  Recently, we have established and used the generalized Littlewood theorem
concerning contour integrals of the logarithm of analytical function to obtain
new criteria equivalent to the Riemann hypothesis. Later, the same theorem was
applied to calculate certain infinite sums and study the properties of zeroes
of a few analytical functions. In this Note, we discuss what, in a sense, are
inverse applications of this theorem. We first prove an easy Lemma that if two
meromorphic on the whole complex plane functions $f(z)$ and $g(z)$ have the
same zeroes and poles, taking into account their orders, and have appropriate
asymptotic for large $|z|$, then for some integer $n$, $d^n \ln(f(z)/dz^n =
d^n\ln(g(z)/dz^n$. The use of this Lemma enables easy proofs of many identities
between elliptic functions and their transformation rules. In particular, we
show how for any complex number $a$, $\wp(z)-a$, where $\wp(z)$ is Weierstrass
$\wp$-function, can be presented as a product and ratio of three elliptic
$\theta_1$-functions of certain arguments. We also establish n-tuple rules for
elliptic theta-functions and /rho_z(z) functions.
