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Special Functions
(Math | Calculus | Integrals | Special Functions)

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) = (integral)(0 to inf)t^(x-1) e^(-t)dt (Gamma function)
B(x,y) = (integral)(0 to 1)t^(x-1) (1-t)^(y-1)DT
(Beta function)
Ei(x) = (integral)(x to inf)e^(-t)/t DT (exponential integral) or it's variant, NONEQUIVALENT form:

Ei(x) = + ln(x) + (integral)(0 to x)(e^t - 1)/t DT = gamma + ln(x) + (sum)(n=1..inf)x^n/(n*n!)
li(x) = (integral)(2 to x)1/ln(t) DT (logarithmic integral)
Si(x) = (integral)(x to inf)sin(t)/t DT (sine integral) or it's variant, NONEQUIVALENT form:
Si(x) = (integral)(0 to x)sin(t)/t DT = PI/2 - (integral)(x to inf)sin(t)/t DT

Ci(x) = (integral)(x to inf)cos(t)/t DT (cosine integral) or it's variant, NONEQUIVALENT form:
CI(x) = - (integral)(x to inf)COs(t)/t DT = gamma + ln(x) + (integral)(0 to x) (COs(t) - 1) / t DT (cosine integral)

Chi(x) = gamma + ln(x) + (integral)(0 to x)(cosh(t)-1)/t DT (hyperbolic cosine integral)
Shi(x) = (integral)(0 to x)sinh(t)/t DT (hyperbolic sine integral)
Erf(x) = 2/PI^(1/2)(integral)(0 to x)e^(-t^2) DT = 2/sqrtPI (sum)(n=0..inf) (-1)^n x^(2n+1) / ( n! (2n+1) ) (error function)
FresnelC(x) = (integral)(0 to x)COs(PI/2 t^2) DT
FresnelS(x) = (integral)(0 to x)sin(PI/2 t^2) DT
dilog(x) = (integral)(1 to x)ln(t)/(1-t) DT
Psi(x) = (d/dx)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) = (sum)(n=1..inf) 1/n^s

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

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


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