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Exponential Function

Result

e^x

Plots

Roots

(no roots exist)

Properties as a real function

R (all real numbers)

{y element R : y>0} (all positive real numbers)

injective (one-to-one)

Periodicity

periodic in x with period 2 i π

Series expansion at x = 0

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

Derivative

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

Indefinite integral

integral e^x dx = e^x + constant

Limit

lim_(x->-∞) e^x = 0

Alternative representations

e^x = z^x for z = e

e^x = w^a for a = x/log(w)

e^x = 1 + 2/(-1 + coth(x/2))

Series representations

e^x = sum_(k=0)^∞ x^k/(k!)

e^x = sum_(k=-∞)^∞ I_k(x)

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

Definite integral

integral_(-∞)^0 e^x dx = 1

Definite integral over a half-period

integral_0^(i π) e^x dx = -2