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, \(J_v(z)\).
See also
jve\(J_v\) with leading exponential behavior stripped off.
spherical_jnspherical 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/