Exponential function: Difference between revisions

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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'''.

Most of the definitions of the exponential function can be used verbatim for definiting the complex exponential function, and the proof of their equivalence is the same as in the real case.

The complex exponential function can be defined in several equivalent ways that are the same as in the real case.

The most''complex commonexponential'' definition ofis the unique complex exponential function parallelsthat theequals powerits series[[complex definitionderivative]] for real arguments,and wheretakes the realvalue variable{{tmath|1}} isfor replacedthe by a complexargument one{{tmath|0}}:

<math display="block">\exp frac{de^z := \sum_}{k dz}= 0}e^z\inftyquad\fractext{zamd}\quad e^k}{k!}0=1.</math>

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

<math display="block">\exp(w+e^z) =\exp w\exp z \text sum_{k for all= 0} w,z^\ininfty\mathbbfrac{z^k}{Ck!}.</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">e^z = \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n</math>

The functional equation

Alternatively, the complex exponential function may be defined by modelling the limit definition for real arguments, but with the real variable replaced by a complex one:

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