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
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
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
The exponential function has the simple Maclaurin series
