maxframe.tensor.special.ive#

maxframe.tensor.special.ive(v, z, out=None)[source]#

Exponentially scaled modified Bessel function of the first kind.

Defined as:

ive(v, z) = iv(v, z) * exp(-abs(z.real))

For imaginary numbers without a real part, returns the unscaled Bessel function of the first kind iv.

Parameters:

v (array_like of float) – Order.
z (array_like of float or complex) – Argument.
out (ndarray, optional) – Optional output array for the function values

Returns:

Values of the exponentially scaled modified Bessel function.

Return type:

scalar or ndarray

See also

iv: Modified Bessel function of the first kind
i0e: Faster implementation of this function for order 0
i1e: Faster implementation of this function for order 1

Notes

For positive v, the AMOS [1] zbesi routine is called. It uses a power series for small z, the asymptotic expansion for large abs(z), the Miller algorithm normalized by the Wronskian and a Neumann series for intermediate magnitudes, and the uniform asymptotic expansions for \(I_v(z)\) and \(J_v(z)\) for large orders. Backward recurrence is used to generate sequences or reduce orders when necessary.

The calculations above are done in the right half plane and continued into the left half plane by the formula,

\[I_v(z \exp(\pm\imath\pi)) = \exp(\pm\pi v) I_v(z)\]

(valid when the real part of z is positive). For negative v, the formula

\[I_{-v}(z) = I_v(z) + \frac{2}{\pi} \sin(\pi v) K_v(z)\]

is used, where \(K_v(z)\) is the modified Bessel function of the second kind, evaluated using the AMOS routine zbesk.

ive is useful for large arguments z: for these, iv easily overflows, while ive does not due to the exponential scaling.

References