# scipy.special.jn¶

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

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

Parameters: v : array_like Order (float). z : array_like Argument (float or complex). J : ndarray Value of the Bessel function, $$J_v(z)$$.

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

Notes

For positive v values, the computation is carried out using the AMOS [R394] zbesj routine, which exploits the connection to the modified Bessel function $$I_v$$,

$\begin{split}J_v(z) = \exp(n\pi\imath/2) I_v(-\imath z)\qquad (\Im z > 0)\end{split}$$\begin{split}J_v(z) = \exp(-n\pi\imath/2) I_v(\imath z)\qquad (\Im z < 0)\end{split}$

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).

References

 [R394] (1, 2) Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”, http://netlib.org/amos/

scipy.special.jv

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