任一非零解为非有理函数的多项式系数的线性偏微分方程

THE LINEAR PARTIAL DIFFERENTIALEQATIONS WITH POLYNOMIAL COEFFICIENTS,WHOSE ARBITRARY SOLUTION IS A NON-RATIONAL FUNCTION

本文用初等方法证明了如果在方程中，｛ （ｚ＿１，ｚ＿２）｝全是ｚ＿１和ｚ＿２的多项式，且｜ （ｚ＿１）ｚ＿２、）｜ （ｚ＿１，ｚ＿２）≠０．当存在ｂ＞Ｏ使得时，此方程的任一非零解是非有理函数，其中Ｄ＝｛｜ｚ＿１｜＝ｒ，｜ｚ＿２｜＝ｒ／２ｂ，ｚ＿１，ｚ＿２∈Ｃ￣２｝．

he linear partial equation with polynomial coefficientswhere {}are polynomials in the z_1 and the z_2 ≡u(z_1,z_2)and.(1) can has the soluin of nonzero rati-onl function.For examPle,the equationhas a solution l-1/(z_1+z_2)￣2.Therefore to find the condition of the solutionswhat all be ron-rationl functios is interest.Let D_r={|Z_1|=R,|Z_2|=R/2b,z_1,z_2∈c￣2}.We have