

z = r(cos# + ism0), (4.1.1)

where r = y/x^2 + y^2 = \z\ is the modulus or absolute value of z, and

9 = arg(z) is the argument of z. Similar to the polar angle at the origin,

the argument is not defined for z = 0.

Note that if 9 is the argument of z, so is 9 + 2nk for every integer k.

Accordingly, the principal value of the argument Arg(z) is defined as

the value of argz in the interval (—n, w].

EXERCISE 4.1.2? Verify the following formula for the principal value of the

argument:

Arg(z) = -

tan^1 (y/x),

7T + ta,n~^1 (y/x),

—TT + t&n~^1 (y/x)

x>0,y^0;

x<0,y>0;

x < 0, y < 0.

(4.1.2)