Bessel function of first kind

`J = besselj(nu,Z)`

J = besselj(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*.

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

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

where Γ(*a*) is
the gamma function.

*Y*_{ν}(*z*) is
a second solution of Bessel's equation that is linearly independent
of *J*_{ν}(*z*).
It can be computed using `bessely`

.

`J = besselj(nu,Z)`

computes
the Bessel function of the first kind, *J*_{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.

`J = besselj(nu,Z,1)`

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

.

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