We briefly recap complex numbers. Through Euler's formulae, the usage of complex numbers is a crucial tool to describe periodic functions, e.g. plane waves and harmonic oscillations. Most textbooks spend only a few words on how this is done, while here we want to give an elaborate introduction.
The complex numbers are a field $\mathbb{C}$ of numbers
where $x \in \mathbb{R}$ and $y \in \mathbb{R}$ are real numbers, with the following definitions for addition $+$ and multiplication $-$:
The real part $\Re\left\{z\right\} \in \mathbb{R}$, imaginary part $\Im\left\{z\right\} \in \mathbb{R}$, complex conjugate $z^* \in \mathbb{C}$, and magnitude or modulus $|z| \in \mathbb{R}$ of a complex number $z \in \mathbb C$ are definded via
$$ \begin{split} z &=\, x + \imath y &\\ &\Rightarrow\\ \Re{\left\{z\right\}} &=\, x & \\ \Im\left\{z\right\} &=\, y & \\ z^* &=\, x + \imath( -y) &= x - \imath y \quad\left[\Rightarrow \, z_1^*\,{z_2}^* =\, (z_1\,z_2)^*\right]\\ \lvert z\rvert &=\, \sqrt{zz^\ast} &=\, \sqrt{x^2+y^2} \in \mathbb{R} \end{split} $$
The field of real numbers can be interpreted as a subfield of the complex numbers with the definition
and the imaginary unit $\imath$ can be interpreted as a number, the square root of the real number $-1$.
We leave it to the reader to show that with this the complex numbers are a field.
Euler's formula, named after Leonhard Euler⤴ (1707-1783), states that, for any real number $\phi \in \mathbb{R}$,
where $e$ is the base of the natural logarithm, $\cos$ and $\sin$ are the trigonometric functions cosine and sine respectively. Euler's formula can be proven by simply expanding the terms in the formula into a Taylor series⤴.
With the function $exp(\imath \phi), \phi \in \mathbb{R}$ one hence describes the position of a point on the unit circle in the complex plane:
Figure 2.1.1 Unit circle in the complex plane. Credit: Wikipedia⤴
Remembering that for any angle $\phi \in \mathbb{R}$
This means that any complex number $z \in \mathbb{C}$ can be written in the form
where $\phi \in \mathbb{R}$ is called the argument of $z$. Note also that $\lvert z \rvert \in \mathbb{R}$ and $\lvert \exp ^{\imath\phi}\rvert$=1
Using Euler's formula ($\S$ 2.1.2), one finds that a (co-)sinusoidal real function
can be re-written as
where
with $z_1\in \mathbb{R}$ and $z_2 \in \mathbb{R}$. Inverting the relation leads to (assuming $a>0$):
We leave it to the reader to calculate this for a sine periodic function.
Another possibility to express a real periodic function in terms of a product of a complex number and an exponential is to identify $a$ with the complex number $z_a$:
and hence (see definition )
This is the most common convention.
To translate from one format to the other just use equation :
The conversion can be made more general, in that $a \in \mathbb{R}$ can be replaced by any complex number $z^\prime$ with the same absolute value but different argument, in other words $a$ (or $z_a$) multiplied with a number on the unit circle $x$:
$$ \begin{split} f(t) & = a\,\cos (\omega t+\phi) \\ & = \Re \left \{{z^\prime \, e^{\imath(\omega t+\phi)}}\right\}\\ z^\prime &= z^\prime_1 + \imath z^\prime_2\\ &=x \cdot a\\ &= (x_1 + \imath\,x_2) \cdot a\\ \lvert z^\prime \rvert & = \lvert a \rvert \Rightarrow \\ \lvert x \rvert & = 1 \Rightarrow \\ x &= e^{\imath\alpha}\\ x_1 &= \cos \alpha\\ x_2 &= \imath\,\sin \alpha\\ z^\prime_1 &= a \cdot \cos \alpha\\ z^\prime_2 &= \imath\,a\cdot\sin \alpha \end{split} \qquad . $$
Again, to figure out $\alpha$:
$$ \begin{split} a & > 0 \Rightarrow \\ \alpha & = atan2(z^\prime_1,z^\prime_2) &= \left \{ \begin{array}{lll} \arctan{\frac{z^\prime_2}{z_1}}&,&z^\prime_1 > 0\\ \arctan{\frac{z^\prime_2}{z^\prime_1}} - sign(z^\prime_2)\pi&,&z^\prime_1 < 0\\ sign(z^\prime_2)\frac{\pi}{2}&,&z^\prime_1 = 0 \land z^\prime_2 \neq 0 \end{array} \right. \end{split} \qquad . $$
Then, the conversion becomes, by simply implementing the identity:
For any complex number $z^\prime$ with $\lvert a \rvert = \lvert z^\prime \rvert $, one can hence formulate:
and $\alpha$ calculated according to equation . The most common convention is to enforce $\Im\left\{z^\prime\right\} = 0$, and hence formulation .
In this section we used the conventional symbols $t$ and $\omega$ usually employed for time and angular frequency for the case of an oscillation in time. This can of course be adapted to other periodic functions or other arguments. The equations above do not change when we replace the term $\omega t$ with $2\pi \nu t$ or $2\pi \frac{t}{T}$ or $\bf{k}\cdot\bf{x}-\omega t$ or $2\pi(\frac{l}{\lambda}x+\frac{m}{\lambda}y+\frac{n}{\lambda}z- \nu t)$