Abstract When creating an initial model of an accelerator, one usually has to resort to a hard edge model for the quadrupoles and higher order multipoles at the start of the project. Ordinarily, it is not until much later on that one has a field map for the given multipoles. This can be rather inconvenient when one is dealing with particularly thin elements or elements which are rather close together in a beamline as the hard edge model may be inadequate for the level of precision desired. For example, in the EMMA project, the two types of quadrupoles used are so close together that they are usually described by a single field map or via hard edge models. The first method has the desired accuracy but was not available at the start of the project and the second is known to be a rough approximation. In this paper, an analytic expression is derived and presented for fringe fields for a multipole of any order with a view to applying it to cases like EMMA. FRINGE FIELDS FOR DIPOLES In order to have fringe fields, given by a → B which satisfy Maxwell's equations, it is important to write all equations down explicitly. For Dipoles, it is sufficient to consider a two dimensional version of the equations Now, if we take B x = 0, we are left with together with which excludes all dependence on x. Further, we seek fringe fields which have a possible fall-off on axis given by the six parameter Enge function [1] with E(z) given by and all a i constants determined by models and/or experiment, or any function which decays sufficiently rapidly. Maxwell's equations (1) imply z . Both wave equations (for B y and B z ) can be easily solved to give Hence, if we ask that equations (1) be solved as well, we end up with If we further restrict ourselves to real magnetic fields, we obtain so B y and B z are given by twice the real and imaginary parts of the function e(z + iy) respectively. A possibility for having a magnetic field whose B y component fall off on axis is given by the six parameter Enge function [1] as which would force B z to have the form for some complex function E(z + iy). If we consider the simple case E(z + iy) = z + iy then equation
