therefore its Fourier transform is well defined. Next de- fine the functions ˜ b$_{m,s}$ and ˜ b$_{m,c}$ by out. Moreover, this filtering action is improved by mak- ing R as large as possible. This filtering, or smoothing , feature will be discussed in more detail in Section V. We close this subsection with the remark that if one wishes to extract the C [ n ] 0 $_{,c}$( z ) (monopole) on-axis gradi- ents from field data, as is required for example in the case of a solenoid, it may be preferable to use the longitudi- nal component B$_{z}$ ( R, φ, z ) on the surface of the cylinder rather than the normal component B$_{ρ}$ ( R, φ, z ) [4]. In the previous subsection we employed a cylinder with circular cross section, and observed mathematically that it is desirable for error insensitivity to use a cylinder with a large radius R . Physically, this is because we want the field data points employed to be as far from the axis as possible since the effect of inhomogeneities (noise) in the data decays with distance from the inhomogeneity. Evidently the use of a large circular cylinder is optimal for beam-line elements with a circular bore. However, for dipoles or wigglers with small gaps and wide pole faces, use of a cylinder with elliptical cross section should give improved error insensitivity. See Fig. 3. In this subsection we will set up the machinery required for the use of elliptical cylinders, and apply it to the calculation of on-axis gradients based on field data [3, 4]. We have found an expression for the on-axis gradients in terms of field data (normal component) on the surface of the cylinder. Equation (22) may be viewed as the con- volution of Fourier surface data ˜ b$_{m,α}$ ( R, k ) with the in- verse Laplacian kernel k n + m − $^{1}$/I ′ $_{m}$( kR ). Moreover, this kernel has a very desirable property. The Bessel functions I ′ $_{m}$( kR ) have the asymptotic behavior Since I ′ $_{m}$( kR ) appears in the denominator of (22), we see that the integrand is exponentially damped for large | k | . Now suppose there is uncorrelated point-to-point noise in the surface data. Such noise will result in anomalously large | k | contributions to the ˜ b$_{m,α}$ ( R, k ). But, because of the exponential damping arising from I ′ $_{m}$( kR ) in the denominator, the effect of this noise is effectively filtered Elliptic coordinates in the x, y plane are described by the relations [6] This relation for G$_{m,α}$ ( k ) can be employed in (8) to give the result from which it follows that Now substitute (19) into the right sides of (17) and per- form the indicated integrations to get the results from which it follows, using the representation (5), that We know that 1. Background 2. Elliptic Coordinates 6 B. Use of Field Data on Surface of Elliptical Cylinder ˜ b$_{m,s}$ ( R, k ) = 1 2 π 2 ∫ 2 π 0 dφ sin mπ$^{∫}$ ∞ −∞ dz e − $^{ikz}$B$_{ρ}$ ( R, φ, z ) , (17a) ˜ b$_{m,c}$ ( R, k ) = 1 2 π 2 ∫ 2 π 0 dφ cos mπ$^{∫}$ ∞ −∞ dz e − $^{ikz}$B$_{ρ}$ ( R, φ, z ) , (17b) ˜ b$_{0}$$_{,c}$ ( R, k ) = 1 4 π 2 ∫ 2 π 0 dφ$^{∫}$ ∞ −∞ dz e − $^{ikz}$B$_{ρ}$ ( R, φ, z ) . (17c) B$_{ρ}$ ( R, φ, z ) = ∂ψ ( ρ, φ, z ) ∂ρ ∣ ∣ ρ = R , (18) for m ≥ 1 and B$_{ρ}$ ( R, φ, z ) = ∞ ∑ m =0 ∫ ∞ −∞ dk kI ′ $_{m}$( kρ ) e ikz [ G$_{m,s}$ ( k ) sin mφ + G$_{m,c}$ ( k ) cos mφ ] . (19) ˜ b$_{m,α}$ ( R, k ) = G$_{m,α}$ ( k ) kI ′ $_{m}$( kR ) , (20) G$_{m,α}$ ( k ) = ˜ b$_{m,α}$ ( R, k ) kI ′ $_{m}$( kR ) . (21) C [ n ] $_{m,α}$( z ) = i n 2 $^{m}$m ! ∫ ∞ −∞ dk e $^{ikz}$kn + m − $_{1}$˜ b$_{m,α}$ ( R, k ) I ′ $_{m}$( kR ) . (22) | I ′ $_{m}$( kR ) | ∼ exp( | k | R ) / $^{√}$2 π | k | R as | k | → ∞ . (23) FIG. 3: An elliptical cylinder, centered on the z -axis, fitting within the bore of a wiggler, and extending beyond the fringe- field regions at the ends of the wiggler. x = f cosh( u ) cos( v ) , (24a) y = f sinh( u ) sin( v ) . (24b)