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The distribution of the zeros of Jacobian elliptic functions with respect to the parameter k. (English) Zbl 1183.30023
In the paper under review, the author studies the size of the complete elliptic integral and the conjugate elliptic integral. Then after, he shows that if for given z, we denote by n(r) the number of zeros of the function msn(z|m) (or any other Jacobian function) inside the disc |m|r, then Ar(logr) -2 n(r)Br for some constants A and B and for sufficiently large r.
MSC:
30D15Special classes of entire functions; growth estimates
33E05Elliptic functions and integrals
References:
[1]M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, volume AMS 55, NBS, U. S. Government Printing Office, Washington, DC, USA, 1964.
[2]A. F. Beardon, Complex Analysis, Wiley, Chichester, UK, 1979.
[3]K. Chandrasekharan, Elliptic Functions, volume 281 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, Germany, 1985.
[4]H. E. Fettis, On the reciprocal modulus relation for elliptic integrals, SIAM J. Math. Anal. 1 (1970), 524–526. · Zbl 0202.34202 · doi:10.1137/0501045
[5]P. L. Walker, The analyticity of Jacobian functions with respect to the parameter k, Proc. R. Soc. Lond. A 8 (2003), 2569–2574. · Zbl 1052.33018 · doi:10.1098/rspa.2003.1157