Exponential function: Difference between revisions

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More geneally, the solutions of every linear differential equation with constant coefficients can be expressed in terms of exponential functions and, when they are not homogeneous, antiderivatives. This true also for systems of linear differential equations with constant coefficients.

==Complex exponential==

== Complex exponential <span class="anchor" id="On the complex plane"></span><span class="anchor" id="Complex plane"></span> ==

{{see alsoanchor|Euler'sOn formula#Definitions ofthe complex exponentiationplane|Complex plane}}

[[File:The exponential function e^z plotted in the complex plane from -2-2i to 2+2i.svg|alt=The exponential function e^z plotted in the complex plane from -2-2i to 2+2i|thumb|The exponential function e^z plotted in the complex plane from -2-2i to 2+2i]]

[[Image:Exp-complex-cplot.svg|thumb|right|A [[Domain coloring|complex plot]] of <math>z\mapsto\exp z</math>, with the [[Argument (complex analysis)|argument]] <math>\operatorname{Arg}\exp z</math> represented by varying hue. The transition from dark to light colors shows that <math>\left|\exp z\right|</math> is increasing only to the right. The periodic horizontal bands corresponding to the same hue indicate that <math>z\mapsto\exp z</math> is [[periodic function|periodic]] in the [[imaginary part]] of <math>z</math>.]]

The exponential function can be naturally extended to a [[complex function]], which is a function with the [[complex number]]s as [[___domain of a function|___domain]] and [[codomain]], such that its [[restriction (mathematics)|restriction]] to the reals is the above-defined exponential function, called ''real exponential function'' in what follows. This function is also called ''the exponential function'', and also denoted {{tmath|e^z}} or {{tmath|\exp(z)}}. For distinguishing the complex case from the real one, the extended function is also called '''complex exponential function''' or simply '''complex exponential'''.

As in the [[real number|real]] case, the exponential function can be defined on the [[complex plane]] in several equivalent forms.

TheMost mostof commonthe definitiondefinitions of the complex exponential function parallelscan the powerbe seriesused definitionverbatim for realdefiniting argumentsthe complex exponential function, whereand the realproof of their variableequivalence is replacedthe bysame aas complexin the real one:case.

<math display="block">\exp z := \sum_{k = 0}^\infty\frac{z^k}{k!}</math>

Alternatively, theThe complex exponential function maycan be defined by modellingin theseveral limitequivalent definitionways forthat realare arguments,the butsame withas in the real variable replaced by a complex one:case.

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

The ''complex exponential'' is the unique complex function that equals its [[complex derivative]] and takes the value {{tmath|1}} for the argument {{tmath|0}}:

<math display="block">\frac{de^z}{dz}=e^z\quad\text{amd}\quad e^0=1.</math>

The ''complex exponential function'' is the sum of the [[series (mathematics)|series]]

<math display="block">\exp e^z := \sum_{k = 0}^\infty\frac{z^k}{k!}.</math>

This series is [[absolutely convergent]] for every complex number {{tmath|z}}. So, the complex differential is an [[entire function]].

The complex exponential function is the [[limit (mathematics)|limit]]

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

The functional equation

<math display="block">e^{w+z}=e^we^z</math>

holda 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}}.

For the power series definition, term-wise multiplication of two copies of this power series in the [[Cauchy product|Cauchy]] sense, permitted by [[Cauchy product|Mertens' theorem]], shows that the defining multiplicative property of exponential functions continues to hold for all complex arguments:

<math display="block">\exp(w+z)=\exp w\exp z \text { for all } w,z\in\mathbb{C}</math>

The definition of the complex exponential function in turn leads to the appropriate definitions extending the [[trigonometric functions]] to complex arguments.