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On the Real Zeros of a certain Trigonometric Function

Published online by Cambridge University Press:  24 October 2008

P. Stein
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Extract

Suppose that a, c and A are real, and that

Let n (X) be the number of zeros of

in ; then it is not difficult to prove that

for some K = K (A, a, c, θ). The problem in this paper is to determine the function K explicitly in terms of its variables.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 31 , Issue 4 , October 1935 , pp. 455 - 467
Copyright
Copyright © Cambridge Philosophical Society 1935

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References

† γ is restricted by 0≤γ<1, while η is not so restricted. Corresponding, however, to any η satisfying the conditions of the Lemma there is a γ for which either γ = η or γ = η ± 1.

‡ For if γ and z are real, then

‡ The result may be obtained more easily by an appeal to the maximum modulus principle for a complex function

and is bounded, since η, β 1 and a 1 are bounded.

† See (1.1.4).

† In the case A = A′, f(x) has a double zero at x = x 1.

‡ In the case b 2 ≠ 0, f′(x 1) = 0, f″(x 1) = 0, it may be shown that f‴(x 1)╪0. Hence f(x) has a triple zero at x = x 1.

§ Neglecting the trivial case θ = 1.

† For this gives 0≤ξ<q.

‡ By Lemma 5.