For $w=f(z)$, we have two differential operators $\partial_z$ and $\partial_\bar{z}$, and we consider $z$ and $\bar{z}$ as "independent" variable for differentiation to get Cauchy-Riemann equation.

I'm confused in understanding this. It seems that we shall write $(w,\bar{w})=f(z,\bar{z})$. But for complex case it is of redundance. Why can't we just think complex numbers "intrinsically" of just 1-dimension, although ususally their real-imaginary expressions are easier to handle.

p.s. I sense that the problem is how to treat conjugate operation "intrinsically"...

Any help will be appreciated.

  • 1
    $\begingroup$ $\partial_{\bar{z}}$ is just a notation, standing for $(\partial_x + i \partial_y)/2$. What makes complex numbers special as $2$-dimensional real vectors is their multiplicative structure. Whether you treat it as $1$- or $2$-dimensional is not that important. $\endgroup$ – Qiyu Wen Oct 3 '16 at 6:27
  • $\begingroup$ Forget about $\partial_z$ and show instead $f(z) = \sum_{k=0}^K c_k z^k$ is complex differentiable, that is $f(z) = f(z_0) + f'(z_0)(z-z_0) + o(|z-z_0|)$, while $g(z) = \sum_{k=0}^K c_k \overline{z}^k$ is not (but its complex conjugate is) $\endgroup$ – reuns Oct 3 '16 at 7:03

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