Editing Exponential function (section)

===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.