Question on derivative of conjugate

[A. Bahat]

While on the topic of dz and dz-bar, it's nice to note that using differential forms in complex analysis brings out the deep connections with multivariable calculus. (After all, C is just R^2 with multiplication.)

First, note that if f is holomorphic in some region, then f dz is closed in that region. Upon applying Stokes' theorem, Cauchy's integral theorem follows as a trivial corollary. (The integral of f dz over the boundary equals the integral of d(f dz)=0 over the interior.) Once this connection is established, it gives the theorem a wonderful physical interpretation: a conservative force does no work over a closed path.

If f is instead meromorphic in a region, then the residue theorem follows similarly: this time the integral of f dz over a loop gives the sum of the integrals of small loops around the poles, which give the residues (up to winding numbers and a multiplicative constant).

 

[mathwonk]

some of the confusion is the distinction between ∂/∂z and d/dz. As defined in usual complex analysis courses, df/dz does not exist unless f is holomorphic, but ∂f/∂z exists for every smooth f.

Hence if f(z) = zbar, then df/dz does not exist, but ∂f/∂z = 0.

 

[A. Bahat]

Moreover, now the Cauchy-Riemann equations take the suggestive form ∂f/∂z-bar=0, i.e. f is only a function of z.

 

[mathwonk]

this is what i tried to say at the bottom of post 29.