Exponential Function

Also found in: Dictionary, Thesaurus, Wikipedia.
Related to Exponential Function: exponential equation

exponential function

[‚ek·spə′nen·chəl ′fəŋk·shən]
The function ƒ(x) = e x , written ƒ(x) = exp (x).
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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

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, ez + 2πi = ez or e2πi = 1. The derivative of the exponential function is equal to the function itself: (ez)ʹ = ez.

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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
It can be observed that the structure of solution (3.13) to (3.12) mirrors that of ordinary linear differential equations with constant coefficients, where exponential functions are replaced with Mittag-Leffler functions.
After obtaining the [e.sup.u] exponential function, the TSTF is calculated by applying it to the expression given in Eq.
If the inner solution is an exponential function, then the proposed schemes give the exact solution since the formulae used for the first and second derivative are exact for exponential functions.
Akter and Akbar [44] utilized the modified simple equation method to obtain exact solutions of (11) which were written as fractions of exponential functions. They can be transformed into the tanh and coth functions for which the arbitrary constants are selected appropriately as shown in their paper.
Caption: FIGURE 7: Prediction model for the settlement of the road shoulder: (a) hyperbolic function; (b) logarithmic function; (c) power function; (d) exponential function.
To solve (1), we need to calculate an exponential function about tensor A.
The reason is that we hypothesize the value function of principal as an exponential function under private information.
In an additional analysis, it was investigated if the fit with an exponential function related to the Titius-Bode law [12] in the form f(n) = [alpha] + [beta] [2.sup.n] was better or worse at describing the data than the exponential function of type f (n) = [alpha] [exp.sup.[beta]n] (with [alpha] and [BETA] free parameters), as also used by Naficy et al.
This latency needs to be removed before fitting the exponential function to the experimental data.
In Section 3.1, for concrete with compressive strength ranging from 15 Mpa to 35 Mpa, the recommended exponential function by this study [f.sub.ts] = 1.02 [f.sub.cu.sup.0.36] is applied to describe the compressive-tensile strength correlation within acceptable error, regardless of the specific deteriorated period among several decades.
Based on the characteristics of the exponential function, the variation of the echo intensity is not determined solely by m.
Besides, the MTNC, with good real-time capability and low computational complexity, does not contain the exponential function, so it is easy to achieve real-time control.

Full browser ?