How to through physical model and mathematical model to seek out and solve mathematical methods of solid physical problems in the study of solid state physics is an important content. This paper through the simple mathematical model analysis of the grid work and understand clearly the lattice vibrations some basic concepts; Through the crystal heat capacity improvement of physical model to solve let under low temperature and the actual problem not; Through the generalized integrals, the delta function, Fourier series, and other special function method to solid state physics, the electronic heat capacity let, Fermi band are solved, and analyses three kind of physical quantity of the physical meaning each contains.