Function: besselyzero
Section: transcendental
C-Name: besselyzero
Prototype: GD1,L,b
Help: besselyzero(nu,{k=1}): k-th zero of the Y-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 $Y$-Bessel function of index \var{nu}, close
 to $\pi(\nu/2 + k - 3/4)$, usually noted $y_{\nu,k}$.
 \bprog
 ? besselyzero(0) \\ @com{first zero of $Y_{0}$}
 %1 = 0.89357696627916752158488710205833824123
 ? bessely(0, %)
 %2 = 1.8708573650996561952 E-39
 ? besselyzero(0, 2) \\ @com{second zero}
 %3 = 3.9576784193148578683756771869174012814
 ? besselyzero(I) \\ @com{also works for complex orders, here $Y_{i}$}
 %4 = 1.03930... + 1.3266...*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 $Y_{\nu}$. For general $\nu$, the result
 is not well defined, given by Newton iteration with $\pi(\nu/2 + k - 3/4)$
 as a starting value. (N.B. Using this method for large real $\nu$ would give
 completely different results than the $y_{\nu,k}$ unless $k$ is large enough.)
