jv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

jv(v, z)

Bessel function of the first kind of real order and complex argument.


v : array_like Order (float). z : array_like Argument (float or complex).


J : ndarray Value of the Bessel function, :math:J_v(z).


For positive v values, the computation is carried out using the AMOS [1]_ zbesj routine, which exploits the connection to the modified Bessel function :math:I_v,

.. math:: J_v(z) = \exp(v\pi\imath/2) I_v(-\imath z)\qquad (\Im z > 0)

J_v(z) = \exp(-v\pi\imath/2) I_v(\imath z)\qquad (\Im z < 0)

For negative v values the formula,

.. math:: J_{-v}(z) = J_v(z) \cos(\pi v) - Y_v(z) \sin(\pi v)

is used, where :math:Y_v(z) is the Bessel function of the second kind, computed using the AMOS routine zbesy. Note that the second term is exactly zero for integer v; to improve accuracy the second term is explicitly omitted for v values such that v = floor(v).

Not to be confused with the spherical Bessel functions (see spherical_jn).

See also

jve : :math:J_v with leading exponential behavior stripped off. spherical_jn : spherical Bessel functions.


.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/