Finding a polynomial sequence approximating certain function on complex plane

Question

I'm trying to find a polynomial sequence on complex plane, which converges to $1$ on the upper-half plane, $0$ on the real line, and $-1$ on the lower-half plane...but just don't have a clue.

Thanks for any help!

Answer 1

Typically a polynomial will not converge to a finite limit. Consider any polynomial of the form $\sum_{i=0}^na_iz^i$. You will always be able to find values of $z$ such that it diverges as for large enough values of $|z|$ it will be dominated by the highest power.

If are after a power series (effectively an infinite polynomial) then the taylor series of either:

$$\frac{2}{\pi}\arctan(\Im z)$$

or

$$\frac{2}{1+e^{-\Im z}}-1$$

would fit your criteria.

However I get the impression that probably wasn't what you were after. Perhaps you can clarify your question.