k1e

pmrf.math.bessel.k1e(x: Array) → Array

Exponentially scaled \(K_1(x)e^{x}\), for complex \(x\).

Mathematical Formulation

Below \(|x|=12\), \(K_1\) is evaluated from the logarithmic series used by k0_over_k1() and multiplied by \(e^{x}\); above it the ten-term large-argument expansion

\[K_1(x)e^{x}\sim\sqrt{\frac{\pi}{2x}} \left(1+\frac{3}{8x}-\frac{15}{128x^2}+\frac{105}{1024x^3} -\frac{4725}{32768x^4}+\ldots\right)\]

is used instead.

Validity

Measured against scipy.special.kve() over \(|x|\in[10^{-3},10^4]\) worst relative error 8e-8 on the \(45^\circ\) ray of a good conductor, at the switch point, rising to 2.5e-7 at \(30^\circ\) where the logarithmic series suffers more cancellation. Like every asymptotic branch in this module it degrades as \(\arg x\to90^\circ\). \(x=0\) is a genuine pole and is the caller’s to handle.

Parameters:

x (jnp.ndarray) – Argument, real or complex.

Returns:

\(K_1(x)e^{x}\).

Return type:

jnp.ndarray

References

Olver, F. W. J., et al. (eds.). NIST Digital Library of Mathematical Functions, 10.31.1, 10.40.2.