# Special functions The 19 definitions of the special functions library, each with its Epsil spelling, its MathJSON name, its signature and its full description. Each definition is listed under its Epsil spelling (the MathJSON name when it has none), with its signature in the engine's type syntax. The [Standard Library](/library/) page is the one-page index of every category. ## Definitions ### agm MathJSON `AGM` · `(complex | infinity, (complex | infinity)?) -> number` Arithmetic-geometric mean. AGM(z) is shorthand for AGM(1, z) (Fungrim convention). ### appellF1 MathJSON `AppellF1` · `(complex | infinity, complex | infinity, complex | infinity, complex | infinity, complex | infinity, complex | infinity) -> number` Appell hypergeometric function F₁(a; b₁, b₂; c; x, y), double series for |x|, |y| < 1. ### barnesG MathJSON `BarnesG` · `(complex | infinity) -> number` The Barnes G-function, the double gamma function G(z+1) = Γ(z)·G(z), G(1) = 1. G(n) is the superfactorial 1!·2!⋯(n−2)! at a positive integer n; G is entire, with zeros at the non-positive integers. ```epsil barnesG(5) // ➔ 12 ``` ```epsil N(barnesG(1/2)) // ➔ 0.603244281209446206191 ``` ### clausenCl MathJSON `ClausenCl` · `(integer, real) -> number` Clausen function Clₙ(θ) of integer order n ≥ 1 and real θ: Im Liₙ(e^{iθ}) = Σ sin(kθ)/kⁿ for even n, Re Liₙ(e^{iθ}) = Σ cos(kθ)/kⁿ for odd n. A real θ follows the engine precision. ```epsil [clausenCl(2, 1), clausenCl(3, 0), N(clausenCl(2, 1))] // ➔ [ClausenCl(2, 1),Zeta(3),1.01395913236076850429] ``` ### dedekindEta MathJSON `DedekindEta` · `(complex | infinity) -> number` Dedekind eta function η(τ), Im(τ) > 0. ### eisensteinE MathJSON `EisensteinE` · `(number, complex | infinity) -> number` Normalized Eisenstein series Eₛ(τ) of even weight s ≥ 2, Im(τ) > 0. ### ellipticE MathJSON `EllipticE` · `(complex | infinity, (complex | infinity)?) -> number` Elliptic integral of the second kind: complete E(m) with one argument, incomplete E(φ|m) with two (amplitude first, parameter convention m = k², as in Mathematica). ### ellipticF MathJSON `EllipticF` · `(complex | infinity, complex | infinity) -> number` Incomplete elliptic integral of the first kind F(φ|m) (amplitude first, parameter convention m = k², as in Mathematica). F(π/2|m) = K(m). ### ellipticK MathJSON `EllipticK` · `(complex | infinity) -> number` Complete elliptic integral of the first kind K(m), parameter convention m = k². ### ellipticPi MathJSON `EllipticPi` · `(complex | infinity, complex | infinity, (complex | infinity)?) -> number` Elliptic integral of the third kind: complete Π(n|m) with two arguments, incomplete Π(n; φ|m) with three (characteristic first, amplitude second, parameter convention m = k², as in Mathematica). ### expIntegralEi MathJSON `ExpIntegralEi` · `(complex | infinity) -> number` Exponential integral Ei(x) = PV ∫_{−∞}^x eᵗ/t dt. ### hypergeometric1F1 MathJSON `Hypergeometric1F1` · `(complex | infinity, complex | infinity, complex | infinity) -> number` Kummer confluent hypergeometric function ₁F₁(a; b; z) = M(a, b, z). ### hypergeometric2F1 MathJSON `Hypergeometric2F1` · `(complex | infinity, complex | infinity, complex | infinity, complex | infinity) -> number` Gauss hypergeometric function ₂F₁(a, b; c; z). ### jacobiTheta MathJSON `JacobiTheta` · `(number, complex | infinity, complex | infinity, number?) -> number` Jacobi theta function θⱼ(z, τ), j ∈ {1,2,3,4}, nome q = e^{iπτ} (Fungrim convention). ### logBarnesG MathJSON `LogBarnesG` · `(complex | infinity) -> number` The logarithm of the Barnes G-function, continued analytically with `LogGamma`: its imaginary part is not principal on the negative axis. −∞ at the zeros of G, the non-positive integers. ```epsil logBarnesG(5) // ➔ 2ln(2) + ln(3) ``` ```epsil N(logBarnesG(-1/2)) // ➔ (-1.7709451779743404 + 3.141592653589793i) ``` ### logGamma MathJSON `LogGamma` · `(complex | infinity) -> number` The analytic continuation of ln Γ(z), with its branch cut on (−∞, 0]; not `GammaLn`, the principal logarithm of Γ(z), which jumps by 2πi across the zeros of Im Γ. ```epsil logGamma(5) // ➔ 3ln(2) + ln(3) ``` ```epsil N(logGamma(-2.5 + 1.5i)) // ➔ (-3.7175134511917927 - 7.713065525834192i) ``` ### logIntegral MathJSON `LogIntegral` · `(complex | infinity) -> number` Logarithmic integral li(x) = PV ∫₀ˣ dt/ln t = Ei(ln x). ### polyLog MathJSON `PolyLog` · `(complex | infinity, complex | infinity) -> number` Polylogarithm Liₛ(z) = Σ_{k≥1} zᵏ/kˢ, at any real or complex order s. ### stieltjesGamma MathJSON `StieltjesGamma` · `(integer, number?) -> number` Generalized Stieltjes constants γₙ(a), the Laurent coefficients of ζ(s, a) at s = 1: ζ(s, a) = 1/(s−1) + Σₙ (−1)ⁿ γₙ(a)(s−1)ⁿ/n!. StieltjesGamma(n) is γₙ = γₙ(1), and γ₀ is Euler's constant. ```epsil stieltjesGamma(0) // ➔ "EulerGamma" ``` ```epsil N(stieltjesGamma(1)) // ➔ -0.0728158454836767248606 ``` ```epsil N(stieltjesGamma(2, 1/2)) // ➔ 0.968864475220290711422 ```