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Bessel function of first kind

`J = besselj(nu,Z)`

J = besselj(nu,Z,1)

`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)))`

.

The Bessel functions are related to the Hankel functions, also called Bessel functions of the third kind, by the formula

$$\begin{array}{l}{H}_{\nu}^{(1)}(z)={J}_{\nu}(z)+i{Y}_{\nu}(z)\\ {H}_{\nu}^{(2)}(z)={J}_{\nu}(z)-i{Y}_{\nu}(z)\end{array}$$

where $${H}_{\nu}^{(K)}(z)$$is `besselh`

, *J*_{ν}(*z*) is `besselj`

,
and *Y*_{ν}(*z*) is `bessely`

.
The Hankel functions also form a fundamental set of solutions to Bessel's
equation (see `besselh`

).