Many of the problems facing physicists， engineers， and applied mathematicians involve difficulties as governing partial differential equations which have odd order cross derivatives with respective to multiply spatial variables. The classical method of mathematical physics which is based on the method of separation of variable， failed in analyzing these partial differential equations. In solving partial differential equations for the mechanical response of the laminated plate， the author and Professor Yang guangsong have developed a new complex series method （NCST） . NCST succeeds in solving the boundary value problem of various partial deferential equations with constant coefficients. Many results can be found in the book of "a new type complex series method for composite structure mechanics and mathematical physics " by the author. In further investigation， the new complex series method is evolved into the method of separation of complex variable in this book and the real principle in mathematical physics is suggested.
　　The partial differential equations in cylindrical coordinates and spherical coordinates have always variable coefficients . Thus， another difficulty arose in solving the boundary value problem of partial deferential equations， which have odd order cross derivatives， in cylindrical coordinates and spherical coordinates. By the method of separation of complex variable， the author developed the series of complex special functions to solve these mathematical physics equations， which have always the background of anisotropic physics. These complex special functions include two series， one kind is associated with cylindrical coordinates and another kind is associated with spherical coordinates . Several ordinary differential equations are suggested. These complex special functions suggested newly in cylindrical coordinates include that the complex cylindrical polynomial， complex cylindrical-annular function， the modified complex cylindrical polynomial， complex spherical cylindrical polynomial， modified complex spherical cylindrical polynomial， complex cylinder function， complex cylindrical surface function and so on.lt is noted that many special functions are orthogonal with particular weight over particular domain and these corresponding theorems of complex cylindrical functions. expansions are suggested in this book. In solving the problem of isotropic physics equation these complex special functions in cylindrical coordinates can be reduced to the corresponding Bessel functions respectively.