Abstract: At present，the analytical solution for the steady-state temperature field of the plane double connected domain mainly includes the separation variable method，the virtual heat source method，the integral transformation method and the Green's function method． These methods are only applicable to the cases with simple geometric shapes and unobstructed boundary conditions． By means of the mapping function，this study used complex variable function method to map the double connected domain with arbitrary shape in the physical plane onto an axisymme- tric circular ring in the image plane，and then transformed the partial differential equations into ordinary differential equations． By the Dirichlet boundary condition in heat transfer，the steady-state temperature field of the axisym- metric circular ring can be obtained in the image plane． Then，returning the result to the physical plane by the mapping function，the steady-state temperature field of the double connected domain in the physical plane could be obtained． For the problems of the double connected domain with known the mapping function，such as the eccen- tric circular ring and a semi-infinite domain containing a circular hole，the proposed method provided an analytical solution of the steady-state temperature field． For the problem of the general double connected domain，this study gives the method of solving the mapping function through optimization techniques，and obtains the analytical solu- tion of the steady-state temperature field of the complex double connected domain．

Key words: plane double connected domain, steady-state temperature field, complex function method, analytical solution