Appendix C. Complex Numbers
Each complex number z is a point in the complex plane, which is spanned by the real axis and the imaginary axes:
Two coordinate systems are commonly used for a (two-dimensional) plane: Cartesian and polar coordinates. For every complex number there exist two equivalent representations:
Here
is the “imaginary unit.”
We can transform between those representations as follows:
| Real part |
| Imaginary part |
| Magnitude |
| Phase |
Basic Operations
Complex numbers are added and multiplied component by component while taking into account that i2 = –1. If z1 = x1 + iy1 and z2 = x2 + iy2, then
Each complex number z has a “complex conjugate,” denoted
, which is the same as
except that the sign of the imaginary part has been reversed.
Thus,
The product of a complex number and its conjugate is always real. The square root of this product is also real. It is called the magnitude of z and is denoted by |z|.
The following identities hold:
We can use the complex conjugate to assign meaning to the reciprocal of a complex number:
Polar Coordinates
Every complex number z can also be expressed in polar coordinates:
where
The radius r can be included in the argument of the exponential:
In polar coordinates, multiplication of two complex numbers amounts to multiplying the magnitudes and adding the phases:
Taking the complex conjugate of a complex number is equivalent to changing the sign of its phase:
When multiplying z and z★, the exponential terms cancel and so leave the purely real number r2.
The Complex Exponential
If a complex number z is expressed in polar coordinates,
then all information about the magnitude of the number is contained in the radius r. The exponential term eiϕ provides information about the number’s angular orientation as a point in the complex plane. For this reason, the exponential term is also known as the phase factor.
A phase factor is an exponential term with a purely imaginary exponent. Its magnitude is always 1:
Geometrically, the phase factor describes a point on the unit circle. A straightforward geometric construction allows us to express it in terms of trigonometric functions:
For multiples of π/2 = 90 degrees, the phase factor takes on special values:
Of particular importance is the case when the phase angle grows steadily with time: ϕ = ωt, where ω is a real constant (the “angular frequency”).
In this case, the phase factor describes a purely oscillatory behavior at constant amplitude. The two trigonometric functions simply “wiggle” and exhibit no growth or decay. The greater is ω, the faster are the wiggles: