Fungrim Identities
The Compute Engine ships a library of special-function identities derived from the Fungrim "Mathematical Functions Grimoire". These identities drive symbolic simplification, expansion, and equation solving for functions such as the elliptic integrals, Jacobi theta functions, Bessel functions, the Riemann zeta function, and many more.
This reference catalogues the 1380 identities behind the engine's 1385 Fungrim rules (a few identities back both a simplification and a solving rule), organized into the areas below. Each identity shows the formula, the conditions under which it holds, the symbols it involves, how the engine uses it, and a link to the authoritative upstream Fungrim entry (whose page carries the full prose description, proof sketch, and references).
This page is generated from the compiled Fungrim artifact by scripts/fungrim/gen-reference-doc.ts (upstream snapshot 953c2afd2822, translator grim2mathjson 0.1.0). Do not edit it by hand. The corpus is MIT-licensed; see data/fungrim/LICENSE.
Areas
Elementary functions (205)
Exponential function (16) · Golden ratio (5) · Inverse tangent (44) · Lambert W-function (8) · Natural logarithm (11) · Pi (4) · Powers (8) · Sinc function (24) · Sine (59) · Square roots (26)
Complex numbers (40)
Complex parts (21) · Complex plane (3) · Imaginary unit (16)
Gamma and related functions (123)
Barnes G-function (27) · Beta function (11) · Digamma function (39) · Factorials and binomial coefficients (22) · Gamma function (24)
Orthogonal polynomials (74)
Chebyshev polynomials (51) · Gaussian quadrature (1) · Legendre polynomials (22)
Bessel and hypergeometric functions (115)
Airy functions (9) · Bessel functions (57) · Confluent hypergeometric functions (11) · Coulomb wave functions (10) · Error functions (12) · Gauss hypergeometric function (16)
Elliptic integrals (304)
Arithmetic-geometric mean (28) · Carlson symmetric elliptic integrals (183) · Legendre elliptic integrals (80) · Weierstrass elliptic functions (13)
Modular forms and theta functions (311)
Dedekind eta function (23) · Illustrations of Eisenstein series (41) · Jacobi theta functions (204) · Modular j-invariant (19) · Modular lambda function (24)
Zeta and L-functions (79)
Dirichlet characters (15) · Hurwitz zeta function (40) · Multiple zeta values (7) · Riemann zeta function (17)
Number theory (67)
Greatest common divisor (56) · Prime numbers (1) · Totient function (10)
Combinatorial and integer sequences (62)
Bell numbers (1) · Bernoulli numbers and polynomials (9) · Fibonacci numbers (34) · Integer sequences (8) · Partition function (6) · Stirling numbers (4)