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Abstract
A refined asymptotics of the Jacobi theta functions and their logarithmic derivatives have been received. The asymptotics of the Nevanlinna characteristics of the indicated functions and the arbitrary elliptic function have been found. The estimation of the type of the Weierstrass sigma functions has been given.
Highlights
In this paper, we investigate issues concerning the refined asymptotics and the distribution of values of the well-known Jacobi theta functions θ j+1 (z), j = 0, 3 [1]and the closely related Weierstrass functions σ (z), ζ (z), ℘(z) [1]
We have found the estimation of the type of the function σ (z) and proved that none of the numbers a, a ∈ C, is the exceptional value for the functions θ j+1 (z), θ0j+1 (z)/θ j+1 (z)( j = 0, 3) and for the arbitrary elliptic functions f, f 6= const in Nevanlinna’s sense
The obtained asymptotic formulas can be applied for an investigation of properties for the solutions of differential equations and their systems, in which the functions θ j+1 (z), θ0j+1 (z)/θ j+1 (z)( j = 0, 3) play a role, similar to the main facts of the Nevanlinna theory used in the papers [13,14,15,16]
Summary
In the works [8,9], it has been revealed that exceptional sets (outside which Formulas (5)–(7) are true) can be significantly narrowed but due to less accurate estimate of their remainder This is true for the functions θ j+1 (z), θ0j+1 (z)/θ j+1 (z)( j = 1, 3). The obtained asymptotic formulas can be applied for an investigation of properties for the solutions of differential equations and their systems, in which the functions θ j+1 (z), θ0j+1 (z)/θ j+1 (z)( j = 0, 3) play a role, similar to the main facts of the Nevanlinna theory used in the papers [13,14,15,16]
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