Exponential function: Difference between revisions

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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 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>

The convergence can be improved:

<math>\exp(x)=\lim_{n \to \infty} \left(1 + \frac{x}{n + \frac{1}{6} \sin\left(\frac{\pi}{n}\right)}\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>

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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: