Bessel function of second kind

`Y = bessely(nu,Z)`

Y = bessely(nu,Z,1)

The differential equation

$${z}^{2}\frac{{d}^{2}y}{d{z}^{2}}+z\frac{dy}{dz}+\left({z}^{2}-{\nu}^{2}\right)y=0,$$

where *ν* is a real constant, is called *Bessel's
equation*, and its solutions are known as *Bessel
functions*.

A solution *Y*_{ν}(*z*) of
the second kind can be expressed as

$${Y}_{\nu}(z)=\frac{{J}_{\nu}(z)\mathrm{cos}(\nu \pi )-{J}_{-\nu}(z)}{\mathrm{sin}(\nu \pi )}$$

where *J*_{ν}(*z*) and *J*_{–ν}(*z*) form
a fundamental set of solutions of Bessel's equation for noninteger *ν*

$${J}_{v}(z)={\left(\frac{z}{2}\right)}^{\nu}{\displaystyle \sum _{k=0}^{\infty}\frac{{\left(-\frac{{z}^{2}}{4}\right)}^{k}}{k!\Gamma (\nu +k+1)}},$$

and Γ(*a*) is
the gamma function. *Y*_{ν}(*z*) is
linearly independent of *J*_{ν}(*z*).

*J*_{ν}(*z*) can
be computed using `besselj`

.

`Y = bessely(nu,Z)`

computes
Bessel functions of the second kind, *Y*_{ν}(*z*),
for each element of the array `Z`

. The order `nu`

need
not be an integer, but must be real. The argument `Z`

can
be complex. The result is real where `Z`

is positive.

If `nu`

and `Z`

are arrays
of the same size, the result is also that size. If either input is
a scalar, it is expanded to the other input's size.

`Y = bessely(nu,Z,1)`

computes `bessely(nu,Z).*exp(-abs(imag(Z)))`

.

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