What are the Cauchy-Riemann equations? - Complex Analysis

We start with the definition of the derivative in complex analysis, and by looking at the real and imaginary parts separately, we deduce the Cauchy-Riemann equations.

[dantemlima]

Your excitement with this beatiful piece of math is contagious! Thank you.

[dansantner]

Thanks for explaining this. I've been self studying Complex Analysis using Zill and Shanahan and tripped on this section. Your explanation makes it very simple to understand.

[stevengubkin]

Hey Jim! I am teaching complex analysis this semester and I am using your phase plotter, so I linked them your video.

I think you already know my perspective on this stuff, but I will comment here regardless.

A function f: C --> C has a total derivative which is a real-linear function Df(p): C --> C. When I write that derivative as a matrix with respect to the basis {1, i} we get the regular Jacobian matrix. Saying that a function is complex differentiable is equivalent to saying that the map Df(p) is complex linear, not just real linear. This happens if and only if the Cauchy-Riemann equations hold. Also note that a complex linear map from C --> C is just multiplication by a fixed complex number. This justifies just writing f'(p) as a complex number instead of a linear map.

Even cooler: the space Lin(C, C) of real-linear maps is a 4 dimensional real vector space (4 real entries in a 2x2 matrix) but it is also (naturally!) a 2 dimensional complex vector space (you can obviously scale a map C --> C by a complex scalar). A natural basis to choose is the identity map and the complex conjugation map (both are real linear). Expressing a real-linear map L:C --> C in this basis breaks it into a complex-linear and a complex-antilinear part. When you apply this decomposition to Df(p) you get (df/dz) dz + (df/zbar) dzbar. The Cauchy-Riemann equations are then equivalent to Df(p) being complex linear, which means the dzbar term vanishing.

[bungeruwu]

Thank you! this video helped me so much<3 Great explanation, animations and speed:)

[DissidentReport]

This was a wonderful explanation, thank you so much! I just started complex analysis and had a hard time understanding this from the textbook.

[Douglas.Kennedy]

Is there some intuitive reason for the negative sign in one equation, but not the other? Maybe I should try to think up a simple example to show myself it is true... Great video :)

[alejandrogillo]

Thank you for this information highly appreciated the effort of explaining the topic 💛

[anjipati]

Put a little English subtitles sir it will help us to understand easily
its my humble request sir

[dmeech]

beautiful explanation, gave me some extra clarify in my complex variables class

[baarrijaama]

best lecture, thanks, where find a note book

[hakeemnaa]

the conclusion
the change in i output when you wiggle i input
= the change in real output when you wiggle real input
//
the change in i output when you wiggle real input
= negative(-) the change in real output when you wiggle i input


still we need more deep explanation
better than other videos

[mohamedelaminenehar]

We want complex calculus course 😢
Help please

[abinashmuduli]

What is the use of this lesson in the day to day life sir!

[pauselab]

this is amazing, need to learn complex analysis to understand a proof in my book using liouville's theorem.