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Q-exponential Function
3D plot
Contour plot
Root
q!=0, x = i (2 π n - i log(q)), n element Z
Integer root
x = 0, q = 1
Properties as a real function
R (all real numbers)
{y element R : y injective (one-to-one)
Periodicity
periodic in x with period 2 i π
Root for the variable x
x = log(q) + 2 i π c_1
Series expansion at x = 0
(q - 1) - x - x^2/2 - x^3/6 - x^4/24 + O(x^5) (Taylor series)
Derivative
d/dx(q - exp(x)) = -e^x
Indefinite integral
integral(-e^x + q) dx = q x - e^x + constant
Limit
lim_(x->-∞)(-e^x + q) = q
Series representation
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)!)