# Exponential Function

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## exponential function

[‚ek·spə′nen·chəl ′fəŋk·shən]
(mathematics)
The function ƒ(x) = e x , written ƒ(x) = exp (x).

## Exponential Function

the important elementary function f(z) = ez; sometimes written exp z. It is encountered in numerous applications of mathematics to the natural sciences and engineering. For any real or complex value of z, the exponential function is defined by the equation

It is obvious that e0 = 1. When z = 1, the value of the function is equal to e, which is the base of the system of natural logarithms. Basic properties of the function are

ez1ez2 = ez1 + z2 (ez1)z2 = ez1z2

for any values of z1 and z2. Moreover, on the real axis (Figure 1), ex > 0. As x → ∞, the function increases faster than any power of x; when x → – ∞, it decreases faster than any power of 1/x:

no matter what the exponent n. The logarithmic function is the inverse of the exponential function: if w = ez, then z = 1n w.

The function az, where the base a > 0 is different from e, is also called an exponential function. For example, in school mathematics courses such exponential functions as 2x and (1/2)x are discussed for real values of z = x. The relation between the exponential function az and the exponential function ez is given by the equation

az = ez 1n a

The exponential function ex is an integral transcendental function. It can be expanded in the power series

which converges throughout the z-plane. Equation (1) can also serve as a definition of the exponential function.

Letting z = x + iy, L. Euler obtained (1748) the formula

(2) ez = ex + iy =ex (cos y + i sin y)

which connects the exponential function with the trigonometric functions. The equations