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.
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.
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.
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 :)
Great question. Yes, there's some geometry to do which yields some intuition for this, which basically boils down to the real input and imaginary output being separated by 90 degrees (just like the imaginary input and real output!), but the "sense" is different: in one case, these are clockwise, and in the other, counterclockwise.
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
