Finding Harmonics Analytically I've always obtained harmonics using simulation and never by hand.

My question is how can someone find the harmonics for a signal (say a square wave) by hand? What equations come in handy for finding the magnitudes at which each harmonic and sub-harmonic occur? 

I want equations or techniques that are useful to find a closed form equation that gives us the magnitude of the harmonic and sub-harmonic for a given harmonic n or sub-harmonic $n\pm \alpha  \ \ \ \  \alpha \in  \mathbb{N}$
What you're asking for is:


  given a periodic signal (e.g. square wave), where are the discrete frequency components (i.e. harmonics)?


You'll find that the harmonics you're looking for are the coefficients of the Fourier series expansion of your signal.

So, if you have a signal that has a known Fourier series expansion, or can be constructed from such signals, it's easy.

But how to find the coefficients in general?

This is answered exactly by the Fourier transform.

So, the relevant formulas are, given a signal $s(t)$ with period $P$


the coefficients $c_n$ of the Fourier series expansion $s(t) = \sum\limits_n c_n e^{j2\pi n\frac tP}$: $$c_n=\frac{1}{P}\int_{t_0}\limits^{t_0+P} s(t)\cdot e^{-j \tfrac{2\pi nt}{P}}\ dt$$ describing the $n$th harmonic. You just plug in your square wave for $s(t)$ and get out the coefficients of your harmonics.


and in the general, you don't get discrete harmonics, but a continuous spectrum, and thus, instead of a discrete Fourier series the Fourier transform.

Not posting info on the Fourier Transform here; it's pretty much common knowledge, and you wouldn't benefit – your profile indicates that you're studying physics, and thus, you've already met the Fourier transform, for example when describing electrons in periodic crystals.