Let /(z) represent a uniform analytic function with no essential singularity in the finite part of the plane, and periodic, either simply or doubly. There exists in certain cases a number ra, different from unity, such that /(raz) is a rational function of/(z). It is desired to determine all of those cases. Thus a part of our problem is the determination of all elliptic functions which admit of complex multiplication. As far as the writer knows, this question has not been investigated before. It will be seen that we must have |w| ^ 1, and that the cases with |ra| = 1 are easy to handle. The problem obtains its interest when |ra| exceeds unity, and then it can be stated in a form which shows its relation to a result of Poincaré.t He has shown that if R(z) is a rational function such that P(0) =0, and that R'(0), which we shall denote by m, has a modulus greater than unity, there exists a function /(z), meromorphic or entire, such that

Periodic Functions with a Multiplication Theorem

Differential geometry and differential algebra are two formalisms which can be used to study systems of partial differential equations. Cartan's characters are numbers which naturally appear in the former case ; stairs of characteristic sets are pictures naturally drawn in the latter. In this paper, we clarify the relationship between these two notions. We prove also some invariant properties of characteristic sets.

https://link.springer.com/content/pdf/10.1007%2F3-540-45624-4_38.pdf

Cartan's Characters and Stairs of Characteristic Sets

Joseph Fels Ritt

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Joseph Fels Ritt