# Math2.org Math Tables: Special Functions

(Math)

Some of these functions I have seen defined under both intervals (0 to x) and (x to inf). In that case, both variant definitions are listed.
gamma = Euler's constant = 0.5772156649...

(x) = tx-1 e-t dt (Gamma function)

B(x,y) = tx-1 (1-t)y-1dt (Beta function)

Ei(x) = e-t/t dt (exponential integral) or it's variant, NONEQUIVALENT form:

Ei(x) = + ln(x) + (e^t - 1)/t dt = gamma + ln(x) + (n=1..inf)x^n/(n*n!)
li(x) = 1/ln(t) dt (logarithmic integral)
Si(x) = sin(t)/t dt (sine integral) or it's variant, NONEQUIVALENT form:
Si(x) = sin(t)/t dt = PI/2 - sin(t)/t dt

Ci(x) = cos(t)/t dt (cosine integral) or it's variant, NONEQUIVALENT form:
Ci(x) = - cos(t)/t dt = gamma + ln(x) + (cos(t) - 1) / t dt (cosine integral)

Chi(x) = gamma + ln(x) + (cosh(t)-1)/t dt (hyperbolic cosine integral)
Shi(x) = sinh(t)/t dt (hyperbolic sine integral)

(error function)

dilog(x) = ln(t) (1-t)-1 dt

Psi(n,x) = nth derivative of Psi(x)

W(x) = inverse of x ex

(laguerre polynomial degree n. (n) meaning nth derivative)

Dirichlet's beta function

Theorems with hyperlinks have proofs, related theorems, discussions, and/or other info.