Transfer matrices and synchrotron integrals for arbitrary dipole fields

Transfer matrices (TMs) and synchrotron integrals for magnetic device with a varying dipole strength along the azimuth s (i.e. bending magnets with endfields or wigglers) are calculated as a power series. The ratio between length and minimum bending radius of the device serves as an expansion parameter. The determinant is 1 to any order to the expansion. A lower bound on the radius of convergence is given. Results for the examples mentioned are calculated to first order. The method is based on the approximations of the magnetic field by a series of small rectangular dipole magnets. The method is particularly useful for the design of low emittance storage rings with their many short magnets and consequently many endfields as well as for wiggler or undulator insertions.