Let \(\displaystyle \{ z_k \}\) be a sequence of distinct points in a domain \(\displaystyle D\) that accumulates on \(\displaystyle \partial D\). Let \(\displaystyle \{ m_k \}\) be a sequence of positive integers, and for each \(\displaystyle k\), let \(\displaystyle a_{k0}, \ldots, a_{km_k}\) be complex numbers. Show that there is an analytic function \(\displaystyle f(z)\) on \(\displaystyle D\) such that \(\displaystyle f^{(j)}(z_k)=a_{kj}\) for \(\displaystyle 0 \leq j \leq m_k\) and for all \(\displaystyle k\).
In this section we have covered Runge's Theorem. However, I am still not sure how to prove this. I need some hints on doing this. Thanks in advance.
In this section we have covered Runge's Theorem. However, I am still not sure how to prove this. I need some hints on doing this. Thanks in advance.