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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) = Gamma(x)
=   t^(x-1)
e^(-t)dt (Gamma function)
B(x,y) =  t^(x-1)
(1-t)^(y-1)DT (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)
Erf(x) = 2/PI^(1/2)e^(-t^2)
DT = 2/ PI (n=0..inf)
(-1)^n x^(2n+1) / ( n! (2n+1) ) (error
function)
FresnelC(x) = COs(PI/2
t^2) DT
FresnelS(x) = sin(PI/2
t^2) DT
dilog(x) =  ln(t)/(1-t)
DT
Psi(x) =  ln(Gamma(x))
Psi(n,x) = nth derivative of Psi(x)
W(x) = inverse of x*e^x
L sub n (x) = (e^x/n!)( x^n e^(-x)
) (n) (laguerre polynomial degree n. (n)
meaning nth derivative)
Zeta(s) = (n=1..inf) 1/n^s
Dirichlet's beta function B(x) = (n=0..inf)
(-1)^n / (2n+1)^x
Theorems with hyperlinks have proofs, related theorems, discussions, and/or
other info.
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