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SciPy

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scipy.special.yve

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

Exponentially scaled Bessel function of the second kind of real order.

Returns the exponentially scaled Bessel function of the second kind of real order v at complex z:

yve(v, z) = yv(v, z) * exp(-abs(z.imag))
Parameters:

v : array_like

Order (float).

z : array_like

Argument (float or complex).

Returns:

Y : ndarray

Value of the exponentially scaled Bessel function.

Notes

For positive v values, the computation is carried out using the AMOS [R483] zbesy routine, which exploits the connection to the Hankel Bessel functions H(1)v and H(2)v,

Yv(z)=12ı(H(1)vH(2)v).

For negative v values the formula,

Yv(z)=Yv(z)cos(πv)+Jv(z)sin(πv)

is used, where Jv(z) is the Bessel function of the first kind, computed using the AMOS routine zbesj. 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

[R483](1, 2) 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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