# 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).
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 It might be outdated or ideologically biased.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

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.

References in periodicals archive ?
"Bringing the summit to Dubai reflects Diamandis' passion and mission to help entrepreneurs and leaders in the region understand the impact of exponential technologies, and to give the region's leaders a hands-on experience of the rate of change that is accelerating at an accelerating rate," said Aman Merchant, chief disruptor at d'labs, A360's exclusive strategic partner in the region.
Applied on HDR images, certain combinations of logarithmic and exponential functions determine a compression of the dynamic range.
Pereira, "Using the matrix pencil method to estimate the parameters of a sum of complex exponentials," IEEE Antennas and Propagation Magazine, vol.
The most known operational identities that involve the exponential of the first derivative are formulas for the shift and dilation operators
which coincides with the class [E.sup.{0}] (A) of entire vectors of A of exponential type [18], is dense in X as well.
The Jacobian logarithm computes the logarithm of a sum of two exponentials [mathematical expression not reproducible] using the max (*) operator and adding a correction term; that is,
In the special case where the spectra of the diagonal blocks ([G.sub.n,n]) are separated, Parlett's method [13] yields--in principle--an efficient computational scheme: compute the exponentials [F.sub.0,0] := exp([G.sub.0,0]) and [F.sub.1,1] := exp([G.sub.1,1]) separately, then the missing (1,2) block of exp([G.sub.1]) is given as the unique solution X to the Sylvester equation
We investigated a novel approach to motion detection based on energy exponentials. Based on presented research, energy exponentials can be concluded not to produce better results than pure energy algorithms.
Cherednikob and Van der Veken offer a systematic pedagogical introduction to the quantum field theory approach to the quantitative analysis of Wilson path-ordered exponentials in quantum chromodynamics and the applications of this formalism to the study of gauge-invariant quark and gluon correlation functions, which can be associated with the three-dimensional transverse momentum-dependent parton density functions, commonly known now as TMD pdfs or simple TMDs.
The strength of discrete logarithm is because of the fact that the forward process of calculation of exponentials modulo prime is easier even for very larger primes using fast modular exponentiation.
Sedletski, "Bases of exponentials, sines, and cosines in weighted spaces on a finite interval," Doklady Akademii Nauk, vol.
As usual for this sort of noncommutative spacetime, in order to have an explicit description of the relativistic properties one must adopt an ordering convention for the basis of exponentials. We find that it is convenient to adopt time-to-the-right ordering, so that elements of the basis of exponentials are written as follows:

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