The static or quasistatic magnetic field in accelerator magnets must be determined by measurements and by numeric codes for calculating the fields in realistic models. These empirical data must be condensed into compact analytic representations, as e.g. multipole expansions, with adjustable parameters so that these data can be checked for their accuracy and consistency and can be communicated to other users for design and beam dynamics calculations. Potential and field theory as well as Green's functions give insight which function systems are admissible, useful and complete; how well and where series or integral representations converge and how the convergence can be improved. Formulas are derived so that the adjustable parameters can be determined from the output of measurements. Numeric evaluations comparing the results obtained from these representations with the empirical data permit one to estimate the accuracy of both the data and the finite sums used to represent them and to check the consistency. For the purposes described above various curvilinear coordinate systems not used up to now are investigated. The systems of particular solutions of the potential equation in these coordinates are found in order to generalize the concept of multipoles; their completeness is verified. As a first step we treat the two-dimensional field in the central region of long magnets with a rectangular (gap and particle beam) cross section (width-to-height ratio about 2:1). The field is expanded with respect to circular and to elliptic multipoles. The latter are a new tool obtained by solving the polential equation in plane elliptic coordinates. Its advantage is that the corresponding expansion is valid in an area larger than the reference circle of the common circular multipoles, namely in the ellipse circumscribed to the circle. Numeric comparisons between the empiric field values
fourd by field computation codes and those obtained from the finite expansions have been performed and give very good agreement. Extensions to three-dimensional field distributions are under investigation. The effect of the curvature on multipoles in curved magnets is also a subject of our efforts.