#SPHERICAL HARMONICS IN SPHERICAL POLAR COORDINATES
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#Legendre polynomials as functions of x
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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):
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#Simplified forms of Legendre polynomials.
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P1:=unapply(P01,x):
P2:=unapply(P02,x):
P3:=unapply(P03,x):
P4:=unapply(P04,x):
P5:=unapply(P05,x):
P6:=unapply(P06,x):
P7:=unapply(P07,x):
P8:=unapply(P08,x):
P9:=unapply(P09,x):
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#Harmonic axisymmetric bounded potentials in terms of R,beta
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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)))):
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#Harmonic axisymmetric singular potentials in terms of R,beta
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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)))):
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