Abstract
A Bank-Laine function is an entire function E such that E(z) = 0 implies that E’(z) = ±1. Such functions arise as the product of linearly independent solutions of a second order linear differential equation ω″ + A(z)ω = 0 with A entire. Suppose that
where R is a rational function, g is a polynomial, and the αj and βj,k are non-zero complex numbers, and that E’(z) = ±1 at all but finally many zeros z of E. Then the quotients αj/αj′ are all rational numbers and E is a Bank-Laine function and reduces to the form E(z) = P0 (eαz) eQ 0(z) with α a non-zero complex number and P0 and Q0 polynomials.
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ElZaidi, S.M., Langley, J.K. Bank-Laine functions with periodic zero-sequences. Results. Math. 48, 34–43 (2005). https://doi.org/10.1007/BF03322894
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DOI: https://doi.org/10.1007/BF03322894