Exponential function: Difference between revisions

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The '''exponential function''' is one of the most important [[function|functions]] in [[mathematics]]. It is written as exp(''x'') or <math>e^x</math> (where <math>e</math> is the [[e - base of natural logarithm|base of the natural logarithm]]) and can be defined in two equivalent ways, the first an [[infinite series]], the second a [[limit]]:

: <math>\exp(x) = \sum_{n = 0}^{\infty} {x^n \over n!}</math>

: <math>\exp(x) = \lim_{n \to \infty} \left( 1 + {x \over n} \right)^n</math>

(see [[limit]] and [[infinite series]]). Here <math>n!</math> stands for the [[factorial]] of <math>n</math> and <math>x</math> can be any [[real number|real]] or [[complex number|complex]] number, or even any element of a [[Banach algebra]] or the field of [[p-adic numbers|<i>p</i>-adic numbers]].

If ''x'' is real, then exp(''x'') is positive and strictly increasing. Therefore its [[inverse function]], the [[natural logarithm]] ln(''x''), is defined for all positive ''x''. Using the natural logarithm, one can define more general exponential functions as follows: