Final answer:
The question inquires about expanding a function as an orthonormal series of Sturm-Liouville problem characteristic functions, and involves understanding dimensional consistency and the probabilistic interpretation of quantum mechanics.
Step-by-step explanation:
The question asks how to find the formal expansion of the function f(x)=1 on (1≤x≤e^π) as a series of orthonormal characteristic functions of a Sturm-Liouville problem. This involves solving the differential equation d/dx[x dy/dx]-λ/x y=0 with boundary conditions y(1) = 0 and y(e^π) = 0. The expansion would typically be done in terms of eigenfunctions of the Sturm-Liouville problem, which satisfy the given boundary conditions and form an orthonormal set with respect to a given weight function over the specified interval.
The interpretation of this problem also involves understanding concepts like dimensional consistency, which ensures that all terms in an equation must have the same dimension. In power series expansions, this would justify why the arguments of the mathematical functions should be dimensionless. Moreover, probabilistic interpretations, such as the Born interpretation, are crucial when solving such Sturm-Liouville problems in quantum mechanics where the square of the wave function describes a probability density.