Exponential function: Difference between revisions

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In [[mathematics]], the '''exponential function''' is the unique [[real function]] which maps [[0|zero]] to [[1|one]] and has a [[derivative (mathematics)|derivative]] everywhere equal to its value. The exponential of a variable {{tmath|x}} is denoted {{tmath|\exp x}} or {{tmath|e^x}}, with the two notations used interchangeably. It is called ''exponential'' because its argument can be seen as an [[exponent (mathematics)|exponent]] to which a constant [[e (mathematical constant)|number {{math|''e'' ≈ 2.718}}]], the base, is raised. There are several other definitions of the exponential function, which are all equivalent although being of very different nature; a common and useful alternative definition defines it as the sum function of a [[Power series#Geometric series, exponential function and sine|power series]].

The exponential function converts sums to products: it maps the [[additive identity]] {{math|0}} to the [[multiplicative identity]] {{math|1}}, and the exponential of a sum is equal to the product of separate exponentials, {{tmath|1=\exp(x + y) = \exp x \cdot \exp y }}. Its [[inverse function]], the [[natural logarithm]], {{tmath|\ln}} or {{tmath|\log}}, converts products to sums: {{tmath|1= \ln(x\cdot y) = \ln x + \ln y}}.

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===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

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The two first characterizations are equivalent, since, if {{tmath|1=b=e^k}} and {{tmath|1= k=\ln b}}, one has

s.<math display=block>e^{kx}= (e^k)^x= b^x.</math>

The basic properties of the exponential function (derivative and functional equation) implies immediately the third and thsthe last condititoncondition.

Suppose that the third condition is verified, and let {{tmath|k}} be the constant value of <math>f'(x)/f(x).</math> Since <math display = inline>\frac {\partial e^{kx}}{\partial x}=ke^{kx},</math> the [[quotient rule]] for derivation

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<math display="block">e^z = \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n</math>

As with the real exponential function (see {{slink||Functional equation}} above), the complex exponential satisfies the functional equation

The functional equation

<math display="block">e^{w+\exp(z}+w)=e^we^ \exp(z)\cdot \exp(w).</math>

Among complex functions, it is the unique solution which is [[holomorphic]] at the point {{tmath|1= z = 0}} and takes the derivative {{tmath|1}} there.<ref>{{cite book |last=Hille |first=Einar |year=1959 |title=Analytic Function Theory |volume=1 |place=Waltham, MA |publisher=Blaisdell |chapter=The exponential function |at=§ 6.1, {{pgs|138–143}} }}</ref>

holds for every complex numbers {{tmath|w}} and {{tmath|z}}. The complex exponential is the unique [[continuous function]] that satisfies this functional equation and has the value {{tmath|1}} for {{tmath|1=z=0}}.

The [[complex logarithm]] is a [[left inverse function|right-inverse function ]] of the complex exponential:

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<math display="block">\overline{e^z}=e^{\overline z}.</math>

Its modulus is

<math display="block">|e^z|= e^{|\Re (z)|},</math>

where {{tmath|\Re(z)}} denotes the real part of {{tmath|z}}.

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<math display="block">e^{it} =\cos(t)+i\sin(t). </math>

This formula provides the decomposition of complex exponentialexponentials into [[real and imaginary parts]]:

<math display="block">e^{x+iy} = e^{x}e^{iy} = e^x\,\cos y + i e^x\,\sin y.</math>

The trigonometric functions can be expressed in terms of complex exponentials: