Termotanque
Aug11-11, 01:45 PM
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?
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?