Mathematics, Department of


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Published in Bull. Amer. Math. Soc. 39 (1933) 923-929. Used by permission.


Hermite has shown that a meromorphic function which satisfies periodicity relations of the form (1) F (z + 2ω) = μF (z), F (z + 2ω’) = μ’F (z), where ω’/ω = a+ib, b>0, and μ, μ' are independent of z, maybe expressed in terms of the function v{z + X) (2) G(z) = σ(z + λ) ÷ σ(z) + σ(λ) eρz, and its derivatives, in which λ, ρ are suitably determined constants and σ(u) is the Weierstrass sigma function. The class of functions which satisfy conditions (1) has been called by Hermite doubly periodic of the second kind. We shall exclude from the present considerations Mittag-Leffler's singular case for this category of functions. He has shown that if F(z) =f(z)eρz, where ƒ(z) is an elliptic function, then the suitable element of decomposition is eρzζ(z), where ρz is the zeta function of Weierstrass.

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