(* NON-AXISYMMETRIC POTENTIALS FOR THE CYLINDER  *)

<<hol0.m;

(*  Bounded polynomials of the form Fni*Cos[n*θ]  *)

(*  n=1  *)

F11=-D[F2,r]
F12=-D[F3,r]
F13=-D[F4,r]
F14=-D[F5,r]
F15=-D[F6,r]
F16=-D[F7,r]
F17=-D[F8,r]
F18=-D[F9,r]

(*  n=2  *)

F22=-Expand[Simplify[D[F13,r]-F13/r]]
F23=-Expand[Simplify[D[F14,r]-F14/r]]
F24=-Expand[Simplify[D[F15,r]-F15/r]]
F25=-Expand[Simplify[D[F16,r]-F16/r]]
F26=-Expand[Simplify[D[F17,r]-F17/r]]
F27=-Expand[Simplify[D[F18,r]-F18/r]]

(*  n=3  *)

F33=-Expand[Simplify[D[F24,r]-2*F24/r] ]
F34=-Expand[Simplify[D[F25,r]-2*F25/r] ]
F35=-Expand[Simplify[D[F26,r]-2*F26/r] ]
F36=-Expand[Simplify[D[F27,r]-2*F27/r]]

(*  Singular potentials of the form Sni*Cos[n*θ]  *)

(*  n=1  *)

S10=Expand[Simplify[D[S0,r]]]
S11=Expand[Simplify[D[S1,r]]]
S12=Expand[Simplify[D[S2,r]]]
S13=Expand[Simplify[D[S3,r]]]
S14=Expand[Simplify[D[S4,r]]]
S15=Expand[Simplify[D[S5,r]]]
S16=Expand[Simplify[D[S6,r]]]
S17=Expand[Simplify[D[S7,r]]]
S18=Expand[Simplify[D[S8,r]]]
S19=Expand[Simplify[D[S9,r]]]

(*  n=2  *)

S20=Expand[Simplify[D[S10,r]-S10/r]]
S21=Expand[Simplify[D[S11,r]-S11/r]]
S22=Expand[Simplify[D[S12,r]-S12/r]]
S23=Expand[Simplify[D[S13,r]-S13/r]]
S24=Expand[Simplify[D[S14,r]-S14/r]]
S25=Expand[Simplify[D[S15,r]-S15/r]]
S26=Expand[Simplify[D[S16,r]-S16/r]]
S27=Expand[Simplify[D[S17,r]-S17/r]]
S28=Expand[Simplify[D[S18,r]-S18/r]]
S29=Expand[Simplify[D[S19,r]-S19/r]]

(*  n=3  *)

S30=Expand[Simplify[D[S20,r]-2*S20/r]]
S31=Expand[Simplify[D[S21,r]-2*S21/r]]
S32=Expand[Simplify[D[S22,r]-2*S22/r]]
S33=Expand[Simplify[D[S23,r]-2*S23/r]]
S34=Expand[Simplify[D[S24,r]-2*S24/r]]
S35=Expand[Simplify[D[S25,r]-2*S25/r]]
S36=Expand[Simplify[D[S26,r]-2*S26/r]]
S37=Expand[Simplify[D[S27,r]-2*S27/r]]
S38=Expand[Simplify[D[S28,r]-2*S28/r]]
S39=Expand[Simplify[D[S29,r]-2*S29/r]]