of known, orthogonal, and complete functions. In the case of bent-bore magnetic elements, such as dipoles with large sagitta, it is not possible to obtain a suitable Green function. New techniques have therefore been developed for treating cases of such general geometries, and these techniques will be presented in a subsequent paper. Our emphasis will be on the accurate representation of fields in terms of scalar and vector potentials. Once these representations are available, there are a variety of methods for computing the associated design orbits and transfer maps about these orbits. An Appendix summa- rizes how this can be done when canonical coordinates are employed and the associated Lie-algebraic structure is exploited. A brief outline of this paper is as follows: Section II reviews circular cylinder harmonic expansions and intro- duces a collection of functions, known as on-axis gra- dients , which uniquely characterize the magnetic scalar potential. It also describes how the vector potential can be obtained from the scalar potential. The on-axis gra- dients themselves are generally unspecified functions of z . In some simple cases they can be found analytically. However, in general they must be determined numeri- cally. Section III describes how this can be done us- ing known magnetic field values determined at points on some regular 3-dimensional grid. In Section IV, we treat an analytically soluble model problem, that of a magnetic monopole doublet, which is used to benchmark these methods. In Section V, we discuss the advantages of these surface-fitting methods that result from numer- ical smoothing. In Section VI, we use these methods to study a proposed ILC damping ring wiggler. The paper concludes with a summary and an Appendix. Further detail may be found in [3, 4]. Note that the term containing G$_{0}$$_{,s}$ ( k ) does not con- tribute to the above sum, and we may set G$_{0}$$_{,s}$ = 0 with- out loss of generality. By utilizing the Taylor series of the modified Bessel function I$_{m}$ , we may write ψ in the form of a circular cylinder harmonic (multipole) expansion: where for α = s , c , The functions C [ n ] $_{m,α}$, known as on-axis gradients [3, 4], are defined by We see that even powers of ρ and the combinations ρ $^{m}$cos mφ and ρ $^{m}$sin mφ are analytic (in fact, polyno- mial) functions of x and y . A general solution ψ satisfying the Laplace equation and analytic near the axis ρ = 0 takes the form Note that We will require an expansion of (6) in the transverse variables x and y . Define the polynomials for integer l ≥ 0 and m ≥ 0. In Cartesian coordinates we then have: Note that each polynomial F l,m α is homogeneous of degree 2 l + m . from which it follows that, for non-negative integers l and m , We note for future use that (2) can be written in the form Since we wish to describe straight beam-line elements, it is convenient to work initially in circular cylindrical coordinates ρ , φ , and z with In a current-free region the magnetic field B is curl free, and can therefore be described most simply in terms of a magnetic scalar potential ψ . Because B is also di- vergence free, ψ must obey the Laplace equation 2 ρ $^{m}$cos mφ = R e [( x + iy ) $^{m}$] , (4b) ρ $^{m}$sin mφ = I m [( x + iy ) $^{m}$] . (4c) ψ ( ρ, φ, z ) = ∞ ∑ m =0 ∫ ∞ −∞ dk I$_{m}$ ( kρ ) e ikz [ G$_{m,s}$ ( k ) sin mφ + G$_{m,c}$ ( k ) cos mφ ] . (5) ψ ( ρ, φ, z ) = ∞ ∑ m =0 [ ψ$_{m,s}$ ( ρ, z ) sin mφ + ψ$_{m,c}$ ( ρ, z ) cos mφ ] , (6) ψ$_{m,α}$ ( ρ, z ) = ∞ ∑ l =0 ( − 1) $^{l}$m ! 2 $^{l}$l !( l + m )! C [2 l ] $_{m,α}$( z ) ρ 2 l + $^{m}$. (7) C [ n ] $_{m,α}$( z ) = i n 2 $^{m}$m ! ∫ ∞ −∞ dk e $^{ikz}$km + $^{n}$G$_{m,α}$ ( k ) . (8) C [ n ] $_{m,α}$( z ) = d $^{n}$C [0] $_{m,α}$( z ) /dz $^{n}$. (9) F l,m s = ( x 2 + y $^{2}$) $^{l}$I m ( x + iy ) $^{m}$, (10a) F l,m c = ( x 2 + y $^{2}$) $^{l}$R e ( x + iy ) $^{m}$, (10b) ψ ( x, y, z ) = ∞ ∑ l =0 ∞ ∑ m =0 ( − 1) $^{l}$m ! 2 $^{l}$l !( l + m )! $^{[}$C [2 l ] $_{m,s}$( z ) F l,m s ( x, y ) + C [2 l ] $_{m,c}$( z ) F l,m c ( x, y ) $^{]}$. (11) ρ 2 ℓ = ( x 2 + y $^{2}$) $^{ℓ}$, (4a) x + iy = ρe iφ (3) x = ρ cos φ, y = ρ sin φ. (2) ∇ $^{2}$ψ = 0 . (1) II. CYLINDRICAL HARMONIC EXPANSIONS A. Scalar Potential