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How do i interprete $f(\bar z)$?

Is there any relationship between $f(z)$ and $f(\bar z)$?

For example, if $f(z)$ is analytic on domain $D$, then under what conditions are $f(\bar z)$ analytic?

Little Rookie
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    For your example, $f(\bar z)$ is analytic if it is constant. – Arthur Sep 12 '17 at 03:58
  • By Cauchy Riemann it is necessary that $\partial f(\bar z)/\partial {\bar z}=0$ if it is analytical. – Vim Sep 12 '17 at 04:07
  • is $z\cdot \bar z$ considered a function of $\bar z$? – Little Rookie Sep 12 '17 at 04:31
  • @LittleRookie All complex functions can be thought of as functions of $z$ and $\bar z$ (just like they can be thought of as functions of $x=\Re(z)$ and $y=\Im(z)$... think of it as a change of coordinates). Holomorphic functions have trivial dependence on $\bar z.$ So your example $|z|^2 = z \bar z$ is not holomorphic. See also https://en.wikipedia.org/wiki/Wirtinger_derivatives – spaceisdarkgreen Sep 12 '17 at 04:34
  • @LittleRookie People are sometimes uncomfortable at this cause it seems like $z$ and $\bar z$ aren't independent. But consider that you can't reconstruct $x$ and $y$ in an algebraic way with just $z$ alone.. you need $\bar z$ too, e.g. $x = (z+\bar z)/2$ – spaceisdarkgreen Sep 12 '17 at 04:43
  • $\overline{f(\overline z)}$ is analytic when $f$ is. – Angina Seng Sep 12 '17 at 06:48

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The notation $f(\bar{z})$ stands for

the image of the complex conjugate of $z$ under the function $f$

You might also see this notated as “$f(z^*)$” by physicists or engineers.

In general, complex conjugation is not analytic/holomorphic (synonyms) except in special cases as indicated in the comments.

gen-ℤ ready to perish
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