We recall Euler’s formula
and our polar representation of a complex number ,
Let us have a look at the polar diagram: is represented in the complex plane –plane, i.e. the – –plane, where
is the modulus of , i.e. geometrically the length of the vector . The argument is the angle between the vector and the –axis.
Motion along a line: we keep fixed and change from small to larger values. The corresponding complex numbers move along a straight line that has a fixed angle with the –axis.
Motion along a circle: we keep fixed and change from to larger values. We recognize that the corresponding complex numbers move along a circle with constant radius .
For , the complex numbers
are all situated on a circle with radius around the origin in the complex –plane.
Coming back to the complex exponential, we notice
where we wrote for (both are the same, this is only a different notation). We obtain sin and cos from the exponential of imaginary argument. What happens if we calculate the exponential for a general complex argument that has both a real and an imaginary part? We calculate
Please note that the real part only determines the modulus of ,
In particular, this means
Since for real , we see that
The circle is useful to describe, e.g., the motion of a particle on a circle. If we increase the angle linearily as a function of time ,
where is a fixed angular frequency(in s ), we have
where we allowed an arbitrary radius again. The real part of describes the –position, the imaginary part of describes the –position of the particle on the circle. The complex number itself is not a measurable quantity, but it contains useful information ( and –position of the particle).
Another important use of the complex exponent is in describing oscillations. This is due the fact that we can calculate the harmonic (sinusoidal) oscillations in terms of a complex exponent, when we argue that the real part is to be taken. The standard place to use such tecniques is in cicuit theory. A voltage is then represented by , and the real part in the expression will be conveniently forgotten for a while. The response of a capacitor to such an applied voltage, then leads to the complex equation (with ) , i.e. we have a simple relation between applied potential and current, just like the relation for a resistor. We can add such complex resistances, and take the rela part of the currents or volatges at the end; the phase of the complex numbers gives the phase difference betwen voltage and current.
A number is called an –th root of a complex number if . We write carelessly .
The solutions of the equation where is a positive integer are called –th roots of unity. We have
therefore we have to solve
Since
we have different solutions, i.e., solutions where the phase is between 0 and ,