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Helmholtz方程的解中，在圆弧弯波导每对面上满足第一或二类齐次边值条件的函数构成四个函数列。去掉每个函数沿圆周的传播因子则称为算子在横截面内的特征函数。它们井不是本征函数。本文证明，任一在横截面内有界交差且在其周界每对边上满足第一或二类齐次边值条件的函数必能按满足相同边值条件的特征函数列展开成级数，并且给出了展开式收敛于被展开函数的意义。
In the solution of Helmholtz equation, the function that satisfies the first- or second-degree homogeneous boundary conditions on each pair of arc-curved waveguides constitutes four function columns. The factor of propagation along the circumference of each function is called the eigenfunction of the operator in the cross section. They are not eigenfunctions. This paper proves that any function satisfying the first or second type of homogeneous boundary conditions on each pair of edges of the boundary with bounded intersections in the cross section must be expanded into series according to the sequence of eigenvalues satisfying the same boundary condition , And gives the meaning of expansion converged to the function being expanded.