Fourier series

A new parameter $q$, called the nome, gives a nice formal expression for the Fourier series of an elliptic function. The nome $q$ is defined by

$$q = \exp -\pi\frac{{\bf K}'}{\bf K}$$

and modulus $k$ can be viewed as an implicit function of nome $q$.

Because sn, cn and dn have real periods of 4K, it is convenient to normalize them to the periods of sine and cosine by transforming argument $u$ to $$\frac{2{\bf K}x}{\pi}$$. The Fourier series that Jacobi obtained for sn, cn and dn [Jacobi1829, §39, equations 19, 21, and 25] become:

$$\frac{2{\bf K}}{\pi} \; {\rm sn}(\frac{2{\bf K}x}{\pi}, k) =... ..._{n=0}^{\infty} \frac{q^{(2n+1)/2}}{1-q^{2n+1}} \sin (2n+1)x$$

$$\frac{2{\bf K}}{\pi} \; {\rm cn}(\frac{2{\bf K}x}{\pi}, k) =... ..._{n=0}^{\infty} \frac{q^{(2n+1)/2}}{1+q^{2n+1}} \cos (2n+1)x$$

$$\frac{2{\bf K}}{\pi} \; {\rm dn}(\frac{2{\bf K}x}{\pi}, k) = 1 + 4 \sum_{n=1}^{\infty} \frac{q^{n}}{1+q^{2n}} \cos (2n)x$$

Jacobi also established similar results for various quotients and powers of elliptic functions.[Jacobi1829, §39, 41-46].