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What does it mean to take a complex derivative? (visually explained)

The complex derivative, from differentials to the Cauchy-Riemann Equations Support me on Patreon!   / vcubingx   Sign up for Brilliant (sponsored link)! https://brilliant.org/vcubingx The source code for the animations can be found here: https://github.com/vivek3141/videos A huge thanks to ‪@3blue1brown‬ , ‪@Aleph0‬ , ‪@alfcnz‬ , Sumedh Shenoy, Nikhil Maserang and Oliver Ni for helping me review the video! If you're interested in learning why conformal doesn't necessarily mean differentiable, read Chapter 4.VI "Conformal = Analytic" of Tristan Needham's "Visual Complex Analysis", which you can find here: http://usf.usfca.edu/vca/ This book is incredible, some of the ideas in this video were taken from this book. You can find all my sources here: https://docs.google.com/document/d/13... Some extra info: At 20:47, I mention that a function is holomorphic if it satisfies the cauchy-riemann equations. This is not always true, the partial derivatives have to be continuous as well. For example, f(z) = {0 if z=0, z^5/|z^4| if z!=0} satisfies the cauchy-riemann equations but is not differentiable at z=0. Thanks to Ge for pointing this out! Mistakes: 3:15: The upper number line should still be labeled as "x" instead of "x^2" 14:56 It should be nz^(n-1) These animation in this video was made using 3blue1brown's library, manim: https://github.com/3b1b/manim Chapters: 0:00 Intro 1:53 The Real Derivative, Revisited 4:31 Differential View 9:17 Transformation View 13:02 Conformality 16:01 Cauchy-Riemann Equations 22:15 Brilliant Ad, Stereographic Projection 23:57 Outro, deriv of e^z Follow me! Website: https://vcubingx.com Twitter:   / vcubingx   Github: https://github.com/vivek3141 Instagram:   / vcubingx   Patreon:   / vcubingx   Music (in order): Knowmadic - Faces https://chll.to/892bc12e Philanthrope, mommy - embrace https://chll.to/7e941f72 GameChops - Route 113 Knowmadic - Faces https://chll.to/892bc12e Philanthrope, Idealism - Still https://chll.to/a110849c GameChops - Azalea Town What does it mean to take a complex derivative? (visually explained) Some tags: complex analysis, cauchy-riemann, imaginary, derivative, vcubingx