Conformal Mapping in Complex Variables

Introduction to #ConformalMapping with the definition and conformal mapping theorem (i.e. a conformal map is a function that preserves angles when it is used to transform one complex variable to another). I use the fact that a conformal map has preservation of angles to derive the requisite condition for a complex function f(z) to be a conformal map: that is to say, f(z) must be analytic and have a non-zero derivative in the area of interest.

Questions/requests? Let me know in the comments!

Special thanks to my Patrons:
Cesar Garza
Daigo Saito
Alvin Barnabas
Patapom
Yenyo Pal
Lisa Bouchard
Eugene Bulkin
Rene Gastelumendi
Borgeth
Jose Antonio Sanchez-Migallon

This video was sponsored by Skillshare.

[slog]

These videos have come about just in time for my undergrad dissertation. Big thanks! :)

[mamig]

Wow I just finished your Complex functions playlist. Thank you, your videos are incredibly good and you have a superb teaching style.

[grantsmith]

This video was awesome. Us members of the math community are lucky to have you :)

[marcovillalobos]

Great video, looking forward future videos using this mappings to solve the laplace equation!!!

[ronanm]

Brilliant stuff! Very clear and concise

[charlesgantz]

Great video. I've never really understood how preserving angles leads to something as important as Conformal Mapping.
It would also be really nice if you could do a video or videos on the application of Conformal mapping. Something like deriving fringe effects of capacitors, or electric fields at plate boundaries. Conformal Mapping is one of the few parts of Complex Theory that has intuitive, practical applications.

[mrunmayeemhatre]

Thank you so much for explaining so nicely!

[eulefranz]

Nice, tbh I never understood why conformal (field theories) are THE buzz, because I never got truely what conformality is. Thanks as always:)

[arielfuxman]

11:48 Why is it not enough to require differentiability at z_0? Why do you need it to be analytic, i.e differentiable at z_0 and in a neighbourhood of z_0?

[Fjorge]

Good explainer! do you acelerate your recordings or do you record your screen and your voice separately?

[Czeckie]

12:40 gotta say I'm not convinced. We proved that analytic function with nonzero derivative is conformal, but what's written down is the converse. And frankly for general angle conforming maps we don't even know if it needs to be differentiable. It might be true but this proof can't work, right?

[giuliocasa]

If the derivative is zero there could be only a constant shift between the two angles. But if the function is analytic (and the derivative of the two curves is always defined and continuous) the constant must be zero on a single point or even on a interval included in the domain. So the only condition IMO is that the function is analytic and the derivative is not *always* zero. Is my reasoning correct?

[SuperDeadparrot]

What’s the difference between a mapping and a transformation?