Periodicity of elliptic functions

Both Abel and Jacobi found that the elliptic functions sn, cn and dn were not only periodic and in fact had two independent periods. (See [Jacobi1829, §19]). Further, if the modulus $k$ is real and strictly between 0 and 1, one of the periods is real and the other pure imaginary. For sn, the real period turns out to be 4K and the imaginary period 2iK' where the values K and K' happen to be certain elliptic integrals, namely:

$${\bf K} = {\bf K}(k) = F(1\vert k)$$

and

$${\bf K'} = {\bf K}'(k) = {\bf K}(k') = F(1\vert\sqrt{1-k^2})$$

Since the functions have real periods when $k$ is between 0 and 1, Jacobi did what any analyst would do - he asked whether they could be expanded into Fourier series. The answer fortunately turned out to be yes.