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We describe a purely-multiplicative method for extending an analyticfunction. It calculates the value of an analytic function at a point, merely bymultiplying together function values and reciprocals of function values atother points closer to the origin. The function values are taken at the pointsof geometric sequences, independent of the function, whose geometric ratios arearbitrary. The method exposes an "elastic invariance" property of all analyticfunctions. We show how to simplify and truncate multiplicative functionextensions for practical calculations. If we choose each geometric ratio to bethe reciprocal of a power of a prime number, we obtain a prime functionalidentity, which contains a generalization of the M\"obius function (with thesame denominator as the Rieman zeta function), and generates prime numberidentities.

Extending a Function Just by Multiplying and Dividing Function Values: Smoothness and Prime Identities