Editing Exponential function (section)

==Definitions and fundamental properties==
{{see also|Characterizations of the exponential function}}
There are several equivalent definitions of the exponential function, although of very different nature.

===Differential equation===
[[Image:Exp tangent.svg|thumb|right |The derivative of the exponential function is equal to the value of the function. Since the derivative is the [[slope]] of the tangent, this implies that all green [[right triangle]]s have a base length of 1.]]

One of the simplest definitions is: The ''exponential function'' is the ''unique'' [[differentiable function]] that equals its [[derivative]], and takes the value {{math|1}} for the value {{math|0}} of its variable.

This "conceptual" definition requires a uniqueness proof and an existence proof, but it allows an easy derivation of the main properties of the exponential function.

''Uniqueness: ''If {{tmath|f(x)}} and {{tmath|g(x)}} are two functions satisfying the above definition, then the derivative of {{tmath|f/g}} is zero everywhere because of the [[quotient rule]]. It follows that {{tmath|f/g}} is constant; this constant is {{math|1}} since {{tmath|1=f(0) = g(0)=1}}.

''Existence'' is proved in each of the two following sections.

===Inverse of natural logarithm===
''The exponential function is the [[inverse function]] of the [[natural logarithm]].'' The [[inverse function theorem]] implies that the natural logarithm has an inverse function,  that satisfies the above definition. This is a first proof of existence. Therefore, one has 
:<math>\begin{align}
\ln (\exp x)&=x\\
\exp(\ln y)&=y
\end{align}</math>
for every [[real number]] <math>x</math> and every positive real number <math>y.</math>

===Power series===

''The exponential function is the sum of the [[power series]]''<ref name="Rudin_1987"/><ref name=":0">{{Cite web|last=Weisstein| first=Eric W.|title=Exponential Function|url=https://mathworld.wolfram.com/ExponentialFunction.html|access-date=2020-08-28| website=mathworld.wolfram.com|language=en}}</ref> 
<math display=block>
\begin{align}\exp(x) &= 1+x+\frac{x^2}{2!}+ \frac{x^3}{3!}+\cdots\\
&=\sum_{n=0}^\infty \frac{x^n}{n!},\end{align}</math>
[[Image:Exp series.gif|right|thumb|The exponential function (in blue), and the sum of the first {{math|''n'' + 1}} terms of its power series (in red)]]
where <math>n!</math> is the [[factorial]] of {{mvar|n}} (the product of the {{mvar|n}} first positive integers). This series is [[absolutely convergent]] for every <math>x</math> per the [[ratio test]]. So, the derivative of the sum can be computed by term-by-term differentiation, and this shows that the sum of the series satisfies the above definition. This is a second existence proof, and shows, as a byproduct, that the exponential function is defined for every {{tmath|x}}, and is everywhere the sum of its [[Maclaurin series]].

===Functional equation===
''The exponential satisfies the functional equation:''
<math display=block>\exp(x+y)= \exp(x)\cdot \exp(y).</math>
This results from the uniqueness and the fact that the function
<math> f(x)=\exp(x+y)/\exp(y)</math> satisfies the above definition. 

It can be proved that a function that satisfies this functional equation has the form {{tmath|x \mapsto \exp(cx)}} if it is either [[continuous function|continuous]] or [[monotonic function|monotonic]]. It is thus [[differentiable function|differentiable]], and equals the exponential function if its derivative at {{math|0}} is {{math|1}}.

===Limit of integer powers===
''The exponential function is the [[limit (mathematics)|limit]], as the integer {{mvar|n}} goes to infinity,<ref name="Maor"/><ref name=":0" />
<math display=block>\exp(x)=\lim_{n \to +\infty} \left(1+\frac xn\right)^n.</math>
By continuity of the logarithm, this can be proved by taking logarithms and proving 
<math display=block>x=\lim_{n\to\infty}\ln \left(1+\frac xn\right)^n= \lim_{n\to\infty}n\ln \left(1+\frac xn\right),</math>
for example with [[Taylor's theorem]].

===Properties===
''[[multiplicative inverse|Reciprocal]]:'' The functional equation implies {{tmath|1=e^x e^{-x}=1}}. Therefore {{tmath|e^x \ne 0}} for every {{tmath|x}} and 
<math display=block>\frac 1{e^x}=e^{-x}.</math>

''Positiveness:'' {{tmath|e^x>0}} for every real number {{tmath|x}}. This results from the [[intermediate value theorem]], since {{tmath|1=e^0=1}} and, if one would have {{tmath|e^x<0}} for some {{tmath|x}}, there would be an {{tmath|y}} such that {{tmath|1=e^y=0}} between {{tmath|0}} and {{tmath|x}}. Since the exponential function equals its derivative, this implies that the exponential function is [[monotonically increasing]].

''Extension of [[exponentiation]] to positive real bases:'' Let {{mvar|b}} be a positive real number. The exponential function  and the natural logarithm being the inverse each of the other, one has <math>b=\exp(\ln b).</math> If {{mvar|n}} is an integer, the functional equation of the logarithm implies 
<math display=block>b^n=\exp(\ln b^n)= \exp(n\ln b).</math>
Since the right-most expression is defined if {{mvar|n}} is any real number, this allows defining {{tmath|b^x}} for every positive real number {{mvar|b}} and every real number {{mvar|x}}:
<math display=block>b^x=\exp(x\ln b).</math>
In particular, if {{mvar|b}} is the [[Euler's number]] <math>e=\exp(1),</math> one has <math>\ln e=1</math> (inverse function) and thus <math display=block>e^x=\exp(x).</math> This shows the equivalence of the two notations for the exponential function.