Now, employ the orthogonality properties of the Mathieu functions to obtain the relations In view of (47), define the function ˜ F ( v, k ) by the rule and define functions ˜ F c $_{r}$( k ) and ˜ F s $_{r}$( k ) by the rules Note the similarity to (17), where cos( rφ ) and sin( rφ ) are replaced by ce$_{r}$( v, q ) and se$_{r}$( v, q ). We will call the functions ˜ F α r ( k ) Mathieu coefficient functions in analogy to the Fourier coefficients that arise in Fourier analysis. With these definitions, the relations (47) can be rewrit- ten in the form Finally, employ (50) in (42). So doing gives the result We have obtained an elliptical cylinder harmonic expan- sion for ψ in terms of surface field data. Of course, what we really want are the on-axis gradi- ents. Again, once these gradients are known, we may use We have found expressions for the on-axis gradients in terms of field data (normal component) on the surface of an elliptic cylinder. Finally, in view of (8) and (54), we have the desired re- sults and Upon comparing (53) with (5), we conclude that there are the relations For further reference, we will call the quantities α r $_{m}$( k ) and β r $_{m}$( k ) Mathieu-Bessel connection coefficients [3, 4]. Using these results, (51) can be rewritten in the form (11) and (12) to compute the associated scalar and vec- tor potentials. The gradients can be found by employing two remarkable connections (identities) between elliptic and circular cylinder functions of the form [3] 9 c$_{r}$ ( k )Ce ′ $_{r}$( U, q ) = 1 2 π 2 ∫ 2 π 0 dv ce$_{r}$( v, q )$^{∫}$ ∞ −∞ dz e − $^{ikz}$F ( U, v, z ) , (47a) s$_{r}$ ( k )Se ′ $_{r}$( U, q ) = 1 2 π 2 ∫ 2 π 0 dv se$_{r}$( v, q )$^{∫}$ ∞ −∞ dz e − $^{ikz}$F ( U, v, z ) . (47b) Ce$_{r}$( u, q ) ce$_{r}$( v, q ) = ∞ ∑ m =0 α r $_{m}$( k ) I$_{m}$ ( kρ ) cos( mφ ) , (52a) Se$_{r}$( u, q ) se$_{r}$( v, q ) = ∞ ∑ m =1 β r $_{m}$( k ) I$_{m}$ ( kρ ) sin( mφ ) . (52b) ˜ F ( v, k ) = 1 2 π ∫ ∞ −∞ dz e − $^{ikz}$F ( U, v, z ) , (48) ˜ F c $_{r}$( k ) = 1 π ∫ 2 π 0 dv ce$_{r}$( v, q ) ˜ F ( v, k ) = 1 2 π 2 ∫ 2 π 0 dv ce$_{r}$( v, q )$^{∫}$ ∞ −∞ dz e − $^{ikz}$F ( U, v, z ) , (49a) ˜ F s $_{r}$( k ) = 1 π ∫ 2 π 0 dv se$_{r}$( v, q ) ˜ F ( v, k ) = 1 2 π 2 ∫ 2 π 0 dv se$_{r}$( v, q )$^{∫}$ ∞ −∞ dz e − $^{ikz}$F ( U, v, z ) . (49b) ψ ( x, y, z ) = ∞ ∑ m =0 ∫ ∞ −∞ dk e $^{ikz}$I$_{m}$ ( kρ ) cos( mφ ) ∞ ∑ r =0 α r $_{m}$( k )[ ˜ F c $_{r}$( k ) / Ce ′ $_{r}$( U, q )] + ∞ ∑ m =1 ∫ ∞ −∞ dk e $^{ikz}$I$_{m}$ ( kρ ) sin( mφ ) ∞ ∑ r =1 β r $_{m}$( k )[ ˜ F s $_{r}$( k ) / Se ′ $_{r}$( U, q )] . (53) G$_{m,c}$ ( k ) = ∞ ∑ r =0 α r $_{m}$( k )[ ˜ F c $_{r}$( k ) / Ce ′ $_{r}$( U, q )] , (54a) G$_{m,s}$ ( k ) = ∞ ∑ r =1 β r $_{m}$( k )[ ˜ F s $_{r}$( k ) / Se ′ $_{r}$( U, q )] . (54b) c$_{r}$ ( k ) = ˜ F c $_{r}$( k ) Ce ′ $_{r}$( U, q ) , s$_{r}$ ( k ) = ˜ F s $_{r}$( k ) Se ′ $_{r}$( U, q ) . (50) C [ n ] $_{m,c}$( z ) = i n 2 $^{m}$m ! ∫ ∞ −∞ dk e $^{ikz}$kn + m ∞ ∑ r =0 α r $_{m}$( k )[ ˜ F c $_{r}$( k ) / Ce ′ $_{r}$( U, q )] , (55a) C [ n ] $_{m,s}$( z ) = i n 2 $^{m}$m ! ∫ ∞ −∞ dk e $^{ikz}$kn + m ∞ ∑ r =1 β r $_{m}$( k )[ ˜ F s $_{r}$( k ) / Se ′ $_{r}$( U, q )] . (55b) ψ ( x, y, z ) = ∞ ∑ r =0 ∫ ∞ −∞ dk e $^{ikz}$[ ˜ F c $_{r}$( k ) / Ce ′ $_{r}$( U, q )]Ce$_{r}$( u, q ) ce$_{r}$( v, q ) + ∞ ∑ r =1 ∫ ∞ −∞ dk e $^{ikz}$[ ˜ F s $_{r}$( k ) / Se ′ $_{r}$( U, q )]Se$_{r}$( u, q ) se$_{r}$( v, q ) . (51)