– 10 – 5 5 10 z (cm) – 0.30 – 0.25 – 0.20 – 0.15 – 0.10 – 0.05 B (T) y – 10 – 5 5 10 z (cm) 0.005 0.010 0.015 B (T) x – 10 – 5 5 10 z (cm) – 0.04 – 0.02 0.02 0.04 B (T) z FIG. 9: The field component B$_{x}$ on the line ρ = 1 / 2, φ = π/ 4, z ∈ [ −∞ , ∞ ] for the monopole doublet in the case that a = 2 . 5 cm and g = 1 Tesla-(cm) $^{2}$. The coordinate z is given in centimeters. FIG. 10: The field component B$_{z}$ on the line ρ = 1 / 2, φ = π/ 4, z ∈ [ −∞ , ∞ ] for the monopole doublet in the case that a = 2 . 5 cm and g = 1 Tesla-(cm) $^{2}$. The coordinate z is given in centimeters. FIG. 8: The on-axis field component B$_{y}$ ( x = 0 , y = 0 , z ) for the monopole doublet in the case that a = 2 . 5 cm and g = 1 Tesla-(cm) $^{2}$. The coordinate z is given in centimeters. nomial series (11-12) representing the field in terms of the functions (60) is given by the condition $^{√}$x 2 + y 2 < √ z 2 + a $^{2}$. In particular, the domain of convergence is a region of circular cross-section whose radius increases as we move longitudinally away from the location of the monopoles at z = 0. Suppose we wish to calculate a transfer map through 7 th order. Then, as shown in the Appendix, to do so requires knowledge of the C [ n ] $_{m,α}$( z ) with ( m + n ) ≤ 7 when m = 0 or m is odd, and knowledge of the C [ n ] $_{m,α}$( z ) with ( m + n ) ≤ 8 when m is even. Graphs of a selected few of these functions, for the monopole doublet in the case that a = 2 . 5 cm and g = 1 Tesla-(cm) $^{2}$, are shown in Figs. 11 through 13. In these plots z has units of centimeters. Evidently the C [0] m,s be- come ever more highly peaked with increasing m . For- tunately, when working through some fixed degree, we need fewer derivatives with increasing m . Note that we expect that the function C [ n ] $_{m,s}$( z ) should have n zeroes. This is indeed the case. The procedure discussed in Section III A has been benchmarked using the field of a monopole doublet in the case that a = 2 . 5 cm and g = 1 Tesla-(cm) $^{2}$. We set up a regular grid in x , y , z space, where we let each variable range over the intervals x ∈ [ − 4 . 4 , 4 . 4] with spacing h$_{x}$ = 0 . 1, y ∈ [ − 2 . 4 , 2 . 4] with h$_{y}$ = 0 . 1, and z ∈ [ − 300 , 300] with h$_{z}$ = 0 . 125 (in units of cm). The known values of the three components of the field are computed using (59) at each grid point. Consider a cylinder of radius R = 2 cm and length 600 cm. We use bicubic interpolation to interpolate B at these grid points onto 49 selected angular points on the cylinder, for each of the 4801 selected values of z . The angular in- tegration in (17) is performed using a Riemann sum with N = 49. (This is necessary to ensure sufficient conver- gence of the angular integrals to within 10 − $^{4}$.) We evalu- ate the Fourier transform at 401 values of k in the range [ − K$_{c}$,K$_{c}$ ] with K$_{c}$ = 20, using a spline-based Fourier transform algorithm [3, 4]. We use these same points in k space to evaluate the inverse Fourier transform, provid- ing a set of numerically determined functions C [ n ] $_{m,α}$( z ). A comparison between the exact on-axis gradients (60) and those obtained from grid data is provided in Figs. 11-13 for the functions C [0] 1 $_{,s}$, C [6] 1 $_{,s}$, and C [0] 7 $_{,s}$. Evidently the agreement is excellent. Fig. 14 illustrates the differ- ence between the on-axis gradient C [0] 1 ,s as obtained from grid data and the exact on-axis gradient obtained from (60) with m = 1. The maximum error attained rela- tive to peak is 1 . 7 × 10 − $^{4}$. Further detailed study shows that numerical results agree with exact results to within relative errors less than a few parts in 10 4 for all the relevant C [ n ] $_{m,α}$( z ). By using exact data on the cylinder B. Circular Cylinder Results 11