by MathLover
Last Updated September 20, 2018 13:20 PM

I encountered following problem

and I solved it by using the hint provided. Thinking of it I noticed that I am able to solve it even if I use the following function: $$ F(z)=1/f(1/z)),\quad |z|> 1$$ $$ =f(z) , \quad |z|\leq 1 $$

What is the problem if I use this function to solve the problem? I can extend it to the whole $\mathbb{C}$ as well: I know that analytic continuation of any function is unique, but I am thinking where is problem if I choose to use this function.

Any Help will be appreciated.

**To be continuous on unit circle.** when $|z|=1$ then $\bar{z}=\dfrac{1}{z}$ therefore
on unit circle
$$\dfrac{1}{\overline{f(1/\bar{z})}}=\dfrac{1}{\overline{f(z)}}=f(z)$$
with definition $f(z)$ in $|z|<1$ and $\dfrac{1}{\overline{f(1/\bar{z})}}$ in $|z|>1$. in your case this continuation will not be continouess.

- ServerfaultXchanger
- SuperuserXchanger
- UbuntuXchanger
- WebappsXchanger
- WebmastersXchanger
- ProgrammersXchanger
- DbaXchanger
- DrupalXchanger
- WordpressXchanger
- MagentoXchanger
- JoomlaXchanger
- AndroidXchanger
- AppleXchanger
- GameXchanger
- GamingXchanger
- BlenderXchanger
- UxXchanger
- CookingXchanger
- PhotoXchanger
- StatsXchanger
- MathXchanger
- DiyXchanger
- GisXchanger
- TexXchanger
- MetaXchanger
- ElectronicsXchanger
- StackoverflowXchanger
- BitcoinXchanger
- EthereumXcanger