TOPICS
Search

Bourget's Hypothesis


When n is an integer >=0, then J_n(z) and J_(n+m)(z) have no common zeros other than at z=0 for m an integer >=1, where J_n(z) is a Bessel function of the first kind.

The theorem was proved as a special case of a more general result due to Siegel (1929; Watson 1966, p. 485).


See also

Bessel function of the First Kind, Siegel's Theorem

Explore with Wolfram|Alpha

References

Bourget, J. "Mémoire sur le mouvement vibratoire des membranes circulaires." Ann. Sci. de l'École norm. sup. 3, 55-95, 1866.Lvovsky, Y. and Mangoubi, D. "Bounded Multiplicity for Eigenvalues of a Circular Vibrating Clamped Plate." 30 Jul 2019. https://arxiv.org/abs/1907.12855.Petropoulou, E. N.; Siafarikas, P. D.; and Stabolas, I. D. "On the Common Zeros of Bessel Functions." J. Comput. Appl. Math 153, 387-393, 2003.Siegel, C. L. "Über einige Anwendungen diophantischer Approximationen." Abh. Preuß. Akad. der Wissensch., Phys.-Math. Kl., No. 1, 58 pp., 1929.Watson, G. N. "Bourget's Hypothesis." §15.28 in A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, pp. 484-485, 1966.

Referenced on Wolfram|Alpha

Bourget's Hypothesis

Cite this as:

Weisstein, Eric W. "Bourget's Hypothesis." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BourgetsHypothesis.html

Subject classifications