Complex representation for armonic waves

No less than sixteen years ago, I was assaulted by a question, posted in my high-school, junior-year phylosophy book: "Show that exp(i*2Pi) = 1 and explain its importance".

It is shocking to notice how many years have been gone having the question unanswered until today: as part of my new studies, I have it understood. What was I thinking to not be able to figure out before? Why could this happen? The fact is not isolated, I am afraid.

There is a couple of very common phenomena in Nature, much shunned and seriously forgotten in Schools and Academia, to which a score of real problemas belong to: diffusion and wave phenomena. Particularly, the wave motion is indeed a perturbation which, no matter its form, can be described as a superposition of sinusoidal, harmonic equations, and that´s the convenience of studying harmonics. In addition, as the mathematics of such waves is rather complicated and inaccesible as we pose realistic and accurate problems, the alternative representation by means of complex exponentials just come so precisely handy and resolutive.

And that´s the advantage of writing a sinusoidal motion as an exponential and its importance. The formula exp(i*2Pi) = 1 indicates a very intuitive and real fact of a harmonic wave: its periodicity. This is because the function exp(z), where z is a complex number is:
exp(z + i*2Pi) = exp(z) * exp(i*2Pi) = exp(z) * 1 = exp(z)                   (1)

Thus, the exponential representation of wave motion gets along well with reality.

Now, why exp(i*2Pi) = 1?

The answer has been a mysterious for a mind like mine, so much untenacious for a number of years. For an extended period of time, the Eulerian relation

exp(iθ) = cosθ + i sinθ                                                                         (2),

was largely retained by heart.

Now, it is all clear. Let´s start from a representation of a complex as

z = x + iy = r (cosθ + i sinθ)                                                                 (3),

where r is the norm of z vector and θ is the director cos(-1) respect to the real axes (angle with that axes).

Let´s take an unitarian complex number z´, for which r = 1. Thus:

z´= cosθ + i sinθ                                                                                   (4)

A differential of z´ is:

dz´= - sinθdθ + i cosθdθ                                                                       (5)

By multiplying in both sides of the equation by i and rearranging:

idz´ = - z´dθ                                                                                             (6)

Now, we separate variables and performe the integration. It results:

ln z´= iθ + C, being C the integration constant.                                       (7)

According to our representation of the complex number, based on Argand´s diagram, C must be 0, as for θ = 0, z´= 1. Thus:

z´= cosθ + i sinθ = exp (iθ)                                                                    (8)

The mystery is solved in 2 minutes!

The equation from which I started this discussion is readily seemed from (8), as z´= 1 when θ = 2Pi.

Furthermore, it is very convenient to treat formally the math of wave motion as complex exponential and at the end of derivations or when eventually the result is available, just take the real part of the complex final form, as that will yield the real solution for the problem.

Further verification:

For instance, prove that ψ(x,t) = Re[A*exp(i(ωt - kx + ε))] (1-D harmonic equation of motion) is equivalent to ψ(x,t) = A cos (ωt - kx + ε), which might clarify why the cosine form is preferred in textbooks to that of sine.

Solution:

From the basic math, the real part of a complex number, z, is calculated as:

Re(z) = 0.5 (z + z*)                                                                              (9),

where z* is the conjugate of z. In this case, then:

z = A [cos(ωt - kx + ε) + isin(ωt - kx + ε)]                                           (10)

Thus:

Re (z) = A/2 * [cos(ωt - kx + ε) + isin(ωt - kx + ε) + cos(ωt - kx + ε) - isin(ωt - kx + ε)]

                                                                                                             (11)

And, then:

Re (z) = Re[A*exp(i(ωt - kx + ε))] = A cos(ωt - kx + ε)                      (12)

q.e.d.