How do I prove the function $f(z) = \sin \bar{z}$ is not differentiable on the disc $D(0,1)$?

Question

How do I prove the function $f(z)=\sin \bar{z}$ is not differentiable on the disc $D(0,1)$?

I originally used cauchy reimann equations to prove it is not differentiable everywhere, but I need to be able to show it's not differentiable specifically for the disc. Any help would be greatly appreciated.

I can't understand why your CR equation method wouldn't satisfy you on the disk if it satisfies you elsewhere. — Jonas Meyer, Apr 02 '15 at 15:46
Possible duplicate of Showing $\sin(\bar{z})$ is not analytic at any point of $\mathbb{C}$ — Jonas Meyer, Apr 02 '15 at 15:46

OR. · Answer 1 · 2015-04-02T15:41:25.380

3

Try the limit $$f'(0)=\lim_{z\to0}\frac{\sin(\overline{z})-\sin(0)}{z-0}=\lim_{z\to0}\frac{\sin(\overline{z})}{z}=\lim_{z\to0}\frac{\overline{z}}{z}\frac{\sin(\overline{z})}{\overline{z}}=\lim_{z\to0}\frac{z}{\overline{z}}$$

edited Apr 02 '15 at 15:41

answered Apr 02 '15 at 15:06

OR.

5,941

score 0 · Answer 2 · answered Apr 02 '15 at 15:05

0

The CR equations tell you at which points $z \in \mathbb{C}$ the function $f$ is differentiable. In particular, applying them shows that it is not differentiable everywhere on $D(0, 1)$.

answered Apr 02 '15 at 15:05

Travis Willse

99,363

How do I prove the function $f(z) = \sin \bar{z}$ is not differentiable on the disc $D(0,1)$?

2 Answers2