Abstract
We survey four families—erf/erfc, Bessel (J, Y, I, K), polylogarithm Li_s, and Lambert W—and lift them to symbolic numerals (m, a) as magnitude maps with alignment carry on clearly stated domains/branches. We give guards, monotonicity/asymptotics, and numerics (log/exp tricks, scaling). Under collapse phi(m,a) = m, all formulas reduce to the classical ones.
General lift policy (unary unless noted)
For input x = (m, a) and a classical real-valued function f:
f*(m, a) = ( f(m) , a ) # alignment carry
- Branches/parameters (e.g., order
nufor Bessel, indexsfor polylog) are classical constants by default and thus tagged(constant, +1)unless you explicitly declare them symbolic. - Collapse check:
phi( f*(m,a) ) = f(m)whereverfis defined on the chosen branch.
Manifest fields used below: branch = <...>, domain_guard = "error" | "clamp" | "nan", precision_targets, scaling.
Error function: erf, erfc
Definitions (classical).
erf(m) = (2/sqrt(pi)) * ∫_0^m exp(-t^2) dt for m in R
erfc(m) = 1 - erf(m)
Lift.
erf*(m, a) = ( erf(m) , a )
erfc*(m, a) = ( erfc(m) , a )
Monotonicity & range.erf is strictly increasing, erf(R) = (-1, +1). erfc is strictly decreasing, in (0, 2).
Asymptotics.
erf(m) ~ 1 - exp(-m^2)/(sqrt(pi)*m) as m -> +inf
erfc(m) ~ exp(-m^2)/(sqrt(pi)*m) as m -> +inf
Numerics.
Use series for small |m|, rational/continued-fraction or erfc for large |m|. Prefer expm1/log1p near zero when forming combinations.
Worked examples.
erf*((0, 0.4)) = ( 0 , 0.4 )
erf*((1, 0.7)) ≈ ( 0.8427 , 0.7 )
erfc*((2, -0.2)) ≈ ( 0.0047 , -0.2 )
Bessel functions: J_nu, Y_nu, I_nu, K_nu (survey)
Treat nu (order) as a classical parameter unless declared otherwise.
Domains (real line, principal).
J_nu(m), I_nu(m): defined for m >= 0
Y_nu(m): defined for m > 0 (singular at m = 0)
K_nu(m): defined for m > 0 (modified Bessel of second kind)
Lifts (unary in m).
J_nu*(m, a) = ( J_nu(m) , a )
Y_nu*(m, a) = ( Y_nu(m) , a ) # domain guard at m=0
I_nu*(m, a) = ( I_nu(m) , a )
K_nu*(m, a) = ( K_nu(m) , a ) # domain guard at m=0
Qualitative behavior (fixed nu).
J_nu: oscillatory, ~sqrt(2/(pi m)) * cos(m - nu*pi/2 - pi/4)for largem.Y_nu: oscillatory with phase shift; singular asm -> 0+.I_nu: monotone increasing form > 0, ~exp(m)/sqrt(2*pi m)for largem.K_nu: decays for largem, ~sqrt(pi/(2 m)) * exp(-m); singular asm -> 0+fornu != 0.
Numerics (guards).
- Use scaled forms for large
|m|: e.g.,I_nu(m) * exp(-|m|),K_nu(m) * exp(+|m|). - Three-term recurrences (up/down in
nu) stabilize evaluation; choose direction per conditioning. - Near
m = 0, switch to series or asymptotics and enforcedomain_guard = "error"forY_nu, K_nuifm <= 0.
Worked examples.
J_0*((0, 0.6)) = ( 1 , 0.6 )
J_1*((0, -0.4)) = ( 0 , -0.4 )
I_0*((3, 0.2)) ≈ ( 4.8808 , 0.2 ) # scaled evaluation recommended
K_0*((3, 0.2)) ≈ ( 0.034 , 0.2 )
Polylogarithm: Li_s(z) (real survey)
We present a real-line slice. Treat s as a classical parameter, use the principal branch in z.
Key cases (real z).
Li_s(z) = sum_{n=1}^inf z^n / n^s for |z| < 1
Li_1(z) = -log(1 - z) for z < 1
Li_s(1) = zeta(s) for s > 1
Lift (unary in z).
Li_s*((z, a)) = ( Li_s(z) , a )
Domains/branches (principal real).
- Safe series for
|z| < 1. - At
z = 1, use special values depending ons. - For
z > 1orz <= 0, analytic continuation introduces branch cuts; defaultdomain_guard = "error"unless explicitly enabled.
Numerics.
- Use acceleration near
z ~ 1-(e.g., Euler transform) or identities (Li_1(z) = -log1p(-z)). - For
s=2(dilog), combine with known constants for checks.
Worked examples.
Li_1*((1/2, 0.5)) = ( -log(1 - 1/2) , 0.5 ) = ( log 2 , 0.5 ) ≈ ( 0.6931 , 0.5 )
Li_2*((1/2, 0.3)) ≈ ( 0.58224 , 0.3 ) # known value for validation
Lambert W: product logarithm W(x) (principal real branches)
Definition.
W(x) * exp( W(x) ) = x
Real branches and domains.
W_0(x): principal branch, defined for x >= -1/e, increasing
W_-1(x): lower branch, defined for -1/e <= x < 0, decreasing
Lift.
W_0*((m, a)) = ( W_0(m) , a ) with m >= -1/e
W_-1*((m, a)) = ( W_-1(m) , a ) with -1/e <= m < 0
Special values & asymptotics.
W_0(0) = 0, W_0(-1/e) = -1
W_-1(-1/e) = -1, W_-1(0-) -> -inf
W_0(x) ~ x - x^2 + (3/2)x^3 for |x| small
W_0(x) ~ log x - log log x for x -> +inf
Guards (manifest).
lambertW.branch = "W0" (default) | "W-1"
lambertW.domain_guard= "error" (default) | "clamp_to_-1/e" (for visualization only)
Worked examples.
W_0*((0, 0.9)) = ( 0 , 0.9 )
W_0*((1, 0.2)) ≈ ( 0.5671 , 0.2 )
W_-1*((-0.1, 0.4)) ≈ ( -2.278 , 0.4 ) # valid since -1/e ≈ -0.3679 < -0.1 < 0
Ordering, collapse, and composition
- Order (magnitude channel). Where a function is monotone on its real domain (e.g.,
erf,I_nuform>0,W_0on[-1/e, +inf)), the lift preserves or reverses order accordingly (see 2.38). - Collapse: For any of the lifts above,
phi( f*(m,a) ) = f(m)on the stated domain/branch. - Composition: Composition with other lifted maps obeys
(g ∘ f)*(m,a) = ( g(f(m)) , a )as usual, provided domains/branches match.
Implementation notes (manifest)
# Common
alignment_policy = "carry" # unary lifts
domain_guard = "error" # default across families
precision_targets = { abs_tol: 1e-12, rel_tol: 1e-12 }
# erf/erfc
erf.strategy = "series_small | erfc_large | continued_fraction"
use_expm1 = true
# Bessel
bessel.nu = <value> # classical unless declared symbolic
bessel.scaling = "scaled_IK" # to avoid overflow/underflow
bessel.recurrence_mode = "up" | "down" # pick by conditioning
bessel.zero_guard = "error" for Y_nu, K_nu when m <= 0
# Polylog
polylog.s = <value> # classical
polylog.branch = "principal_real"
polylog.domain = "|z|<1 plus special points" # error otherwise
polylog.acceleration = "euler | richardson | none"
# Lambert W
lambertW.branch = "W0" | "W-1"
lambertW.domain_guard = "error"
Takeaway
Each family lifts as-is on the magnitude channel while the alignment channel is carried—yielding collapse-safe definitions that interoperate with classical numerics. Clear domain/branch guards, asymptotic guidance, and stable evaluation strategies make these special functions reliable within SSM; when a = +1, you recover the standard functions exactly.
Navigation
Previous → Special functions I — gamma & beta (2.41)
Next → Logistic/sigmoid, softplus, softsign (2.43)
Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: mult_mode = M2, clamp_eps = 1e-6, |a| < 1. All formulas are presented in plain text. Collapse uses phi(m,a) = m.
SHUNYAYA 