Three sets of conditions are given which produce a complete set of eigenfunctions for two classes of analytic operator-valued functions, whose values are compact operators on a separable Hilbert space, H. In the first case, conditions are put on the adjoint and the range of the operator-valued function. In the second case, the inverse of the appropriate operator-valued function has a condition imposed on it. The Fredholm alternative theorem is used to prove the theorem. In the third case, the Fourier transform for operator-valued functions is introduced and used to construct a complete set of eigenfunctions of the operator-valued function. The proof of this theorem is based on the residue theorem and Jordan's lemma for analytic opertator-valued functions. The remainder of the paper is devoted to providing preliminary properties of an operator-valued function and its spectrum.