Complex Exponentials

There is one more complex variable situation we want to discuss because it occurs so frequently in quantum mechanics and, hence, in QIS and QC. This is the complex exponential function built with the fundamental exponential number $e = 2.71828 \cdots$. In particular, the famous Euler formula links the exponential function to sines and cosines: a truly amazing and extremely useful relationship. The Euler formula is

$$e^{i\phi} = \cos\phi + i\sin\phi$$

for any number $\phi$. If we use a real number $\phi$ as the argument, the cosine part of the equation above is the real part of $e^{i\phi}$, while the sine term is the imaginary part.

I always found the Euler formula mind-boggling. It combines algebra (sums and exponentials) with trigonometry (sines and cosines) and complex numbers ($i$). That’s a lot of math in one package!

You can also turn the formula around to write cosine and sine in terms of the complex exponentials:

$$\cos\phi = (\frac{1}{2} e^{i\phi} +e^{−i\phi}) \hspace{0.6cm} \sin\phi = \frac{1}{2i} (e^{i\phi} −e^{−i\phi}).$$

Not only is the Euler formula a useful way to do trigonometry, it also leads to a useful geometric interpretation for complex variables. Multiplying a generic complex variable, for example, our old friend $z = u + i υ$, by $e^{iθ}$ (with $θ$ real) is geometrically equivalent to rotating the vector representing $z$ in the complex plane by the angle $θ$ without changing its length:

$$\begin{aligned} e^{iθ}z & = (\cos θ + i \sin θ) (u + i υ) \\ & = (u \cos θ − υ \sin θ) + i(u \sin θ + υ \cos θ) = z^{\prime}. \end{aligned}$$