What is the graph of x^a when a is not an integer? An unusual look at familiar functions

This is my entry into the Second Summer of Math Exposition. I used to wonder about these questions ever since my first year in university, so in a sense this is a rather personal subject for me. I hope that this video reaches someone else who is also wondering about these issues.

Here's the Desmos document for visualizing the n complex roots of any complex number:
https://www.desmos.com/calculator/1v4...

Videos mentioned in order of apparition:

e to the pi i, a nontraditional take (old version)
   • e to the pi i, a nontraditional take ...  

The 'Everything' Formula - Numberphile
   • The 'Everything' Formula - Numberphile  

Winding numbers and domain coloring
   • Winding numbers and domain coloring  

The 5 ways to visualize complex functions | Essence of complex analysis #3
   • The 5 ways to visualize complex funct...  

Visualizing Complex-Valued Functions
   • Visualizing Complex-Valued Functions  

Jeff Tupper's webpage on the graph of x^x
http://www.peda.com/grafeq/gallery/ro...

A relevant Stack Exchange question and answer that touches on these ideas:
https://mathematica.stackexchange.com...

Chapters:
00:00 Intro and problem motivation
01:49 The R-to-C graph of x^a
04:37 The R-to-C graph of a^x
06:37 Rotations in complex exponentiation
07:27 The R-to-C graph of x^x
08:17 An alternative version of the graph of x^x
12:37 Recap of the various graphs of x^x
13:49 Homework
14:09 Summary and outro

Music by Vincent Rubinetti
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
https://open.spotify.com/album/1dVyjw...