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) = n^th 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 n^th derivative)
Zeta(s) = (n=1..inf) 1/n^{^s}

Dirichlet's beta function B(x) = (n=0..inf) (-1)^{^n} / (2n+1)^{^x}

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