Show that if $f$ is analytic then $g(z)=\overline{f(\overline{z})}$ is analytic and find its derivative

Question

Let $f:\mathbb{C}\to\mathbb{C}$ be an analytic function. Show that $g(z)=\overline{f(\overline{z})}$ is analytic and find its derivative.

$f$ is analytic then C-R equations hold. Suppose $f=u+iv$.

Then $g(z)=u(x,-y)+i(-v(x,-y))$. We can observe that C-R equations hold for $g$ as well since if we set $\hat{u}(x,y):=u(x,-y)$ and $\hat{v}(x,y)=-v(x,-y)$ we get $$ \hat{u}_x=u_x=-v_y=-\hat{v}_y \\-\hat{u}_y=u_y=v_x=-v_x $$ But this is not enough for to show that $g$ is analytic.

easier to directly compute the derivative of $\bar f(\bar z)$ — trisct, Jul 08 '19 at 09:10
Another one: Let $f(z)$ be a holomorphic function on C. Show that $\overline{f(\bar{z})}$ is holomorphic on C — Martin R, Jul 08 '19 at 09:13
Why it's not enough to show that $g$ is analytic? It is enough — Jakobian, Jul 08 '19 at 09:22
Because if $g$ is analytic then C-R equations hold but the other direction is not true, isn't it? @Jakobian — J. Doe, Jul 08 '19 at 09:31
@J.Doe: A function is holomorphic if (and only if) $u$ and $v$ are real-differentiable and the CR equations hold. And holomorphic functions are analytic. — Martin R, Jul 08 '19 at 09:32
It's true with some weak assumptions. For example, Looman-Menchoff theorem demands continuity. Or you could try the more elementary Goursat's theorem which demands function to be $C^1$. — Jakobian, Jul 08 '19 at 09:33

score 1 · Accepted Answer · answered Jul 08 '19 at 09:06

1

If $f(z) = \sum\limits_{k=0}^{\infty} a_nz^{n}$ then $g(z)=\sum\limits_{k=0}^{\infty} \overline {a}_n z^{n}$. You can differentiate the series for $g$ term by term.

$g'$ is obtained from $f'$ the same way $g$ is obatined from $f$.

answered Jul 08 '19 at 09:06

Kavi Rama Murthy

311,013

2

Why did you not post this to one of the duplicate targets? Finding them is not that difficult, and it is sort of a civic duty here. I cannot believe that a trusted user would not think that this question has NOT been handled earlier! – Jyrki Lahtonen Jul 12 '19 at 15:56

Show that if $f$ is analytic then $g(z)=\overline{f(\overline{z})}$ is analytic and find its derivative

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