Curl and divergence of the conjugate of an holomorphic function

  • #1
I noted that if [itex]f : C \to C[\itex] is holomorphic in a subset [itex]D \in C[\itex], then [itex]\nabla \by \hat{f} = 0, \nabla \dot \hat{f} = 0[\itex]. Moreover, those two expressions are equivalent to the Cauchy-Riemann equations.

I'm rewriting this in plaintext, in case latex doesn't render properly. I'm not sure if I'm using it correctly.

If f : C -> C is holomorphic in a subset D in C, then div conj(f) = 0, and rot conj(f) = 0, where conj(f) is the complex conjugate of f. These expressions should be though of formally, admitting that f(x+iy) = u(x,y) + iv(x,y) "=" (u(x, y), v(x, y)). Note that this is exactly the same as saying that the Cauchy-Riemann equations are satisfied.

So does this show up somewhere? Is it an important consequence? Does it have any physical interpretation?
 

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