  We describe a purely-multiplicative method for extending an analytic
function. It calculates the value of an analytic function at a point, merely by
multiplying together function values and reciprocals of function values at
other points closer to the origin. The function values are taken at the points
of geometric sequences, independent of the function, whose geometric ratios are
arbitrary. The method exposes an "elastic invariance" property of all analytic
functions. We show how to simplify and truncate multiplicative function
extensions for practical calculations. If we choose each geometric ratio to be
the reciprocal of a power of a prime number, we obtain a prime functional
identity, which contains a generalization of the M\"obius function (with the
same denominator as the Rieman zeta function), and generates prime number
identities.
