Sines, Cosines, Exponentials

It is helpful to remember the definition of the exponential function in terms of a power series:

$$\begin{eqnarray*}\exp(x) &\equiv& e^x \; \equiv \; \sum_{n=0}^{\infty} {x^n \ove... ...2} + {x^3 \over 6} + {x^4 \over 24} + {x^5 \over 120} + \dots \end{eqnarray*}$$

If we let x be imaginary, $x = i \theta$ (where $\theta$ is real), then this can be written

$$e^{i \theta} = 1 + i \theta - {\theta^2 \over 2} - i {\theta^... ...r 6} + {\theta^4 \over 24} + i {\theta^5 \over 120} - \dots$$

which ought to remind you (doesn't it?) of the series expansions for the sinusoidal functions:

$$\begin{eqnarray*}\cos \theta &=& 1 - {\theta^2 \over 2} + {\theta^4 \over 24} - ... ... &=& \theta - {\theta^3 \over 6} + {\theta^5 \over 120} - \dots \end{eqnarray*}$$

Note that these expansions perform a sort of "leapfrog" between even and odd terms. Putting them all together we can easily see that

$$\hbox{\fbox{\fbox{ ${\displaystyle e^{i \theta} \; = \; \cos \theta \; + \; i \, \sin \theta }$ }} }$$

and, since $\cos(-\theta) = \cos \theta$ while $\sin(-\theta) = - \sin \theta$,

$$e^{- i \theta} \; = \; \cos \theta \; - \; i \, \sin \theta$$

These simple equivalences are, to my mind, among the most astonishing relationships in all of mathematics. Why? Because they show an intimate relationship between two functions which would seem at first glance to have absolutely nothing in common: the monotonically increasing or decreasing exponential function $e^{\pm x}$ and the sinusoidally oscillating $\sin$ and $\cos$ functions!

We can also invert the relationship and obtain a definition for the $\sin$ and $\cos$ functions in terms of exponentials:

$$\hbox{\fbox{ ${\displaystyle \cos \theta \equiv \, {1\over2} \left( e^{i\theta} + e^{-i\theta} \right) }$ } }$$

$$\hbox{\fbox{ ${\displaystyle \sin \theta \equiv {1\over2i} \left( e^{i\theta} - e^{-i\theta} \right) }$ } }$$

Let's review the most important (for Physics, anyway) property of the exponential function: it is its own derivative! ${\displaystyle {d \over dx} e^x = e^x }$. If x = kt then we "pull out an extra factor of k" with each derivative with respect to t:

$$\hbox{\fbox{ ${\displaystyle {d^n \over dt^n} \; e^{kt} \; = \; k^n \, e^{kt} }$ } }$$

This latter property (which, by the way, works just as well for complex k as for real k) establishes the connection between the derivatives of $e^{i \omega t}$ (for instance) and those of $\sin(\omega t)$ and $\cos(\omega t)$:

$$\begin{eqnarray*}{d^2 \over dt^2} \; e^{i \omega t} &=& - \omega^2 \, e^{i \omeg... ... \over dt^2} \; \sin(\omega t) &=& - \omega^2 \, \sin(\omega t) \end{eqnarray*}$$

You will recall how useful these second-derivative properties were in SIMPLE HARMONIC MOTION and WAVES. Well, you ain't seen nothing yet!