hyp2f1_1b_3half_asymptotic#

jeanspy.hyp2f1_jax.hyp2f1_1b_3half_asymptotic

jeanspy.hyp2f1_jax.hyp2f1_1b_3half_asymptotic(b, w, *, n_terms_regular=3, b_half_tol=1e-06)[source]#

Asymptotic evaluation of 2F1(1,b;3/2;w) for w close to 1.

Uses the analytic continuation around w=1:

2F1(1,b;3/2;w)
= A(b) * 2F1(1,b;b+1/2;1-w)

  • B(b) * (1-w)^(1/2-b) / sqrt(w)

where

A(b) = 1 / (1 - 2b), B(b) = Gamma(3/2) * Gamma(b-1/2) / Gamma(b).

For the regular hypergeometric factor we keep only a few terms in 1-w, which is accurate when w is sufficiently close to 1. This continuation is valid for negative non-integer b as well; exact non-positive integers and negative half-integers are handled by the auto selector and should stay on the power-series path.

Notes

Inputs and units. Dimensionless b and w; the shape of b must broadcast to the shape of w. n_terms_regular is the static regular-series term count; b_half_tol selects the b=1/2 limit.

Returns and shape. Approximation to 2F1(1,b;3/2;w), with the shape of w.

Validity. Use sufficiently close to w=1 within 0<w<1, away from poles of the continuation coefficients except for the implemented b=1/2 limit. The helper clips w inside the floating-point interval (0,1); clipping does not extend the asymptotic approximation’s valid region.

Errors. Domain and convergence are not validated; invalid inputs can produce nonfinite or inaccurate values.

Backend. JAX arrays on the configured CPU/GPU, with dtype set before import.

Differentiation. Fixed-loop continuous expressions support autodiff in their valid regions; piecewise thresholds and parameter singularities need checks.

Examples. examples/docs_numerics.py

Parameters:
Return type:

jax.Array