# scipy.special.jv¶

scipy.special.jv(v, z) = <ufunc 'jv'>

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

Parameters
varray_like

Order (float).

zarray_like

Argument (float or complex).

Returns
Jndarray

Value of the Bessel function, $$J_v(z)$$.

jve

$$J_v$$ with leading exponential behavior stripped off.

spherical_jn

spherical Bessel functions.

Notes

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

\begin{align}\begin{aligned}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)\end{aligned}\end{align}

For negative v values the formula,

$J_{-v}(z) = J_v(z) \cos(\pi v) - Y_v(z) \sin(\pi v)$

is used, where $$Y_v(z)$$ 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).

References

1

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

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