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In mathematics, exponentiation, denoted b n, is an operation involving two numbers: the base, b, and the exponent or power, n. [1] When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b n is the product of multiplying n bases: [1] = ⏟.
The polynomials, exponential function e x, and the trigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are not entire include the square root, the logarithm, the trigonometric function tangent, and its inverse, arctan. For these functions the Taylor series do not converge if x is far from b.
Just as the class P is defined in terms of polynomial running time, the class EXPTIME is the set of all decision problems that have exponential running time. In other words, any problem in EXPTIME is solvable by a deterministic Turing machine in O (2 p ( n ) ) time, where p ( n ) is a polynomial function of n .
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
Often the independent variable is time. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential growth is the inverse of logarithmic growth.
Similar asymptotic analysis is possible for exponential generating functions; with an exponential generating function, it is a n / n! that grows according to these asymptotic formulae. Generally, if the generating function of one sequence minus the generating function of a second sequence has a radius of convergence that is larger than ...
The analytic functions have many fundamental properties. In particular, an analytic function of a real variable extends naturally to a function of a complex variable. It is in this way that the exponential function, the logarithm, the trigonometric functions and their inverses are extended to functions of a complex variable.
Two to the power of n, written as 2 n, is the number of values in which the bits in a binary word of length n can be set, where each bit is either of two values. A word, interpreted as representing an integer in a range starting at zero, referred to as an "unsigned integer", can represent values from 0 (000...000 2) to 2 n − 1 (111...111 2) inclusively.