Functions Holomorphic along Holomorphic Vector Fields

The main result of the paper is the following generalization of Forelli's theorem (Math. Scand. 41:358–364, 1977): Suppose 'F' is a holomorphic vector field with singular point at 'p', such that 'F' is linearizable at 'p' and the matrix is diagonalizable with eigenvalues whose ratios are positive reals. Then any function φ that has an asymptotic Taylor expansion at 'p' and is holomorphic along the complex integral curves of 'F' is holomorphic in a neighborhood of 'p'. We also present an example to show that the requirement for ratios of the eigenvalues to be positive reals is necessary.