Show that there is a sequence of polynomials \(\displaystyle \{ p_n(z) \}\) such that \(\displaystyle p_n(z) \rightarrow 1\) if \(\displaystyle \text{Re}(z) >0\), \(\displaystyle p_n(z) \rightarrow 0\) if \(\displaystyle \text{Re}(z) =0\), and \(\displaystyle p_n(z) \rightarrow -1\) if \(\displaystyle \text{Re}(z) <0\).
This section of the book covers Runge's Theorem. However, since we have to show that there is such a sequence of polynomials, we can define a sequence explicitly with these properties. I can not seem to come up with a suitable sequence. I would appreciate some help on this. Thanks.
This section of the book covers Runge's Theorem. However, since we have to show that there is such a sequence of polynomials, we can define a sequence explicitly with these properties. I can not seem to come up with a suitable sequence. I would appreciate some help on this. Thanks.