  We prove that the classical theta function $\theta_4$ may be expressed as $$
\theta_4(v,\tau) = \theta_4(0,\tau) \exp[- \sum_{p\geq 1} \sum_{k\geq 0} \frac
{1}{p} \bigg(\frac {\sin \pi v}{(\sin (k+{1/2})\pi \tau)}\bigg)^{2p}].$$ We
obtain an analogous expansion for the three other theta functions since they
are related. \\
  These results have several consequences. In particular, an expansion of the
Weierstrass elliptic function will be derived. Actions of the modular group and
other arithmetical properties will also be considered. Finally using a new
expression for the Rogers-Ramanujan continued fraction we produce a simple
proof of a Rogers identity. {\it Key words and phrases} : theta functions,
elliptic functions, q-series, Fourier series, continued fractions
