#NON-AXISYMMETRIC SPHERICAL HARMONICS (UP TO m=3)
#
P0:=1:
P01:=x:
P02:=simplify((3*x*P01-1*P0)/2):
P03:=simplify((5*x*P02-2*P01)/3):
P04:=simplify((7*x*P03-3*P02)/4):
P05:=simplify((9*x*P04-4*P03)/5):
P06:=simplify((11*x*P05-5*P04)/6):
P07:=simplify((13*x*P06-6*P05)/7):
P08:=simplify((15*x*P07-7*P06)/8):
P09:=simplify((17*x*P08-8*P07)/9):
#
#Simplified forms of Legendre polynomials.
#
P1:=unapply(P01,x):
P2:=unapply(simplify(P02),x):
P3:=unapply(simplify(P03),x):
P4:=unapply(simplify(P04),x):
P5:=unapply(simplify(P05),x):
P6:=unapply(simplify(P06),x):
P7:=unapply(simplify(P07),x):
P8:=unapply(simplify(P08),x):
P9:=unapply(simplify(P09),x):
#
#
#Harmonic axisymmetric bounded potentials in terms of R,beta
#
F1:=expand(simplify(R*P1(cos(beta))));
F2:=expand(simplify(R^2*P2(cos(beta))));
F3:=expand(simplify(R^3*P3(cos(beta))));
F4:=expand(simplify(R^4*P4(cos(beta)))):
F5:=expand(simplify(R^5*P5(cos(beta)))):
F6:=expand(simplify(R^6*P6(cos(beta)))):
F7:=expand(simplify(R^7*P7(cos(beta)))):
F8:=expand(simplify(R^8*P8(cos(beta)))):
F9:=expand(simplify(R^9*P9(cos(beta)))):
#
#Harmonic axisymmetric singular potentials in terms of R,beta
#
S0:=1/R;
S1:=expand(simplify(R^(-2)*P1(cos(beta))));
S2:=expand(simplify(R^(-3)*P2(cos(beta))));
S3:=expand(simplify(R^(-4)*P3(cos(beta))));
S4:=expand(simplify(R^(-5)*P4(cos(beta)))):
S5:=expand(simplify(R^(-6)*P5(cos(beta)))):
S6:=expand(simplify(R^(-7)*P6(cos(beta)))):
S7:=expand(simplify(R^(-8)*P7(cos(beta)))):
S8:=expand(simplify(R^(-9)*P8(cos(beta)))):
S9:=expand(simplify(R^(-10)*P9(cos(beta)))):
#
#Legendre functions P^1_i.
#
#We first calculate dPn(x)/dx
#
dP1:=unapply(diff(P1(x),x),x):
dP2:=unapply(diff(P2(x),x),x):
dP3:=unapply(diff(P3(x),x),x):
dP4:=unapply(diff(P4(x),x),x):
dP5:=unapply(diff(P5(x),x),x):
dP6:=unapply(diff(P6(x),x),x):
dP7:=unapply(diff(P7(x),x),x):
dP8:=unapply(diff(P8(x),x),x):
dP9:=unapply(diff(P9(x),x),x):
#
P11:=unapply(-sqrt(1-x^2)*dP1(x),x):
P12:=unapply(-sqrt(1-x^2)*dP2(x),x):
P13:=unapply(-sqrt(1-x^2)*dP3(x),x):
P14:=unapply(-sqrt(1-x^2)*dP4(x),x):
P15:=unapply(-sqrt(1-x^2)*dP5(x),x):
P16:=unapply(-sqrt(1-x^2)*dP6(x),x):
P17:=unapply(-sqrt(1-x^2)*dP7(x),x):
P18:=unapply(-sqrt(1-x^2)*dP8(x),x):
P19:=unapply(-sqrt(1-x^2)*dP9(x),x):
#
#The following functions will yield harmonic potentials
#if multiplied by either sin(theta) or cos(theta).
#
#Bounded potentials
#
F11:=simplify(R^1*P11(cos(beta)));
F12:=simplify(R^2*P12(cos(beta)));
F13:=simplify(R^3*P13(cos(beta)));
F14:=simplify(R^4*P14(cos(beta))):
F15:=simplify(R^5*P15(cos(beta))):
F16:=simplify(R^6*P16(cos(beta))):
F17:=simplify(R^7*P17(cos(beta))):
F18:=simplify(R^8*P18(cos(beta))):
F19:=simplify(R^9*P19(cos(beta))):
#
#Singular potentials
#
S11:=simplify(R^(-2)*P11(cos(beta)));
S12:=simplify(R^(-3)*P12(cos(beta)));
S13:=simplify(R^(-4)*P13(cos(beta)));
S14:=simplify(R^(-5)*P14(cos(beta))):
S15:=simplify(R^(-6)*P15(cos(beta))):
S16:=simplify(R^(-7)*P16(cos(beta))):
S17:=simplify(R^(-8)*P17(cos(beta))):
S18:=simplify(R^(-9)*P18(cos(beta))):
S19:=simplify(R^(-10)*P19(cos(beta))):
#
#Repeat for P^2_i.
#
d2P2:=unapply(diff(dP2(x),x),x):
d2P3:=unapply(diff(dP3(x),x),x):
d2P4:=unapply(diff(dP4(x),x),x):
d2P5:=unapply(diff(dP5(x),x),x):
d2P6:=unapply(diff(dP6(x),x),x):
d2P7:=unapply(diff(dP7(x),x),x):
d2P8:=unapply(diff(dP8(x),x),x):
d2P9:=unapply(diff(dP9(x),x),x):
#
P22:=unapply((1-x^2)*d2P2(x),x):
P23:=unapply((1-x^2)*d2P3(x),x):
P24:=unapply((1-x^2)*d2P4(x),x):
P25:=unapply((1-x^2)*d2P5(x),x):
P26:=unapply((1-x^2)*d2P6(x),x):
P27:=unapply((1-x^2)*d2P7(x),x):
P28:=unapply((1-x^2)*d2P8(x),x):
P29:=unapply((1-x^2)*d2P9(x),x):
#
#The following functions will yield harmonic potentials
#if multiplied by either sin(2*theta) or cos(2*theta).
#
#Bounded potentials
#
F22:=simplify(R^2*P22(cos(beta)));
F23:=simplify(R^3*P23(cos(beta)));
F24:=simplify(R^4*P24(cos(beta)));
F25:=simplify(R^5*P25(cos(beta))):
F26:=simplify(R^6*P26(cos(beta))):
F27:=simplify(R^7*P27(cos(beta))):
F28:=simplify(R^8*P28(cos(beta))):
F29:=simplify(R^9*P29(cos(beta))):
#
#Singular potentials
#
S22:=simplify(R^(-3)*P22(cos(beta)));
S23:=simplify(R^(-4)*P23(cos(beta)));
S24:=simplify(R^(-5)*P24(cos(beta)));
S25:=simplify(R^(-6)*P25(cos(beta))):
S26:=simplify(R^(-7)*P26(cos(beta))):
S27:=simplify(R^(-8)*P27(cos(beta))):
S28:=simplify(R^(-9)*P28(cos(beta))):
S29:=simplify(R^(-10)*P29(cos(beta))):
#
#Repeat for P^3_i.
#
d3P3:=unapply(diff(d2P3(x),x),x):
d3P4:=unapply(diff(d2P4(x),x),x):
d3P5:=unapply(diff(d2P5(x),x),x):
d3P6:=unapply(diff(d2P6(x),x),x):
d3P7:=unapply(diff(d2P7(x),x),x):
d3P8:=unapply(diff(d2P8(x),x),x):
d3P9:=unapply(diff(d2P9(x),x),x):
#
P33:=unapply(-(1-x^2)^(3/2)*d3P3(x),x):
P34:=unapply(-(1-x^2)^(3/2)*d3P4(x),x):
P35:=unapply(-(1-x^2)^(3/2)*d3P5(x),x):
P36:=unapply(-(1-x^2)^(3/2)*d3P6(x),x):
P37:=unapply(-(1-x^2)^(3/2)*d3P7(x),x):
P38:=unapply(-(1-x^2)^(3/2)*d3P8(x),x):
P39:=unapply(-(1-x^2)^(3/2)*d3P9(x),x):
#
#The following functions will yield harmonic potentials
#if multiplied by either sin(3*theta) or cos(3*theta).
#
#Bounded potentials
#
F33:=simplify(R^3*P33(cos(beta)));
F34:=simplify(R^4*P34(cos(beta)));
F35:=simplify(R^5*P35(cos(beta)));
F36:=simplify(R^6*P36(cos(beta))):
F37:=simplify(R^7*P37(cos(beta))):
F38:=simplify(R^8*P38(cos(beta))):
F39:=simplify(R^9*P39(cos(beta))):
#
#Singular potentials
#
S33:=simplify(R^(-4)*P33(cos(beta)));
S34:=simplify(R^(-5)*P34(cos(beta)));
S35:=simplify(R^(-6)*P35(cos(beta)));
S36:=simplify(R^(-7)*P36(cos(beta))):
S37:=simplify(R^(-8)*P37(cos(beta))):
S38:=simplify(R^(-9)*P38(cos(beta))):
S39:=simplify(R^(-10)*P39(cos(beta))):