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# Functions

• Combinatorial Functions.
n!, ln(n!), n!!, n!!!, n!!!!, (n)m, Cnm, and Cnm1m2...mk

• Gamma and Beta Functions.
Γ(x), lnΓ(x), γ(x,a) = ∫0xt a-1e-tdt , γ*(x,a) = ∫0xt a-1e-tdt / Γ(a), &psi(x)=(1/Γ(x)) dΓ(x)/dx, B(a,b) = ∫01t a-1(1-t) b-1dt, ln B(a,b), and Ix(a,b) = ∫0xt a-1(1-t) b-1dt

• Probability Distributions and Random Variates.
Beta distribution, Binomial distribution, Cauchy distribution, Chi square distribution, Exponential distribution, F distribution, Gamma distribution, Gaussian distribution, Normal distribution, Geometric distribution, Gumbel's minimumin extreme value distribution, Gumbel's maximum extreme value distribution, Hypergeometric distribution, Kolmogorov's distribution, Kumaraswamy's distribution, Laplace's distribution, Logistic distribution, Log series distribution, Negative binomial distribution, Pareto's distribution, Poisson's distribution, Student's t distribution, Uniform distribution, Weibull's distribution, Bernoulli random variate, Beta random variate, Binomial random variate, Cauchy random variate, Exponential random variate, Gamma random variate, Gaussian random variate, Geometric random variate, Gumbel's minimumin extreme value random variate, Gumbel's maximum extreme value random variate, Kumaraswamy's random variate, Laplace's random variate, Logistic random variate, Log series random variate, Negative binomial random variate, Pareto's random variate, Poisson random variate, Student's T with 2 degrees of freedom random variate, Uniform random variate, Weibull random variate

• Orthogonal Polynomials.
T n, U n, V n, W n, T*n, U*n, V*n, W*n, Cαn, Hn, Hen, Pn(α,β),Ln, Lnα, Pn, Pn*,
Cn(x; α), tn(x; N), and Kn(x; p, N)

• Elliptic Integrals.
F(φ,k), E(φ,k), K(k), E(k), Λ0(&phi,k), and Z(φ,k)

• Elliptic Functions.
am(u,k), sn(u,k), cn(u,k), dn(u,k), (sn,cn,dn), (cs,ds,ns), (sc,dc,nc), (sd,cd,nd),
sn -1(x,k), cn -1(x,k), dn -1(x,k) and q(k) = exp( - &pi K'(k) / K(k) )

• Theta Functions.
θi(ν, x), ϑi(z, q), i = 1,2,3,4, ϑi(u, k) = (2K / π), i = s, c, d, n, θi(0, x), ϑi(0, q),
i = 1,2,3,4
, (dθi/dν)(ν, x), (dϑi / dz)(z, q), i = 1,2,3,4, (d ϑi / du)(u, k), i = s, c, d, n

• Exponential Integrals.
Ei(x) = ∫-∞x( e t / t ) dt, Ein(x) = ∫-∞x ( 1 - e -t ) / t dt, En(x) = ∫1 ( e -xt / t n ) dt, αn(x) = ∫1 tn e -xt dt and βn(x) = ∫-11 tn e -xt dt

• Bernoulli Numbers.
Bn

• Euler Numbers.
En

• Riemann Zeta and Related Functions.
ζ, ζ*, η, η*, λ, λ*, β, β*

• Dawson's Integral.
Daw(x) = exp(-x2) ∫0xexp(t2) dt.

• Sin and Cos Integrals.
Si(x) = ∫0x ( sin(t) / t ) dt, Ci(x) = - ∫x ( cos(t) / t) dt , Cin(x) = ∫0x ( 1 - cos(t) ) / t ) dt, fi(t) = ∫0 ( sin(t) / ( t + x ) ) dt and gi(t) = ∫0 ( cos(t) / ( t + x ) ) dt

• Fresnel Sin and Cos Integrals.
S(x) = √(2 / π) ∫0x sin( t 2 ) dt, C(x) = √(2 / π) ∫0x cos( t 2 ) dt,
g(x) = √(2 / π) ∫0 exp(-2xt) sin( t 2 ) dt, and
f(x) = √(2 / π) ∫0 exp(-2xt) cos( t 2 ) dt