The Bessel Function of the First Kind $J_a(x)$, and the Bessel Function of the Second Kind $Y_a(x)$, at least when $a$, is an integer or half integer are cyclical, as their values go from positive to negative and from negative to positive an infinite number of times as $x$ increases. They are not periodic however as the period length, in which $J_a(x)$ or $Y_a(x)$ are positive, or in which $J_a(x)$ or $Y_a(x)$ are negative is not the same for every cycle. For the nth cycle of $J_a(x)$ or $Y_a(x)$, in which n is an integer, is there a formula for the period length of that cycle?

  • $\begingroup$ Look at the zeros of the two functions. Each "period length" is the difference of two consecutive zeros. $\endgroup$ – Somos May 6 '19 at 11:27
  • $\begingroup$ @Somos I understand that the "period length" is the difference between two consecutive zeros, but is there a formula for calculating the difference between two X-Values, in which the function has a value of $0$ that is more efficient than approximating the difference between two consecutive zeros by looking at a bunch of values and subtracting X-Values for which the function is very close to $0$? $\endgroup$ – Anders Gustafson May 6 '19 at 15:31
  • $\begingroup$ For $a = \pm \frac{1}{2}$, $J_a(x)$ are actually decaying sinusoids for which the zeros are periodic (except at perhaps $x=0$). $\endgroup$ – Andy Walls May 6 '19 at 15:46
  • $\begingroup$ Did you look at the DLMF Chapter 10 entry for zeros? $\endgroup$ – Somos May 6 '19 at 16:51

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