Module special 

Special mathematical functions: gamma, ln_gamma, digamma, erf, erfc, bessel_j0, plus Euler-Mascheroni and Catalan constants. Special mathematical functions for arbitrary-precision floating-point.

Provides pure-Rust implementations of:

• gamma — Gamma function Γ(x)
• ln_gamma — log-Gamma function ln(Γ(x))
• digamma — Digamma function ψ(x) = d/dx ln(Γ(x))
• erf — Error function
• erfc — Complementary error function
• bessel_j0 — Bessel function J₀(x)
• euler_gamma — Euler–Mascheroni constant γ
• catalan — Catalan’s constant G
• free_cache — No-op (MPFR compatibility shim)

All functions work on DBig (decimal arbitrary-precision big float) and accept a precision parameter giving the number of significant decimal digits to carry through the computation.

Algorithms

• Gamma / log-Gamma: Lanczos approximation (g=7, 9 coefficients) for x ∈ (0, 20]; Stirling series for x > 20; reflection formula for x < 0.
• Digamma: recurrence to shift x > 8, then Bernoulli asymptotic series.
• Erf: Taylor series for |x| ≤ 2; asymptotic continued-fraction for |x| > 2.
• Erfc: complement of erf with careful sign handling.
• Bessel J₀: power series for |x| ≤ 12; asymptotic expansion for |x| > 12.
• Euler γ: pre-stored 200-digit decimal string.
• Catalan G: pre-stored 200-digit decimal string.

Functions

bessel_j0

Compute the Bessel function J₀(x) to precision significant decimal digits.

catalan

Compute Catalan’s constant G ≈ 0.9159655941… to precision significant decimal digits (capped at 200).

digamma

Compute the digamma function ψ(x) = d/dx ln(Γ(x)) to precision significant decimal digits.

erf

Compute erf(x) to precision significant decimal digits.

erfc

Compute erfc(x) = 1 - erf(x) to precision significant decimal digits.

euler_gamma

Compute the Euler–Mascheroni constant γ ≈ 0.5772156649… to precision significant decimal digits (capped at 200).

free_cache

No-op cache cleanup shim.

gamma

Compute the Gamma function Γ(x) to precision significant decimal digits.

ln_gamma

Compute ln(Γ(x)) to precision significant decimal digits.