  We produce trigonometric expansions for Jacobi theta functions\\
$\theta_j(u,\tau), j=1,2,3,4$\ where $\tau=i\pi t, t > 0$. This permits us to
prove that\ $\log \frac{\theta_j(u, t)}{\theta_j(0, t)}, j=2,3,4$ and $\log
\frac{\theta_1(u, t)}{\pi \theta'_1(0, t)}$ as well as
$\frac{\frac{\delta\theta_j}{\delta u}}{\theta_j}$ as functions of $t$ are
completely monotonic. We also interested in the quotients $S_j(u,v,t) =
\frac{\theta_j(u/2,i\pi t)}{\theta_j(u/2,i\pi t)}$. For fixed $u,v$ such that
$0\leq u < v < 1$ we prove that the functions $\frac{(\frac{\delta}{\delta
t}S_j)}{S_j}$ for $j=1,4$ as well as the functions
$-\frac{(\frac{\delta}{\delta t}S_j)}{S_j}$ for $j=2,3$ are completely
monotonic for $t \in ]0,\infty[$.\\ {\it Key words and phrases} : theta
functions, elliptic functions, complete monotonicity.
