Editing Exponential function (section)

==Complex exponential==
{{anchor|On the complex plane|Complex plane}}
[[File:The exponential function e^z plotted in the complex plane from -2-2i to 2+2i.svg|alt=The exponential function {{math|''e''{{isup|''z''}}}} plotted in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}|thumb|The exponential function {{math|''e''{{isup|''z''}}}} plotted in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}]]
[[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'''.  

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 ''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{and}\quad e^0=1.</math>

The ''complex exponential function'' is the sum of the [[series (mathematics)|series]]
<math display="block">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">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>
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:
<math display="block">e^{\log z} =z. </math>
However, since the complex logarithm is a [[multivalued function]], one has
<math display="block">\log e^z= \{z+2ik\pi\mid k\in \Z\},</math>
and it is difficult to define the complex exponential from the complex logarithm. On the opposite, this is the complex logarithm that is often defined from the complex exponential.

The complex exponential has the following properties:
<math display="block">\frac 1{e^z}=e^{-z} </math>
and
<math display="block">e^z\neq 0\quad \text{for every } z\in \C .</math>
It is [[periodic function|periodic function]] of period {{tmath|2i\pi}}; that is 
<math display="block">e^{z+2ik\pi} =e^z \quad \text{for every } k\in \Z.</math>
This results from [[Euler's identity]] {{tmath|1=e^{i\pi}=-1}} and the functional identity.

The [[complex conjugate]] of the complex exponential is 
<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}}.

===Relationship with trigonometry===
Complex exponential and [[trigonometric function]]s are strongly related by [[Euler's formula]]:
<math display="block">e^{it} =\cos(t)+i\sin(t). </math>

This formula provides the decomposition of complex exponential into [[real and imaginary parts]]:
<math display="block">e^{x+iy} = e^x\,\cos y + i e^x\,\sin y.</math>

The trigonometric functions can be expressed in terms of complex exponentials:
<math display="block">\begin{align}
\cos x &= \frac{e^{ix}+e^{-ix}}2\\
\sin x &= \frac{e^{ix}-e^{-ix}}{2i}\\
\tan x &= i\,\frac{1-e^{2ix}}{1+e^{2ix}}
\end{align}</math>

In these formulas, {{tmath|x, y, t}} are commonly interpreted as real variables, but the formulas remain valid if the variables are interpreted as complex variables. These formulas may be used to define trigonometric functions of a complex variable.<ref name="Apostol_1974"/>

===Plots===

<gallery caption="3D plots of real part, imaginary part, and modulus of the exponential function" class="center" mode="packed" style="text-align:left" heights="150px">
 Image:ExponentialAbs_real_SVG.svg| {{math|1=''z'' = Re(''e''{{isup|''x'' + ''iy''}})}}
 Image:ExponentialAbs_image_SVG.svg| {{math|1=''z'' = Im(''e''{{isup|''x'' + ''iy''}})}}
 Image:ExponentialAbs_SVG.svg| {{math|1=''z'' = {{abs|''e''{{isup|''x'' + ''iy''}}}}}}
</gallery>

Considering the complex exponential function as a function involving four real variables:
<math display="block">v + i w = \exp(x + i y)</math>
the graph of the exponential function is a two-dimensional surface curving through four dimensions.

Starting with a color-coded portion of the <math>xy</math> domain, the following are depictions of the graph as variously projected into two or three dimensions.

<gallery class="center" mode="packed" style="text-align:left" heights="200px" caption="Graphs of the complex exponential function">
File: Complex exponential function graph domain xy dimensions.svg|Checker board key:<br> <math>x> 0:\; \text{green}</math><br> <math>x< 0:\; \text{red}</math><br><math>y> 0:\; \text{yellow}</math><br><math>y< 0:\; \text{blue}</math>
File: Complex exponential function graph range vw dimensions.svg|Projection onto the range complex plane (V/W). Compare to the next, perspective picture.
File: Complex exponential function graph horn shape xvw dimensions.jpg|Projection into the <math>x</math>, <math>v</math>, and <math>w</math> dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image)|alt=Projection into the                 <nowiki> </nowiki>       x             <nowiki> </nowiki>   {\displaystyle x}    ,                 <nowiki> </nowiki>       v             <nowiki> </nowiki>   {\displaystyle v}    , and                 <nowiki> </nowiki>       w             <nowiki> </nowiki>   {\displaystyle w}    <nowiki> </nowiki>dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image).
File: Complex exponential function graph spiral shape yvw dimensions.jpg|Projection into the <math>y</math>, <math>v</math>, and <math>w</math> dimensions, producing a spiral shape (<math>y</math> range extended to ±2{{pi}}, again as  2-D perspective image)|alt=Projection into the                 <nowiki> </nowiki>       y             <nowiki> </nowiki>   {\displaystyle y}    ,                 <nowiki> </nowiki>       v             <nowiki> </nowiki>   {\displaystyle v}    , and                 <nowiki> </nowiki>       w             <nowiki> </nowiki>   {\displaystyle w}    <nowiki> </nowiki>dimensions, producing a spiral shape. (                <nowiki> </nowiki>       y             <nowiki> </nowiki>   {\displaystyle y}    <nowiki> </nowiki>range extended to ±2π, again as  2-D perspective image).
</gallery>

The second image shows how the domain complex plane is mapped into the range complex plane:
* zero is mapped to 1
* the real <math>x</math> axis is mapped to the positive real <math>v</math> axis
* the imaginary <math>y</math> axis is wrapped around the unit circle at a constant angular rate
* values with negative real parts are mapped inside the unit circle
* values with positive real parts are mapped outside of the unit circle
* values with a constant real part are mapped to circles centered at zero
* values with a constant imaginary part are mapped to rays extending from zero

The third and fourth images show how the graph in the second image extends into one of the other two dimensions not shown in the second image.

The third image shows the graph extended along the real <math>x</math> axis.  It shows the graph is a surface of revolution about the <math>x</math> axis of the graph of the real exponential function, producing a horn or funnel shape.

The fourth image shows the graph extended along the imaginary <math>y</math> axis.  It shows that the graph's surface for positive and negative <math>y</math> values doesn't really meet along the negative real <math>v</math> axis, but instead forms a spiral surface about the <math>y</math> axis.  Because its <math>y</math> values have been extended to {{math|±2''π''}}, this image also better depicts the 2π periodicity in the imaginary <math>y</math> value.