Chapter 2: Complexes

Geometry

Thanks to Euler's formula, \begin{equation} e^{i\theta} = \cos\theta + i\sin\theta \label{cis} \end{equation} polar coordinates can be used to write complex numbers in terms of their norm and a phase angle $\theta$. (A factor of the form $e^{i\theta}$ is called a phase.) That is, any complex number can be written in the form \begin{equation} z = r e^{i\theta} \end{equation} where \begin{equation} r = |z| \end{equation} since $|e^{i\theta}|=1$. Each complex number thus has a direction associated with it in the complex plane, determined by the angle $\theta$.

Euler's formula provides an elegant derivation of the angle addition formulas for sine and cosine. We have \begin{align} (\cos\alpha+i\sin\alpha)(\cos\beta+i\sin\beta) &= e^{i\alpha}e^{i\beta} = e^{i(\alpha+\beta)} \nonumber\\ &= \cos(\alpha+\beta)+i\sin(\alpha+\beta) \label{angleadd} \end{align} so that working out the left-hand side using complex multiplication, and comparing real and imaginary parts, yields the standard formulas for the trigonometric functions on the right-hand side.

We can use (\ref{angleadd}) to provide a geometric interpretation of complex multiplication. We have \begin{equation} (r_1 e^{i\theta_1})(r_2 e^{i\theta_2}) = r_1r_2 e^{i(\theta_1+\theta_2)} \end{equation} so the result of multiplying one complex number, $z_1$ by another, $z_2$, is to stretch $z_1$ by the magnitude $r_2$ of $z_2$, and to rotate it counterclockwise by the phase angle $\theta_2$ of $z_2$. The same product can of course be reinterpreted with the roles of $z_1$ and $z_2$ reversed. A special case is that multiplication by $i$ rotates a complex number counterclockwise by $\pi\over2$, without changing its norm.

Euler's formula is usually proved by comparing the power series expansions of each side. An alternative proof is obtained by noticing that both sides of this equation satisfy the differential equation \begin{align} {d^2\!f\over  d\theta^2} = - f \nonumber \end{align} with the same initial conditions. A special case of Euler's formula is the famous equation \begin{equation*} e^{i\pi}+1=0 \end{equation*} which relates five of the most basic symbols in mathematics!


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