Polar Representation and Euler's Formula
As was seen before, one can represent any complex number as a vector in a two-dimensional plane - the so-called Argand diagram. Commonly, complex numbers are written in terms of rectangular coordinates with the $x$-coordinate being given by the real part and the $y$-coordinate by the imaginary part of the complex number.
Polar Representation
An equivalent way to represent complex numbers is provided by the polar representation. Here each corresponding vector is characterised by its length $|z|=\sqrt{z \bar{z}} \in [0,\infty)$ and the angle $\varphi \in [0,2\pi)$ between the real axis and this vector.^{[1]}
The domains of the coordinates $|z|\in [0,\infty)$ and $\varphi\in [0,2\pi)$ are limited in order to ensure a unique assignment between the complex numbers and the points in space. For instance, raising the angle by $2\pi$ or $360^\circ$ reproduces the same point in space and hence the same complex number. Equivalently, raising the angle by $\pi$ or $180^\circ$ corresponds to a simple multiplication by -1 of the whole number.
One can easily see that the rectangular coordinates $x$ and $y$ are related to the polar coordinates $|z|$ and $\varphi$ by basic trigonometric relationships:^{[2]} \begin{align} x &= \Re{z} = |z| \cos (\varphi) \\ y &= \Im{z} = |z| \sin (\varphi) \\[1ex] \Rightarrow \quad z &= x+iy = |z| \left[ \cos (\varphi) + i \sin (\varphi) \right] \label{eq:square-brackets} \end{align} Accordingly, the polar coordinates can be expressed in terms of the rectangular ones as well: \begin{align} |z| &= \sqrt{z\bar{z}} = \sqrt{x^2+y^2} \\ \tan (\varphi) &=\frac{\sin (\varphi)}{\cos (\varphi)} = \frac{y}{x} \end{align} Since $\tan$ is periodic, one needs to be careful with its inverse function (which is not unique). A given ratio of $y$ and $x$ can correspond to different values of $\varphi$. Thus, one needs to pay special attention when calculating $\varphi$. The correct way of doing that is provided by the so-called atan2 function.
If one now goes ahead and tries to simplify the expression in square brackets in eq. \eqref{eq:square-brackets}, one obtains a result which is quite remarkable:
Euler's Formula
The expression $\cos (\varphi) + i \sin (\varphi)$ can be simplified by replacing the trigonometric functions $\cos$ and $\sin$ with their power series representations and by using the relation $i^2=-1$:^{[3]} \begin{align} \cos (\varphi) + i \sin (\varphi) &= \sum_{i=0}^{\infty} (-1)^n \frac{\varphi^{2n}}{(2n)!} + i \sum_{i=0}^{\infty} (-1)^n \frac{\varphi^{2n+1}}{(2n+1)!} \nonumber \\ &= \underbrace{ \sum_{i=0}^{\infty} i^{2n} \frac{\varphi^{2n}}{(2n)!}}_\text{even terms} + \underbrace{ \sum_{i=0}^{\infty} i^{2n+1} \frac{\varphi^{2n+1}}{(2n+1)!} }_\text{odd terms} \nonumber \\ &= \sum_{n=0}^{\infty} \frac{(i\varphi)^n}{n!} = e^{i\varphi} \end{align}
The result of this short calculation is referred to as Euler's formula:^{[4]}^{[5]} \begin{align} e^{i\varphi} = \cos (\varphi) + i \sin (\varphi) \end{align}
The importance of the Euler formula can hardly be overemphasised for multiple reasons:
- It indicates that the exponential and the trigonometric functions are closely related to each other for complex arguments even though they exhibit a completely different behaviour for real arguments. In particular, one can express the trigonometric functions in terms of complex exponentials by using the definitions of the real and imaginary part of a complex number:^{[6]}^{[7]} \begin{align} \cos(\varphi) &= \Re{e^{i\varphi}} = \frac{e^{i\varphi} + e^{-i\varphi}}{2} \\ \sin(\varphi) &= \Im{e^{i\varphi}} = \frac{e^{i\varphi} - e^{-i\varphi}}{2i} \end{align} In general it is much easier to evaluate expressions that are given in terms of exponentials as compared to trigonometric ones - some examples/applications are given here.
- Evaluating the Euler formula for $\varphi=\pi$ yields a result which is considered as one of the most beautiful mathematical expressions that were ever found: \begin{align} e^{i\pi} + 1 = 0 \end{align} This expression unifies the three very fundamental numbers $e$, $\pi$ and $i$ as well as 0 and 1 within a single and even very simple equation.
- Furthermore, products of complex numbers can be rather unpleasant to evaluate if the numbers are given in rectangular representation. However, such products can be handled very easily when being given in polar form as will be demonstrated below.
Multiplication of Complex Numbers
Let $z_1 = |z_1| e^{i\varphi_1}$ and $z_2 = |z_2| e^{i\varphi_2}$ be two complex numbers in polar representation. Their product is given by:^{[8]}^{[9]} \begin{align} z_1 z_2 = |z_1||z_2| e^{i(\varphi_1+\varphi_2)} \end{align} Hence, its absolute value $|z_1 z_2 | = |z_1| |z_2|$ is the product of the individual absolute values $|z_1|$ and $|z_2|$ and its angle is equal to the sum of the individual angles $\varphi_1$ and $\varphi_2$.
Have a look at the following example \begin{align} z_1 = \frac{3}{2} e^{i\frac{\pi}{6}}& \quad\text{and}\quad z_2 = 2 e^{i \frac{3\pi}{4}} \\ \Leftrightarrow \quad &z_1 z_2 = 3 e^{i\frac{11\pi}{12}} \end{align} which is a straight forward calculation. The result is visualised in figure 2:
The same calculation could be done in rectangular coordinates just as well, but it would definitely be less fun as you can convince yourself: \begin{align} z_1 = \sqrt{\frac{3}{4}} + i \frac{1}{2} \quad\text{and}\quad z_2 = -\sqrt{\frac{1}{2}} + i \sqrt{\frac{1}{2}} \end{align} Therefore it is reasonable to use the polar representation when dealing with products of complex numbers.
References
[1] | Mathematische Methoden in der Physik Springer Spektrum 2016 (ch. 2.1) |
[2] | Mathematische Methoden in der Physik Springer Spektrum 2016 (ch. 2.1) |
[3] | Grundkurs Theoretische Physik 1 Springer 2012 (ch. 2.3) |
[4] | Grundkurs Theoretische Physik 1 Springer 2012 (ch. 2.3) |
[5] | Mathematische Methoden in der Physik Springer Spektrum 2016 (ch. 2.3.1) |
[6] | Grundkurs Theoretische Physik 1 Springer 2012 (ch. 2.3) |
[7] | Mathematische Methoden in der Physik Springer Spektrum 2016 (ch. 2.3.1) |
[8] | Grundkurs Theoretische Physik 1 Springer 2012 (ch. 2.3) |
[9] | Mathematische Methoden in der Physik Springer Spektrum 2016 (ch. 2.3.1) |