Exponential Function

exponential function

In mathematics, a function in which a constant base is raised to a variable power. Exponential functions are used to model changes in population size, in the spread of diseases, and in the growth of investments. They can also accurately predict types of decline typified by radioactive decay (see half-life). The essence of exponential growth, and a characteristic of all exponential growth functions, is that they double in size over regular intervals. The most important exponential function is e^x, the inverse of the natural logarithmic function (see logarithm).

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exponential function [‚ek·spə′nen·chəl ′fəŋk·shən]

(mathematics)

The function ƒ(x) =e^x, written ƒ(x) = exp (x).

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Exponential Function 

the important elementary function f(z) = e^z; 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

e^z₁e^z₂ = e^{z₁ + z₂} (e^z₁)^z₂ = e^z₁z₂

for any values of z₁ and z₂. Moreover, on the real axis (Figure 1), e^x > 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 = e^z, then z = 1n w.

The function a^z, 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 2^x and (1/2)^x are discussed for real values of z = x. The relation between the exponential function a^z and the exponential function e^z is given by the equation

a^z = e^{z 1n a}

The exponential function e^x 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) e^z = e^{x + iy} =e^x (cos y + i sin y)

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

follow from it. The functions

are called hyperbolic functions and have a number of properties similar to those of the trigonometric functions. They and the trigonometric functions play an important role in various applications of mathematics.

Figure 1

It follows from equation (2) that the exponential function of a complex variable z has a period 2πi; that is, e^{z + 2πi} = e^z or e^2πi = 1. The derivative of the exponential function is equal to the function itself: (e^z)ʹ = e^z.

These properties of the exponential function account for its numerous applications. In particular, the exponential function expresses the law of natural growth, which governs the course of processes whose rate is proportional to the current value of a variable quantity. Unimolecular chemical reactions and, under certain conditions, the growth of a bacteria colony are examples. The periodicity of an exponential function of a complex variable, along with the function’s other properties, accounts for the exceptionally important role the function plays in the study of all periodic processes, particularly oscillations and wave propagation.