q!=0, x = i (2 π n - i log(q)), n element Z

x = 0, q = 1

R (all real numbers)

{y element R : y

injective (one-to-one)

periodic in x with period 2 i π

x = log(q) + 2 i π c_1

(q - 1) - x - x^2/2 - x^3/6 - x^4/24 + O(x^5) (Taylor series)

d/dx(q - exp(x)) = -e^x

integral(-e^x + q) dx = q x - e^x + constant

lim_(x->-∞)(-e^x + q) = q

q - exp(x) = q - sum_(k=0)^∞ x^k/(k!)

q - exp(x) = q - sum_(k=-∞)^∞ I_k(x)

q - exp(x) = q - sum_(k=0)^∞ (x^(-1 + 2 k) (2 k + x))/((2 k)!)