F.22 Functions

Gaussian Error Function Approximations (Abramowitz & Stegun) and Probability Distributions

The standard normal cumulative distribution Phi(x) and error function erf(x) satisfy Phi(x) = 1/2 + (1/2) erf(x/sqrt(2)), where erf(x) is approximated to 1.5e-7 accuracy by Abramowitz and Stegun’s 5-term rational-exponential formula.

Interactive GEOMETOR construction for F.22 (Functions). Hover any geometric label in the text or figure to highlight the corresponding element.

O : Point A_1 : Point A_2 : Point B_1 : Point ===== major_axis = segment(A_1, A_2) b = segment(O, B_1) P = point_on_curve(A_2, B_1)

Mathematical Synthesis

Used by Seelig for his Monte Carlo geometric probability simulations (see F.41 and F.49), Topic 22 (SeeligNotebooks) records the interconversion formulas among \(\Phi(x), Q(x), \operatorname{erf}(x), \operatorname{erfc}(x)\), the 5-term rational-exponential approximation from Abramowitz & Stegun (AbramowitzStegun1964, p. 299, Eq. 7.1.26), and summary integrals for the Binomial, Poisson, Bernoulli, Uniform, Exponential, and Normal distributions.

Key Algebraic Identities

\[\Phi(x) = \frac{1}{2} + \frac{1}{2}\operatorname{erf}\left(\frac{x}{\sqrt{2}}\right), \qquad \operatorname{erf}(x) \approx 1 - \left(\sum_{k=1}^5 a_k t^k\right) e^{-x^2}, \quad t = \frac{1}{1 + 0.3275911 x}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.22 Functions — Folios (Folios 0130–0131, 2 scanned manuscript pages in resources/seelig/22_functions)
Euclid’s Elements	Classical Euclidean synthesis (EuclidElements)
GEOMETOR Elements (G-Index)	Foundational construction
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	General Reference (Spreadsheets)

Constructive Dependency Graph