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For example, from the differential equation definition, e x e −x = 1 when x = 0 and its derivative using the product rule is e x e −x − e x e −x = 0 for all x, so e x e −x = 1 for all x. From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity.
The definition of e x as the exponential function allows defining b x for every positive real numbers b, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm ln(x) is the inverse of the exponential function e x means that one has = () = for every b > 0.
The exponential function e x for real values of x may be defined in a few different equivalent ways (see Characterizations of the exponential function). Several of these methods may be directly extended to give definitions of e z for complex values of z simply by substituting z in place of x and using the complex algebraic operations. In ...
In mathematics, the exponential function can be characterized in many ways. This article presents some common characterizations, discusses why each makes sense, and proves that they are all equivalent. The exponential function occurs naturally in many branches of mathematics. Walter Rudin called it "the most important function in mathematics". [1]
The value of the natural log function for argument e, i.e. ln e, equals 1. The principal motivation for introducing the number e, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms. [28] A general exponential function y = a x has a derivative, given by a limit:
The function e (−1/x 2) is not analytic at x = 0: the Taylor series is identically 0, although the function is not. If f ( x ) is given by a convergent power series in an open disk centred at b in the complex plane (or an interval in the real line), it is said to be analytic in this region.
For any real numbers (scalars) x and y we know that the exponential function satisfies e x+y = e x e y. The same is true for commuting matrices. If matrices X and Y commute (meaning that XY = YX), then, + =. However, for matrices that do not commute the above equality does not necessarily hold.
The field of real numbers R, or (R, +, ·, 0, 1) as it may be written to highlight that we are considering it purely as a field with addition, multiplication, and special constants zero and one, has infinitely many exponential functions. One such function is the usual exponential function, that is E(x) = e x, since we have e x+y = e x e y and e ...