The study of parametric partial differential equations (parametric PDEs) has significant practical importance. Given the significance of eigenvalue problems in quantum mechanics, investigating parametric eigenvalue problems is particularly important. Despite their importance, theoretical analyses for parametric nonlinear eigenvalue problems remain insufficiently explored. This thesis, particularly motivated from the application of deep operator networks(DeepONet), focuses on a critical aspect of parametric nonlinear eigenvalue problems: the holomorphic dependence of the ground eigenpair on parameters. This holomorphic dependence provides theoretical justification for employing DeepONet to efficiently solve parametric PDEs avoiding the curse of dimensionality. This thesis is organized into five chapters. The first chapter provided the introduction and the motivations of this research. The second chapter provides a novel framework for establishing parametric holomorphy in the PDE setting which is desired the DeepONet application. Using results from Chapter 2, the third chapter demonstrates the parametric holomorphy of the ground eigenpair for linear eigenvalue problems. The fourth chapter investigates parametric holomorphy in nonlinear eigenvalue problems. Lastly, Chapter 5 discusses applications to quasi-Monte Carlo methods.