# Period of a particular Cycle for a Bessel Function

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?

• Look at the zeros of the two functions. Each "period length" is the difference of two consecutive zeros. – Somos May 6 '19 at 11:27
• @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 $0$? – Anders Gustafson May 6 '19 at 15:31
• For $a = \pm \frac{1}{2}$, $J_a(x)$ are actually decaying sinusoids for which the zeros are periodic (except at perhaps $x=0$). – Andy Walls May 6 '19 at 15:46
• Did you look at the DLMF Chapter 10 entry for zeros? – Somos May 6 '19 at 16:51