i1e

pmrf.math.bessel.i1e(x: Array) → Array

Exponentially scaled \(I_1(x)e^{-x}\), for complex \(x\).

Mathematical Formulation

Below \(|x|=20\) the ascending series \(I_1(x)=\sum_{k\ge0}(x/2)^{2k+1}/(k!\,(k+1)!)\) is summed to 40 terms and multiplied by \(e^{-x}\); above it the large-argument expansion

\[I_1(x)e^{-x}\sim\frac{1}{\sqrt{2\pi x}} \left(1-\frac{3}{8x}-\frac{15}{128x^2}-\frac{105}{1024x^3} -\frac{4725}{32768x^4}-\frac{72765}{262144x^5}\right)\]

is used instead. Scaling is what keeps a thick-wall tube evaluable: \(I_1\) itself overflows well before the physics does.

Validity

Measured against scipy.special.ive() over \(|x|\in[10^{-3},10^4]\) on the \(45^\circ\) ray of a good conductor, worst relative error 1.2e-8 at the switch point. Like i0_over_i1(), the large-argument branch is an asymptotic series about the positive real axis and degrades as \(\arg x\to90^\circ\).

Parameters:

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

Returns:

\(I_1(x)e^{-x}\).

Return type:

jnp.ndarray

References

Olver, F. W. J., et al. (eds.). NIST Digital Library of Mathematical Functions, 10.25.2, 10.40.1.