A Note on the Solutions to a Transcendental Equation


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New Zealand Journal of Mathematics

Vol. 44, (2014), Pages 35-44

Mihaela-Cristina Drignei

Division of Physical and Computational Sciences,

University of Pittsburgh at Bradford,

Bradford, PA 16701, USA


Abstract We show that the transcendental equation \cos z + H\frac{\sin z}{z} = 0, with H a real number has only real solutions, which are countably many, simple, and there exists a positive H0 such that the positive solutions satisfy

z_n(H) = (n-\frac{1}{2})\pi + \frac{H}{(n-\frac{1}{2})\pi} - \frac{H^2}{2(n-\frac{1}{2})^3\pi^3} + \mathcal{O}(\frac{1}{n^3}), as n\rightarrow\infty,

for each H\geq -H_0, and

z_n(H) = (n+\frac{1}{2})\pi + \frac{H}{(n+\frac{1}{2})\pi} - \frac{H^2}{2(n+\frac{1}{2})^3\pi^3} + \mathcal{O}(\frac{1}{n^3}), as n\rightarrow\infty.

Keywords transcendental equation, implicit function theorem.

Classification (MSC2000) 34E05.

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