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I came across this question whilst doing some research into complex analysis, and I just can't see what to do!

Let $f(z)$ be a holomorphic function on $\mathbb{C}$. Show that $\overline{f(\overline{z})}$ is holomorphic, whilst $f(\overline{z})$ is holomorphic if and only if $f(z)$ is constant.

I know that holomorphic means that the function is differentiable everywhere, and I need to apply the Cauchy-Riemann equations somehow, but I'm not sure how to approach this.

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    $\begingroup$ That is not true -- $f(z)=z$ is holomorphic, but $\overline{f(z)}$ is not. $\endgroup$ – Henning Makholm Nov 4 '13 at 11:50
  • $\begingroup$ sorry it should say the conjugate of f acting on the conjugate of z. could you help with that? $\endgroup$ – Tom Nov 4 '13 at 11:52
  • $\begingroup$ @Tom is that better? $\endgroup$ – Tom Oldfield Nov 4 '13 at 11:54
  • $\begingroup$ yes, thanks! any ideas on a solution? $\endgroup$ – Tom Nov 4 '13 at 11:55
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Since $f$ is entire, it has a Taylor expansion (centered at $z_0\in \mathbb{C}$) $$f(z)=\sum_{n\ge 0}a_n(z-z_0)^n$$ Hence, $$\overline{ f(\bar z)}=\overline{\sum_{n\ge 0}a_n(\overline{z}-z_0)^n}=\sum_{n\ge 0}\overline{a_n}(z-\overline{z_0})^n$$ and we know this series converges since you can show it has the same radius of convergence as the original series. Therefore, $\overline{ f(\bar z)}$ is holomorphic.

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So you want to show that if $f(z)$ is holomorphic, then $\overline{f(\bar z)}$ is holomorphic too.

I think it will be easiest not to split into real and imaginary parts -- so no Cauchy-Riemann -- but instead work directly from the definition of differentiability.

A natural guess would be that $\frac{d}{dz} \overline{f(\bar z)}$ would be $\overline{f'(\bar z)}$. Can you show that this is in fact the case?


For the second part, perhaps show that if $g(z)$ and $\overline{g(z)}$ are both holomorphic, then $g$ is constant. (Here, using Cauchy-Riemann feels more promising).

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  • $\begingroup$ thanks, I've done the first part, but I don't see how to do the second part, or why it shows what I want to get $\endgroup$ – Tom Nov 4 '13 at 12:03
  • $\begingroup$ @Tom: If $\overline{f(\overline{z})}$ and $f(\overline{z})$ are analytic, then $f(\overline{z})$ is constant. Cauchy-Riemann equations can help you see that. $\endgroup$ – Francis Nov 4 '13 at 12:10
  • $\begingroup$ We can use the fact that $f$ is entire to represent it by its Taylor expansion. $\endgroup$ – The Substitute Feb 19 '15 at 4:06

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