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Undid revision 1295921430 by 2600:1700:A410:C4E0:A48F:F30C:4B6E:7354 (talk) There are ways to make that true, but there are also ways to make that false. Unless we add some verbiage to distinguish the cases, we should stay away from that quagmire |
m →Complex exponential: use z instead of x |
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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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<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
<math display=
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>
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^{
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
<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:
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