Exponential Function

The most general form of "an" exponential function is a power-law function of the form

(1)

where , , and are real numbers, is a positive real number, and is a real variable. When is positive, is an exponentially increasing function and when is negative, is an exponentially decreasing function.

In contrast, "the" exponential function (in elementary contexts sometimes called the "natural exponential function") is the function defined by

(2)

where e is positive real number is the base of the natural logarithm. The function is also the unique solution of the differential equation with initial condition . In other words, the exponential function is its own derivative, so

(3)

The exponential function defined for complex variable is an entire function in the complex plane.

The exponential function is implemented in the Wolfram Language as Exp[z].

The "natural" and general exponential functions are related to one another by a simple scalings of the variable and multiplicative prefactors via the identity

(4)

where is the natural logarithm.

The exponential function has the simple Maclaurin series

(5)

where is a factorial, and satisfies the limit

(6)

The exponential function satisfies the identity

(7)

It is also related to trigonometric functions via the identities

			(8)
			(9)
			(10)
			(11)

where is the Gudermannian (Beyer 1987, p. 164; Zwillinger 1995, p. 485).

If ,

(12)

Similarly, if

(13)

then

			(14)
			(15)
			(16)

The exponential function has continued fraction

(17)

(Wall 1948, p. 348).

The above plot shows the function (Trott 2004, pp. 165-166).

Integrals involving the exponential function include

			(18)
			(19)

(Borwein et al. 2004, p. 55).