Exact solution of Qian equation of slender toroidal shells
In 1979 Qian Weichang studied the slender toroidal shell systematically and derived a called Qian’s equation, then obtained a series solution with the expression of continued fractions. But Qian did not mention if the series solution can be converted to a well-known special functions. In this paper, a linear transformation has been introduced, which will transfer the equation into a Mathieu equation, whose solution can be expressed in terms of Mathieu functions. This study has revealed a intrinsic relationship between the Qian’s solution and the Mathieu solutions.