Abstract: This paper discusses the problem of the extraction of characteristic roots {λ_t}(λ₀<λ₁<λ₂…) on Bessel-Neumann’s mixed equations. It gives the expressions and the evaluation of the minimum root. The advantage of the method has no use for the table of the multi-figure number Bessel function and it does not need computer but can calculate all the characteristic roots {λ_t}. The precision of these roots is still high.