Function: besseljzero
Section: transcendental
C-Name: besseljzero
Prototype: GD1,L,b
Help: besseljzero(nu,{k=1}): k-th zero of the J-bessel function of index nu. Works for real and complex ordrers but the result is guaranteed only if nu is nonnegative.
Doc: $k$-th zero of the $J$-Bessel function of index \var{nu}, close
 to $\pi(\nu/2 + k - 1/4)$, usually noted $j_{\nu,k}$.
 \bprog
 ? besseljzero(0) \\ @com{first zero of $J_{0}$}
 %1 = 2.4048255576957727686216318793264546431
 ? besselj(0, %)
 %2 = 7.1951595399463653939930598011247182898 E-41
 ? besseljzero(0, 2) \\ @com{second zero}
 %3 = 5.5200781102863106495966041128130274252
 ? besseljzero(I) \\ @com{also works for complex orders, here $J_{i}$}
 %4 = 2.5377... + 1.4753...*I
 @eprog\noindent The function uses a Newton iteration due to Temme.
 If $\nu$ is real and nonnegative, the result is guaranteed and the function really
 returns the $k$-th positive zero of $J_{\nu}$. For general $\nu$, the result
 is not well defined, given by the Newton iteration with $\pi(\nu/2 + k - 1/4)$
 as a starting value. (N.B. Using this method for large real $\nu$ would give
 completely different results than the $j_{\nu,k}$ unless $k$ is large enough.)
