Editing Function (mathematics) (section)

=== Using differential calculus ===
Many functions can be defined as the [[antiderivative]] of another function. This is the case of the [[natural logarithm]], which is the antiderivative of {{math|1/''x''}} that is 0 for {{math|1=''x'' = 1}}. Another common example is the [[error function]].

More generally, many functions, including most [[special function]]s, can be defined as solutions of [[differential equation]]s. The simplest example is probably the [[exponential function]], which can be defined as the unique function that is equal to its derivative and takes the value 1 for {{math|1=''x'' = 0}}.

[[Power series]] can be used to define functions on the domain in which they converge. For example, the [[exponential function]] is given by <math display="inline">e^x = \sum_{n=0}^{\infty} {x^n \over n!}</math>. However, as the coefficients of a series are quite arbitrary, a function that is the sum of a convergent series is generally defined otherwise, and the sequence of the coefficients is the result of some computation based on another definition. Then, the power series can be used to enlarge the domain of the function. Typically, if a function for a real variable is the sum of its [[Taylor series]] in some interval, this power series allows immediately enlarging the domain to a subset of the [[complex number]]s, the [[disc of convergence]] of the series. Then [[analytic continuation]] allows enlarging further the domain for including almost the whole [[complex plane]]. This process is the method that is generally used for defining the [[logarithm]], the [[exponential function|exponential]] and the [[trigonometric functions]] of a complex number.