hyp2f1_1b_3half#

jeanspy.hyp2f1_jax.hyp2f1_1b_3half

jeanspy.hyp2f1_jax.hyp2f1_1b_3half(b, w, *, method='auto', n_terms=192, w_switch=0.65, w_asymptotic=0.6, w_asymptotic_negative=0.9, b_min_quad=1e-06, b_max_quad=1.49, b_half_avoid_asym=0.001, b_integer_avoid_asym=1e-08, n_quad=128, quad_rule='tanh_sinh', asym_n_terms=20)[source]#

2F1(1,b;3/2;w) specialized helper.

Parameters:

  • method (str) –

    • “series”: fixed-term power series.

    • ”quad”: Euler integral quadrature (0<b<3/2).

    • ”asymptotic”: asymptotic expansion around w=1.

    • ”auto”: uses the asymptotic continuation for positive b once it is

      more accurate than quadrature, uses it for negative non-integer b only very near w=1, and otherwise falls back to quadrature or the terminating/power series.

  • n_quad (int) –

    • quad_rule='tanh_sinh': currently ignored (uses precomputed tanh-sinh rule).

    • quad_rule='gauss_kronrod': number of uniform panels for composite GK.

  • w_asymptotic (float) – In method='auto', switch positive b values from quadrature to asymptotic evaluation for w >= w_asymptotic.

  • w_asymptotic_negative (float) – In method='auto', allow negative non-integer b values onto the asymptotic branch only for w >= w_asymptotic_negative.

  • b_half_avoid_asym (float) – In method='auto', avoid asymptotic branch for |b-1/2| < b_half_avoid_asym and keep quadrature in this region.

  • b_integer_avoid_asym (float) – In method='auto', avoid the asymptotic branch when b is close to a non-positive integer pole or negative half-integer pole of the continuation coefficient.

  • asym_n_terms (int) – Number of terms kept for the regular 2F1(1,b;b+1/2;1-w) factor in the asymptotic branch.

  • b (jax.Array | float)

  • w (jax.Array | float)

  • n_terms (int)

  • w_switch (float)

  • b_min_quad (float)

  • b_max_quad (float)

  • quad_rule (str)

Return type:

jax.Array

Notes

  • Intended domain for this project is w in [0,1).

  • The “auto” choice is designed specifically to handle the numerically difficult region around b≈1/2 and w≈1.

Inputs and units. Dimensionless b and w, with 0<=w<1. For the series, asymptotic and auto paths, the shape of b must broadcast to the shape of w; the quad path also supports full array broadcasting. method, n_terms, n_quad, quad_rule, asym_n_terms and the selection thresholds above are static controls. The denominator is 3/2.

Returns and shape. Approximation to 2F1(1,b;3/2;w), with the shape of w except for the quad path’s broadcast output.

Validity. The Euler integral branch requires 0<b<3/2. The auto selector routes around continuation poles and uses the power series where needed; accuracy still depends on the selected terms, rule and thresholds.

Errors. Unsupported methods or quadrature rules raise ValueError when selected. Out-of-domain values can produce nonfinite or inaccurate results.

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