Given a complex Banach space \(E\) with open unit ball \(B_E^\circ\), the authors call a function \(f: B_E^\circ\to {\mathbb C}\) integral if there exists a regular Borel measure \(\mu\) on the closed unit ball of \(E'\), \(B_{E'}\), endowed with the weak\(^*\) topology, such that \[ f(z)=\int_{B_{E'}}{1\over 1-\phi(z)}d\mu(\phi) \] for all \(z\) in \(B_E^\circ\). Defining the integral norm of \(f\), \(\| f\| _I\), as the infimum of \(\| \mu\| \) where \(\mu\) represents \(f\), \(\| f\| _I\) makes the space of all integral functions on \(B_E^\circ\), \({\mathcal H}_I(B_E^\circ)\), into a Banach space. Each of the terms in the Taylor series expansion of an integral holomorphic function about the origin is an integral homogeneous polynomial. As \({\mathcal H}_I(B_E^\circ)\) is the dual of the ball algebra on \(B_{E'}\) (the holomorphic functions on \(B_{E'}^\circ\) which have weak\(^*\)-continuous extension to the closed unit ball of \({E'}\)), it follows that the Taylor series of an integral holomorphic function will not in general converge in \({\mathcal H}_I(B_E^\circ)\). Because of this, the authors introduce two Fréchet spaces of holomorphic functions. Given \(0