Let $\displaystyle f(z)=\sum_{k=0}^\infty a_kz^k$ have a radius of convergence 2, and let $\displaystyle g(z)=\sum_{k=0}^\infty b_k(z-3)^k$ have a radius of convergence 2. Let $\displaystyle I=\{z:f(z),g(z)\text{ both converge}\}$.

**(a)** If $\displaystyle f(z)=g(z)$ for $\displaystyle z\in I$, prove there is an analytic function $\displaystyle w$ for which $\displaystyle w(z)=f(z)$ when $\displaystyle |z|<2$ and $\displaystyle w(z)=g(z)$ when $\displaystyle |z-3|<2$.

**(b)** For the function $\displaystyle w$ of part (a), determine the radius of convergence $\displaystyle R$ for the power series $\displaystyle w(z)=\sum_{k=0}^\infty c_k\left(z-\frac{3}{2}\right)^k$.

**(c)** What relationship exists between the coefficient sequence $\displaystyle \{a_k\}$ and the coefficient sequence $\displaystyle \{b_k\}$?