Books on complex valued functions and solution of PDE

[jonjacson]

Hello folks,

1.- In geometry we study for example the conic sections, their exentricity and properties. I was wondering what part of the mathematical science studies the different properties of complex valued distributions. One example are the spherical armonics. I guess mathematicians have studied this.

Let's say this in other words, in Classical Mechanics the solution of the equations are the orbits, that are normally curves on the plane, but when you study quantum mechanics the solutions are distributions of complex valued functions. I know there are few analytical solutions, but I want to learn about other distributions that are analytical.

2.- When you study mechanics all the energies are allowed, but if you study membranes or chords only certain patterns are allowed. In Quantum mechanics the solutions of the differential equations are a set of discrete states. What part of math in general studies this particular differential equations that have a discrete set of functions as solutions?

 

[vanhees71]

The best introductory book on these questions is

A. Sommerfeld, Lectures on Theoretical Physics vol. 6, Partial Differential Equations, Academic Press 1949