Because is an eigenfunction under differentiation (i.e., the exponential function is its own derivative), it is often profitable to replace it with a generalized exponential function, with maximum degrees of freedom in its parametrization, to see if parameters can be found to fulfill the constraints imposed by differential equations.
In the case of the one-dimensional ideal wave equation (Eq. (C.1)), with no boundary conditions, an appropriate choice of eigensolution is
Defining the wave velocity (or phase velocityC.2) as , the wave equation becomes
is a solution for all . By superposition,
is also a solution, where and are arbitrary complex-valued functions of arbitrary points in the complex plane.