Carlson R_c

(carlson-rc x y)

Carlson's R_c function defined by

 Rc(x,y) = integrate(1/2*(t+x)^(-1/2)*(t+y)^(-1), t, 0, inf)

Some interesting identities:

log(x)   = (x-1)*Rc(((1+x)/2)^2, x), x > 0
asin(x)  = x * Rc(1-x^2, 1),         |x|<= 1
acos(x)  = sqrt(1-x^2)*Rc(x^2,1),    0 <= x <=1
atan(x)  = x * Rc(1,1+x^2)
asinh(x) = x * Rc(1+x^2,1)
acosh(x) = sqrt(x^2-1) * Rc(x^2,1),  x >= 1
atanh(x) = x * Rc(1,1-x^2),          |x|<=1