Frequency-domain effective source for a point mass on a circular orbit in Schwarzschild spacetime

Barry Wardell and Niels Warburton

This notebook gives expressions for the punctures used in computing an effective source for a point mass on a circular orbit around a Schwarzschild black hole with mass M. We parametrize the orbit by its radius (Subscript[r, 0]) and define Δr=r-(Subscript[r, 0]). We label a spherical harmonic mode by l and m.

A definition of the punctures and a description of their calculation was given in the paper with which this notebook is included.

Updates to this notebook will be posted to: http://www.barrywardell.net/research/code/effectivesource.

Low-order punctures

Λ1=((l(l+1))/((2l-1)(2l+3)));
Λ2=(((l-1)l(l+1)(l+2))/((2l-3)(2l-1)(2l+3)(2l+5)));

hP[1]=(Sqrt[2]) (r0+Δr)WignerD[{l,m,0},π,π/2,π/2]((Sqrt[2 π])/(Sqrt[2l+1]))(((8(((r0-2 M))^(3/2)) EllipticK[(M/(r0-2 M))])/((r0^2)π (((r0-3M))^(1/2))))-(1+2 l)((2 (r0-2 M))/((r0^(5/2))(((r0-3M))^(1/2))))(Sqrt[((Δr)^2)])+((4 (((r0-2 M))^(1/2)) Δr )/(π (r0^3) (((r0-3M))^(1/2))))((r0-2 M) EllipticE[-(M/(2 M-r0))]-2 (r0-4 M) EllipticK[-(M/(2 M-r0))]));

hP[2]=If[l<1,0,(Sqrt[2]) (r0+Δr) ((Sqrt[2π])/((Sqrt[l(1+l)])(Sqrt[2l+1]))) (1-((2 M)/(r0+Δr))) ((64(Sqrt[r0-2M])((r0-2 M) EllipticE[-(M/(2 M-r0))]-(r0-3 M) EllipticK[-(M/(2 M-r0))]))/((Sqrt[r0-3M])π (r0^(3/2))(Sqrt[M])))Λ1(-WignerD[{l,m,-1},π,(π/2),(π/2)]+WignerD[{l,m,1},π,(π/2),(π/2)])];

hP[3]=(Sqrt[2]) (r0+Δr) ((Sqrt[2 π])/(Sqrt[2l+1]))(((1-((2 M)/(r0+Δr))))^(-1))(((8(((r0-2 M))^(3/2)) EllipticK[(M/(r0-2 M))])/((r0^2)π (((r0-3M))^(1/2))))-(1+2 l)((2 (r0-2 M))/((r0^(5/2))(((r0-3M))^(1/2))))(Sqrt[((Δr)^2)])+((4 (((r0-2 M))^(1/2)) Δr )/(π (r0^3) (((r0-3M))^(1/2))))((r0-2 M) EllipticE[-(M/(2 M-r0))]-2 (r0-4 M) EllipticK[-(M/(2 M-r0))]))WignerD[{l,m,0},π,(π/2),(π/2)];

hP[4]=If[l<1,0,(Sqrt[2]) (r0+Δr)(Sqrt[l (1+l)])(1/(r0+Δr))((Sqrt[2π])/(Sqrt[2l+1]))(- ((2(2l+1)(Sqrt[M]) (Sqrt[((Δr)^2)]))/(r0(Sqrt[r0-3 M])))-Λ1((64(((r0-2 M))^(3/2))(EllipticE[(M/(r0-2 M))]-EllipticK[(M/(r0-2 M))]))/((r0^(1/2))(M^(1/2))π(((r0-3 M))^(1/2))))+(1/((-1+2 l) (3+2 l)))((48(((r0-2M))^(1/2))(2(r0-2 M) EllipticE[(M/(r0-2 M))]-(2r0-5 M) EllipticK[(M/(r0-2 M))]))/((r0^(1/2))(M^(1/2))π(((r0-3M))^(1/2))))+Δr (Λ1((32(((r0-2M))^(1/2))((r0-4 M) EllipticE[(M/(r0-2 M))]-(r0-5 M) EllipticK[(M/(r0-2 M))]))/((M^(1/2))(r0^(3/2))π(((r0-3M))^(1/2))))-(1/((-1+2 l) (3+2 l)))((24((r0-2M)(2r0-9M) EllipticE[(M/(r0-2 M))]-2(11 (M^2)-7 M r0+(r0^2)) EllipticK[(M/(r0-2 M))]))/((M^(1/2))(r0^(3/2))π(((r0-3M))^(1/2))(((r0-2M))^(1/2)))))) (-WignerD[{l,m,-1},π,(π/2),(π/2)]+WignerD[{l,m,1},π,(π/2),(π/2)])];

hP[5]=If[l<2,0,(Sqrt[2])(r0+Δr)(Sqrt[l (1+l)])(((Sqrt[2π])(Sqrt[l(l+1)]))/((Sqrt[2l+1])(Sqrt[(l-1)l(l+1)(l+2)])))((1 )/(r0+Δr))(1-((2 M)/(r0+Δr)))(((256((4r0-11 M) (r0-2 M) EllipticE[(M/(r0-2 M))]-(4r0-9 M) (r0-3 M) EllipticK[(M/(r0-2 M))]))/( M π(Sqrt[r0-2M])(Sqrt[r0-3M])))Λ2-(((l-1)(l+2))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l)))((640((8r0-23 M)(r0-2 M) EllipticE[(M/(r0-2 M))]-(8r0-19 M)(r0-3 M) EllipticK[(M/(r0-2 M))]))/( M π(Sqrt[r0-2M])(Sqrt[r0-3M]))))(WignerD[{l,m,-2},π,(π/2),(π/2)]+WignerD[{l,m,2},π,(π/2),(π/2)])]+(Sqrt[2])(r0+Δr)(Sqrt[l (1+l)])((Sqrt[2π])/((Sqrt[l(l+1)])(Sqrt[2l+1])))(1-((2 M)/(r0+Δr)))(1/(r0+Δr))(Λ1(64/(π(((r0-2 M))^(1/2))(((r0-3M))^(1/2))))  (-(r0-2 M) EllipticE[-(M/(2 M-r0))]+(r0-3 M) EllipticK[-(M/(2 M-r0))]))WignerD[{l,-m,0},0,(π/2),(π/2)];

hP[6]=(Sqrt[2])(r0+Δr)((Sqrt[2 π])/(Sqrt[1+2 l]))(1/(((r0+Δr))^2)) (-((2 (1+2 l) M (Sqrt[r0]) (Sqrt[((Δr)^2)]))/((Sqrt[r0-3 M]) (r0-2 M)))+((8 M r0 EllipticK[(M/(r0-2 M))])/(π (Sqrt[r0-3 M]) (Sqrt[r0-2 M])))+((4 M Δr (EllipticE[(M/(-2 M+r0))]+2 EllipticK[(M/(r0-2 M))]))/(π (Sqrt[-3 M+r0]) (Sqrt[r0-2 M]))))WignerD[{l,m,0},π,(π/2),(π/2)];

hP[7]=If[l<2,0,(Sqrt[2])(r0+Δr)(Sqrt[(-1+l) l (1+l) (2+l)]) ((Sqrt[2π])/(Sqrt[2l+1]))(1/(((Δr+r0))^2))(-(1+2 l)((M (r0^(1/2)))/((r0-2 M)(((r0-3M))^(1/2))))  (Sqrt[((Δr)^2)])-Λ2((128 r0 (4 (r0-2M)(2r0-5M) EllipticE[(M/(r0-2M))]-(51 (M^2)-40 M r0+8 (r0^2)) EllipticK[(M/(r0-2M))]))/(3 M π (((r0-2 M))^(1/2))(((r0-3M))^(1/2))))+(1/( (-1+2 l) (3+2 l)))((160 r0 (8 (r0-2M)(2r0-5M)  EllipticE[(M/(r0-2M))]-(4r0-9M)(4r0-11M) EllipticK[(M/(r0-2M))]))/(3 M π (((r0-2 M))^(1/2))(((r0-3M))^(1/2))))+Δr (Λ2((64 ((67 (M^2)-48 M r0+8 (r0^2)) EllipticE[(M/(r0-2M))]-2 (39 (M^2)-26 M r0+4 (r0^2)) EllipticK[(M/(r0-2M))]))/(3M π  (((r0-2 M))^(1/2))(((r0-3M))^(1/2))))-(80((131 (M^2)-96 M r0+16 (r0^2)) EllipticE[(M/(r0-2M))]-2 (81 (M^2)-52 M r0+8 (r0^2)) EllipticK[(M/(r0-2M))]))/(3 (-1+2 l) (3+2 l) M π (((r0-2 M))^(1/2))(((r0-3 M))^(1/2)))))(WignerD[{l,m,-2},π,(π/2),(π/2)]+WignerD[{l,m,2},π,(π/2),(π/2)])];

hP[8]=If[l<1,0,(Sqrt[2]) (r0+Δr)(Sqrt[l (1+l)])(1/(r0+Δr))((I(Sqrt[2π]))/(Sqrt[2l+1]))((2 l+1)((2(M^(1/2))(Sqrt[((Δr)^2)]))/((((r0-3M))^(1/2))r0))-Λ1((64 (((r0-2 M))^(1/2)) ((r0-2 M) EllipticE[(M/(r0-2 M))]-(r0-3 M) EllipticK[(M/(r0-2 M))]))/((M^(1/2))(r0^(1/2)) π(((r0-3M))^(1/2))))+(1/((-1+2 l) (3+2 l)))((48 (((r0-2M))^(1/2))(2(r0-2M)EllipticE[(M/(r0-2 M))]-(2r0-5M)EllipticK[(M/(r0-2 M))]))/( (M^(1/2))(r0^(1/2))  π(((r0-3M))^(1/2))))+Δr (Λ1((32((r0-2M)(r0-5M) EllipticE[(M/(r0-2 M))]-(r0-3M)(r0-4M)  EllipticK[(M/(r0-2 M))]))/( ((M )^(1/2))(r0^(3/2)) π(((r0-2M))^(1/2))(((r0-3M))^(1/2))))-(1/((-1+2 l) (3+2 l)))((24((r0-2M)(2r0-9 M) EllipticE[(M/(r0-2 M))]-2(11 (M^2)-7 M r0+(r0^2)) EllipticK[(M/(r0-2 M))]))/( ((M )^(1/2))(r0^(3/2))π (((r0-3M))^(1/2))(((r0-2M))^(1/2))))))(WignerD[{l,m,-1},π,(π/2),(π/2)]+WignerD[{l,m,1},π,(π/2),(π/2)])];

hP[9]=If[l<2,0,(Sqrt[2]) (r0+Δr)(Sqrt[l (1+l)])(((Sqrt[2π])(Sqrt[l(l+1)]))/((Sqrt[2l+1])(Sqrt[(l-1)l(l+1)(l+2)])))(I/(r0+Δr))(1-((2 M)/(r0+Δr)))(((512(Sqrt[r0-3M])(2(r0-2 M)EllipticE[(M/(r0-2 M))]-(2r0-5 M) EllipticK[(M/(r0-2 M))]))/( M π(Sqrt[r0-2M])))Λ2-(((l-1)(l+2))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l)))((640((8r0-23 M) (r0-2 M)EllipticE[(M/(r0-2 M))]-(8r0-19 M) (r0-3 M)EllipticK[(M/(r0-2 M))]))/( M π(Sqrt[r0-2M])(Sqrt[r0-3M]))))(-WignerD[{l,m,-2},π,(π/2),(π/2)]+WignerD[{l,m,2},π,(π/2),(π/2)])];

hP[10]=If[l<2,0,(Sqrt[2])(r0+Δr)(Sqrt[(-1+l) l (1+l) (2+l)])((Sqrt[2π])/(Sqrt[2l+1]))(I/(((Δr+r0))^2))((1+2 l)((M (r0^(1/2))(Sqrt[((Δr)^2)]))/((r0-2 M)(((r0-3M))^(1/2))))-Λ2((512  r0 (((r0-2M))^(1/2))((2r0-5 M) EllipticE[(M/(r0-2M))]-2 (r0-3 M) EllipticK[(M/(r0-2M))]))/(3M π (((r0-3M))^(1/2)))) +(1/((-1+2 l) (3+2 l)))((160 r0 (8 (r0-2M)(2r0-5M) EllipticE[(M/(r0-2M))]-(4r0-9M)(4r0-11M)EllipticK[(M/(r0-2M))]))/(3 M π (((r0-2 M))^(1/2))(((r0-3M))^(1/2))))+Δr (Λ2((256 (2 (r0-4M)(r0-2M) EllipticE[(M/(r0-2M))]-(r0-3M)(2r0-7M) EllipticK[(M/(r0-2M))]))/(3 M π (((r0-2 M))^(1/2))(((r0-3M))^(1/2))))-(1/( (-1+2 l) (3+2 l)))((80((131 (M^2)-96 M r0+16 (r0^2)) EllipticE[(M/(r0-2M))]-2 (81 (M^2)-52 M r0+8 (r0^2)) EllipticK[(M/(r0-2M))]))/(3 M π (((r0-2 M))^(1/2))(((r0-3M))^(1/2))))))(-WignerD[{l,m,-2},π,(π/2),(π/2)]+WignerD[{l,m,2},π,(π/2),(π/2)])];

High-order punctures

hP[1]=(Sqrt[2]) (r0+Δr)WignerD[{l,m,0},π,π/2,π/2]((Sqrt[2π])/(Sqrt[2l+1]))(-((2 (1+2 l) (-2 M+r0)  (Sqrt[((Δr)^2)]))/((r0^(5/2)) (Sqrt[-3 M+r0])))+(((((1+2 l))^2) (Sqrt[-2 M+r0]) ((Δr)^2) EllipticE[(M/(r0-2M))])/(π (r0^3) (Sqrt[-3 M+r0])))+((8 (((-2 M+r0))^(3/2)) EllipticK[(M/(r0-2M))])/(π (r0^2) (Sqrt[-3 M+r0])))+((2 (1+2 l) (Sqrt[-3 M+r0]) Δr  (Sqrt[((Δr)^2)]))/(r0^(7/2)))+((4 (Sqrt[-2 M+r0]) Δr ((-2 M+r0) EllipticE[(M/(r0-2M))]-2 (-4 M+r0) EllipticK[(M/(r0-2M))]))/(π (r0^3) (Sqrt[-3 M+r0])))+(1/(2 π (r0^4) (((-3 M+r0))^(3/2))))(Sqrt[-2 M+r0]) ((Δr)^2) ((14 (M^2)-5 M r0+(r0^2)) EllipticE[(M/(r0-2M))]+5 (-7 M+r0) (-3 M+r0) EllipticK[(M/(r0-2M))])+(4 (((-2 M+r0))^(3/2)) ((17 M-7 r0) (-2 M+r0) EllipticE[(M/(r0-2M))]+(-3 M+r0) (-32 M+7 r0) EllipticK[(M/(r0-2M))]))/((-1+2 l) (3+2 l) π (r0^3) (((-3 M+r0))^(3/2)))+((Sqrt[-2 M+r0]) Δr ((3952 (M^4)-5402 (M^3) r0+2689 (M^2) (r0^2)-510 M (r0^3)+23 (r0^4)) EllipticE[(M/(r0-2M))]+(-9246 (M^4)+9973 (M^3) r0-3767 (M^2) (r0^2)+559 M (r0^3)-23 (r0^4)) EllipticK[(M/(r0-2M))]))/((-1+2 l) (3+2 l) 2 π (r0^4) (((-3 M+r0))^(5/2))))+If[l<2,0,(Sqrt[2]) (r0+Δr)(WignerD[{l,m,-2},π,(π/2),(π/2)]+WignerD[{l,m,2},π,(π/2),(π/2)])((Sqrt[(l-1)l(l+1)(l+2)])/((Sqrt[2l+1])(Sqrt[2π])))(((8((Δr)^2)(Sqrt[r0-2 M]))/(3 M (r0^3) (Sqrt[r0-3 M]))) ((2r0-5 M) EllipticE[(M/(r0-2M))]-2 (r0-3 M) EllipticK[(M/(r0-2M))])- (1/((-1+2 l) (3+2 l)))((192(((r0-2 M))^(3/2)))/(3 M (r0^2) (Sqrt[r0-3 M])))(-2(r0-2 M) EllipticE[-(M/(2 M-r0))]+(2 r0-5 M) EllipticK[(M/(r0-2M))])-((32  (Sqrt[-2 M+r0]) Δr)/(M (r0^3) (Sqrt[-3 M+r0]))) ((42 (M^2)-33 M r0+6 (r0^2)) EllipticE[-(M/(2 M-r0))]-2 (26 (M^2)-18 M r0+3 (r0^2)) EllipticK[(M/(r0-2M))])((1/((-1+2 l) (3+2 l))))+(1/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l)))(32/( M (r0^3))) (((((-2 M+r0)/(-3 M+r0))))^(3/2)) ((1022 (M^3)-913 (M^2) r0+217 M (r0^2)-8 (r0^3)) EllipticE[-(M/(2 M-r0))]+(-1284 (M^3)+1019 (M^2) r0-221 M (r0^2)+8 (r0^3)) EllipticK[(M/(r0-2M))])-(1/((-1+2 l) (3+2 l)))((4 (Sqrt[r0-2 M]) ((Δr)^2))/( M (r0^4) (((r0-3 M))^(3/2)))) ((590 (M^3)-731 (M^2) r0+287 M (r0^2)-36 (r0^3)) EllipticE[(M/(r0-2M))]+(-741 (M^3)+838 (M^2) r0-305 M (r0^2)+36 (r0^3)) EllipticK[(M/(r0-2M))])-(1/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l)))((4 (Sqrt[r0-2 M]) Δr )/(( M (r0^4) (((r0-3 M))^(5/2)))))((104536 (M^5)-153254 (M^4) r0+81693 (M^3) (r0^2)-18606 (M^2) (r0^3)+1451 M (r0^4)+20 (r0^5)) EllipticE[(M/(r0-2M))]+(r0-3 M) (43678 (M^4)-45091 (M^3) r0+14858 (M^2) (r0^2)-1501 M (r0^3)-20 (r0^4)) EllipticK[(M/(r0-2M))]))];

hP[2]=If[l<1,0,(Sqrt[2]) (r0+Δr) ((32 (Sqrt[l (1+l)]) (Sqrt[(2/π)]) (Sqrt[-2 M+r0]) Δr (1-((2 M)/(r0+Δr))) ((-27 (M^2)+19 M r0-3 (r0^2)) EllipticE[-(M/(2 M-r0))]+3 (12 (M^2)-7 M r0+(r0^2)) EllipticK[-(M/(2 M-r0))]))/((-1+2 l) (Sqrt[1+2 l]) (3+2 l) (Sqrt[M]) (r0^(5/2)) (((-3 M+r0))^(3/2)))+((Sqrt[l (1+l)]) (Sqrt[2 π]) (1-((2 M)/(r0+Δr))) (((8 (-3 M+r0) ((Δr)^2) (EllipticE[-(M/(2 M-r0))]-EllipticK[-(M/(2 M-r0))]))/(π (r0^(5/2))))+((64 (-2 M+r0) ((-2 M+r0) EllipticE[-(M/(2 M-r0))]-(-3 M+r0) EllipticK[-(M/(2 M-r0))]))/((-1+2 l) (3+2 l) π (r0^(3/2))))))/((Sqrt[1+2 l]) (Sqrt[M]) (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))+(1/(Sqrt[1+2 l]))(Sqrt[l (1+l)]) (Sqrt[2 π]) (1-((2 M)/(r0+Δr))) (-((((Δr)^2) ((444 (M^3)-618 (M^2) r0+258 M (r0^2)-32 (r0^3)) EllipticE[-(M/(2 M-r0))]+(-3 M+r0) (189 (M^2)-181 M r0+32 (r0^2)) EllipticK[-(M/(2 M-r0))]))/((-1+2 l) (3+2 l) (Sqrt[M]) π (r0^(7/2)) (((-3 M+r0))^(3/2)) (Sqrt[-2 M+r0])))+(8 (Sqrt[-2 M+r0]) ((2 M-r0) (143 (M^3)-22 (M^2) r0-31 M (r0^2)+6 (r0^3)) EllipticE[-(M/(2 M-r0))]+2 (-3 M+r0) (71 (M^3)-12 (M^2) r0-14 M (r0^2)+3 (r0^3)) EllipticK[-(M/(2 M-r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (Sqrt[M]) π (r0^(5/2)) (((-3 M+r0))^(5/2))))) (-WignerD[{l,m,-1},π,(π/2),(π/2)]+WignerD[{l,m,1},π,(π/2),(π/2)])];

hP[3]=If[l<2,0,(1/(Sqrt[1+2 l]))2 (Sqrt[(-1+l) l (1+l) (2+l)]) (Sqrt[π]) (r0+Δr) (((1-((2 M)/(r0+Δr))))^(-1)) (-((16 (Sqrt[-2 M+r0]) Δr (3 (-2 M+r0) (-7 M+2 r0) EllipticE[-(M/(2 M-r0))]-2 (26 (M^2)-18 M r0+3 (r0^2)) EllipticK[-(M/(2 M-r0))]))/((-1+2 l) (3+2 l) M π (r0^3) (Sqrt[-3 M+r0])))-(2 ((Δr)^2) ((-2 M+r0) (2174 (M^3)-2459 (M^2) r0+879 M (r0^2)-100 (r0^3)) EllipticE[-(M/(2 M-r0))]-(-3 M+r0) (1790 (M^3)-2161 (M^2) r0+829 M (r0^2)-100 (r0^3)) EllipticK[-(M/(2 M-r0))]))/((-1+2 l) (3+2 l) M π (r0^4) (((-3 M+r0))^(3/2)) (Sqrt[-2 M+r0]))-(16 (((((-2 M+r0)/(-3 M+r0))))^(3/2)) ((466 (M^3)-767 (M^2) r0+375 M (r0^2)-56 (r0^3)) EllipticE[-(M/(2 M-r0))]+(-588 (M^3)+901 (M^2) r0-403 M (r0^2)+56 (r0^3)) EllipticK[-(M/(2 M-r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π (r0^3))+(2 (Sqrt[-2 M+r0]) Δr ((80360 (M^5)-141274 (M^4) r0+96259 (M^3) (r0^2)-31970 (M^2) (r0^3)+5221 M (r0^4)-340 (r0^5)) EllipticE[-(M/(2 M-r0))]+(-3 M+r0) (33650 (M^4)-44381 (M^3) r0+21110 (M^2) (r0^2)-4371 M (r0^3)+340 (r0^4)) EllipticK[-(M/(2 M-r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π (r0^4) (((-3 M+r0))^(5/2)))+((4 (Sqrt[-2 M+r0]) ((Δr)^2) ((-5 M+2 r0) EllipticE[(M/(-2 M+r0))]-2 (-3 M+r0) EllipticK[(M/(-2 M+r0))]))/(3 M π (r0^3) (Sqrt[-3 M+r0])))-((32 (((-2 M+r0))^(3/2)) (-2 (-2 M+r0) EllipticE[(M/(-2 M+r0))]+(-5 M+2 r0) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) M π (r0^2) (Sqrt[-3 M+r0])))) (WignerD[{l,m,-2},π,(π/2),(π/2)]+WignerD[{l,m,2},π,(π/2),(π/2)])]+(1/(Sqrt[1+2 l]))2 (Sqrt[π]) (r0+Δr) (((1-((2 M)/(r0+Δr))))^(-1)) (-((2 (1+2 l) (-2 M+r0) (Sqrt[((Δr)^2)]))/((r0^(5/2)) (Sqrt[-3 M+r0])))+((2 (1+2 l) (Sqrt[-3 M+r0]) Δr (Sqrt[((Δr)^2)]))/(r0^(7/2)))+(((((1+2 l))^2) (Sqrt[-2 M+r0]) ((Δr)^2) EllipticE[(M/(-2 M+r0))])/(π (r0^3) (Sqrt[-3 M+r0])))+((8 (((-2 M+r0))^(3/2)) EllipticK[(M/(-2 M+r0))])/(π (r0^2) (Sqrt[-3 M+r0])))+((4 (Sqrt[-2 M+r0]) Δr ((-2 M+r0) EllipticE[(M/(-2 M+r0))]-2 (-4 M+r0) EllipticK[(M/(-2 M+r0))]))/(π (r0^3) (Sqrt[-3 M+r0])))+(4 (((-2 M+r0))^(3/2)) ((14 (M^2)-17 M r0+9 (r0^2)) EllipticE[(M/(-2 M+r0))]+(16 M-9 r0) (-3 M+r0) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) π (r0^3) (((-3 M+r0))^(3/2)))+(((Δr)^2) ((2 M-r0) (130 (M^2)-91 M r0+15 (r0^2)) EllipticE[(M/(-2 M+r0))]+(-3 M+r0) (214 (M^2)-157 M r0+21 (r0^2)) EllipticK[(M/(-2 M+r0))]))/(2 π (r0^4) (((-3 M+r0))^(3/2)) (Sqrt[-2 M+r0]))-((Sqrt[-2 M+r0]) Δr ((2960 (M^4)-4294 (M^3) r0+2367 (M^2) (r0^2)-498 M (r0^3)+25 (r0^4)) EllipticE[(M/(-2 M+r0))]+(-7602 (M^4)+8603 (M^3) r0-3433 (M^2) (r0^2)+545 M (r0^3)-25 (r0^4)) EllipticK[(M/(-2 M+r0))]))/(2 (-1+2 l) (3+2 l) π (r0^4) (((-3 M+r0))^(5/2)))) WignerD[{l,m,0},π,(π/2),(π/2)];

hP[4]=If[l<1,0,(1/(Sqrt[1+2 l]))2 (Sqrt[l (1+l)]) (Sqrt[π]) (-((2 (1+2 l) M ( (Sqrt[((Δr)^2)])))/(r0 (Sqrt[M (-3 M+r0)])))-((16 (((-2 M+r0))^(3/2)) (EllipticE[(M/(-2 M+r0))]-EllipticK[(M/(-2 M+r0))]))/((Sqrt[M]) π (Sqrt[r0]) (Sqrt[-3 M+r0])))-(40 (Sqrt[M]) (Sqrt[-2 M+r0]) ((82 (M^2)-39 M r0-(r0^2)) EllipticE[(M/(-2 M+r0))]-2 (51 (M^2)-20 M r0+(r0^2)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) π (r0^(3/2)) (((-3 M+r0))^(3/2)))-(8 (Sqrt[-2 M+r0]) (2 (12 (M^3)-8 (M^2) r0+3 M (r0^2)-(r0^3)) EllipticE[(M/(-2 M+r0))]+(-42 (M^3)+35 (M^2) r0-13 M (r0^2)+2 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) (Sqrt[M]) π (r0^(3/2)) (((-3 M+r0))^(3/2)))+Δr (((8 (Sqrt[M]) ((-2 M+r0) EllipticE[(M/(-2 M+r0))]+2 M EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) π (r0^(3/2)) (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+((8 (Sqrt[-2 M+r0]) ((-4 M+r0) EllipticE[(M/(-2 M+r0))]-(-5 M+r0) EllipticK[(M/(-2 M+r0))]))/((Sqrt[M]) π (r0^(3/2)) (Sqrt[-3 M+r0])))+((120 ((10 (M^2)-7 M r0+(r0^2)) EllipticE[(M/(-2 M+r0))]-(12 (M^2)-7 M r0+(r0^2)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (Sqrt[M]) π (r0^(3/2)) (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))))+((Δr)^2) (-((2 (((1+2 l))^2) ((-4 M+r0) EllipticE[(M/(-2 M+r0))]-(-3 M+r0) EllipticK[(M/(-2 M+r0))]))/(3 (Sqrt[M]) π (r0^(3/2)) (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+((-230 (M^3)+251 (M^2) r0-83 M (r0^2)+8 (r0^3)) EllipticE[(M/(-2 M+r0))]+2 (141 (M^3)-146 (M^2) r0+45 M (r0^2)-4 (r0^3)) EllipticK[(M/(-2 M+r0))])/(π (3 M-r0) (r0^(5/2)) (Sqrt[M (-3 M+r0) (-2 M+r0)]))+(3 (2 (60 (M^3)-80 (M^2) r0+31 M (r0^2)-3 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-162 (M^3)+191 (M^2) r0-65 M (r0^2)+6 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) π (2 M-r0) (r0^(5/2)) (Sqrt[M (-3 M+r0) (-2 M+r0)])))) (-WignerD[{l,m,-1},π,(π/2),(π/2)]+WignerD[{l,m,1},π,(π/2),(π/2)])]+If[l<3,0,(1/(Sqrt[1+2 l]))2 (Sqrt[l (1+l)]) (Sqrt[(-2+l) (-1+l) (2+l) (3+l)]) (Sqrt[π]) (-((64 (((-2 M+r0))^(3/2)) ((17 M-8 r0) EllipticE[(M/(-2 M+r0))]+(-21 M+8 r0) EllipticK[(M/(-2 M+r0))]))/(3 (-1+2 l) (3+2 l) (M^(3/2)) π (Sqrt[r0]) (Sqrt[-3 M+r0])))-(4032 (Sqrt[-2 M+r0]) ((3802 (M^3)-4271 (M^2) r0+1541 M (r0^2)-178 (r0^3)) EllipticE[(M/(-2 M+r0))]+2 (-2355 (M^3)+2429 (M^2) r0-815 M (r0^2)+89 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (Sqrt[M]) π (r0^(3/2)) (((-3 M+r0))^(3/2)))+(32 (Sqrt[-2 M+r0]) (-4 (238 (M^4)+277 (M^3) r0-620 (M^2) (r0^2)+299 M (r0^3)-44 (r0^4)) EllipticE[(M/(-2 M+r0))]+(1182 (M^4)+1475 (M^3) r0-2891 (M^2) (r0^2)+1284 M (r0^3)-176 (r0^4)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (M^(3/2)) π (r0^(3/2)) (((-3 M+r0))^(3/2)))+(32 (Sqrt[-2 M+r0]) ((1886 (M^4)+4907 (M^3) r0-6097 (M^2) (r0^2)+1906 M (r0^3)-160 (r0^4)) EllipticE[(M/(-2 M+r0))]-2 (1155 (M^4)+3158 (M^3) r0-3440 (M^2) (r0^2)+993 M (r0^3)-80 (r0^4)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (M^(3/2)) π (r0^(3/2)) (((-3 M+r0))^(3/2)))+(1/(3 (M^(3/2)) π (r0^(3/2)) (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))32 Δr (-((210 ((398 (M^3)-477 (M^2) r0+187 M (r0^2)-24 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-492 (M^3)+545 (M^2) r0-199 M (r0^2)+24 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l)))+(1/((-1+2 l) (3+2 l)))((344 (M^3)-438 (M^2) r0+181 M (r0^2)-24 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-426 (M^3)+503 (M^2) r0-193 M (r0^2)+24 (r0^3)) EllipticK[(M/(-2 M+r0))])+(3 ((1666 (M^3)-2151 (M^2) r0+899 M (r0^2)-120 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-2064 (M^3)+2473 (M^2) r0-959 M (r0^2)+120 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l)))+((Δr)^2) ((8 ((36 (M^2)-35 M r0+8 (r0^2)) EllipticE[(M/(-2 M+r0))]+(-45 (M^2)+39 M r0-8 (r0^2)) EllipticK[(M/(-2 M+r0))]))/(15 (M^(3/2)) π (r0^(3/2)) (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))-(280 (4 (1862 (M^4)-3335 (M^3) r0+2170 (M^2) (r0^2)-604 M (r0^3)+60 (r0^4)) EllipticE[(M/(-2 M+r0))]+(-9222 (M^4)+15665 (M^3) r0-9633 (M^2) (r0^2)+2536 M (r0^3)-240 (r0^4)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (M^(3/2)) π (r0^(5/2)) (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2)))+(4 ((13398 (M^4)-22723 (M^3) r0+14177 (M^2) (r0^2)-3850 M (r0^3)+384 (r0^4)) EllipticE[(M/(-2 M+r0))]-2 (8289 (M^4)-13299 (M^3) r0+7847 (M^2) (r0^2)-2021 M (r0^3)+192 (r0^4)) EllipticK[(M/(-2 M+r0))]))/(3 (-1+2 l) (3+2 l) (M^(3/2)) π (r0^(5/2)) (((-3 M+r0))^(3/2)) (Sqrt[-2 M+r0]))-(4 (2 (69216 (M^5)-139190 (M^4) r0+110825 (M^3) (r0^2)-43615 (M^2) (r0^3)+8470 M (r0^4)-648 (r0^5)) EllipticE[(M/(-2 M+r0))]+(-171270 (M^5)+328731 (M^4) r0-250216 (M^3) (r0^2)+94323 (M^2) (r0^3)-17588 M (r0^4)+1296 (r0^5)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (M^(3/2)) π (r0^(5/2)) (((-3 M+r0))^(3/2)) (((-2 M+r0))^(3/2))))) (-WignerD[{l,m,-3},π,(π/2),(π/2)]+WignerD[{l,m,3},π,(π/2),(π/2)])];

hP[5]=If[l<2,0,(1/(Sqrt[1+2 l]))2 (Sqrt[(-1+l) l (1+l) (2+l)]) (Sqrt[π]) (1-((2 M)/(r0+Δr))) (Δr ((160 ((486 (M^4)-809 (M^3) r0+489 (M^2) (r0^2)-127 M (r0^3)+12 (r0^4)) EllipticE[-(M/(2 M-r0))]+(-594 (M^4)+942 (M^3) r0-539 (M^2) (r0^2)+133 M (r0^3)-12 (r0^4)) EllipticK[-(M/(2 M-r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π r0 (((-3 M+r0))^(3/2)) (((-2 M+r0))^(3/2)))-(32 ((450 (M^4)-763 (M^3) r0+471 (M^2) (r0^2)-125 M (r0^3)+12 (r0^4)) EllipticE[-(M/(2 M-r0))]-(558 (M^4)-894 (M^3) r0+521 (M^2) (r0^2)-131 M (r0^3)+12 (r0^4)) EllipticK[-(M/(2 M-r0))]))/((-1+2 l) (3+2 l) M π r0 (((-3 M+r0))^(3/2)) (((-2 M+r0))^(3/2))))-((384 ((2 M-r0) EllipticE[(M/(-2 M+r0))]+(-3 M+r0) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+((64 ((22 (M^2)-19 M r0+4 (r0^2)) EllipticE[(M/(-2 M+r0))]+(-27 (M^2)+21 M r0-4 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) M π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+((2240 ((26 (M^2)-21 M r0+4 (r0^2)) EllipticE[(M/(-2 M+r0))]+(-33 (M^2)+23 M r0-4 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) M π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+((Δr)^2) (((16 (Sqrt[-3 M+r0]) (EllipticE[(M/(-2 M+r0))]-EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) π r0 (((-2 M+r0))^(3/2))))+((120 (Sqrt[-3 M+r0]) ((9 M-4 r0) EllipticE[(M/(-2 M+r0))]+(-11 M+4 r0) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π r0 (((-2 M+r0))^(3/2))))-((8 (Sqrt[-3 M+r0]) ((7 M-4 r0) EllipticE[(M/(-2 M+r0))]+(-9 M+4 r0) EllipticK[(M/(-2 M+r0))]))/(3 M π r0 (((-2 M+r0))^(3/2)))))) (WignerD[{l,m,-2},π,(π/2),(π/2)]+WignerD[{l,m,2},π,(π/2),(π/2)])]+(1/(Sqrt[1+2 l]))2 l (1+l) (Sqrt[π]) (1-((2 M)/(r0+Δr))) (-((8 (Sqrt[-3 M+r0]) ((Δr)^2) (EllipticE[-(M/(2 M-r0))]-EllipticK[-(M/(2 M-r0))]))/(π r0 (((-2 M+r0))^(3/2))))+((64 ((2 M-r0) EllipticE[-(M/(2 M-r0))]+(-3 M+r0) EllipticK[-(M/(2 M-r0))]))/((-1+2 l) (3+2 l) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+((32 Δr ((-18 (M^3)+23 (M^2) r0-9 M (r0^2)+(r0^3)) EllipticE[(M/(-2 M+r0))]+(18 (M^3)-24 (M^2) r0+9 M (r0^2)-(r0^3)) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) π r0 (((-3 M+r0))^(3/2)) (((-2 M+r0))^(3/2))))) WignerD[{l,-m,0},0,(π/2),(π/2)];

hP[6]=If[l<2,0,(1/((Sqrt[1+2 l]) (r0+Δr)))2 (Sqrt[(-1+l) l (1+l) (2+l)]) (Sqrt[π]) (-((4 ((Δr)^2) ((5 M-2 r0) EllipticE[(M/(-2 M+r0))]+2 (-3 M+r0) EllipticK[(M/(-2 M+r0))]))/(3 π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2))))+(16 ((1/((-1+2 l) (3+2 l)))+(3/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l)))) Δr ((-3 M+2 r0) EllipticE[(M/(-2 M+r0))]-2 (-2 M+r0) EllipticK[(M/(-2 M+r0))]))/(π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))+((32 r0 (2 (-2 M+r0) EllipticE[(M/(-2 M+r0))]-(-5 M+2 r0) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+(16 ((1022 (M^4)-1697 (M^3) r0+1073 (M^2) (r0^2)-304 M (r0^3)+32 (r0^4)) EllipticE[(M/(-2 M+r0))]+(-1284 (M^4)+2003 (M^3) r0-1197 (M^2) (r0^2)+320 M (r0^3)-32 (r0^4)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π (((-3 M+r0))^(3/2)) (Sqrt[-2 M+r0]))+(2 ((Δr)^2) ((1282 (M^4)-2029 (M^3) r0+1217 (M^2) (r0^2)-324 M (r0^3)+32 (r0^4)) EllipticE[(M/(-2 M+r0))]+(-1563 (M^4)+2354 (M^3) r0-1343 (M^2) (r0^2)+340 M (r0^3)-32 (r0^4)) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) M π r0 (((-3 M+r0))^(3/2)) (((-2 M+r0))^(3/2)))+(2 Δr ((10064 (M^5)-18017 (M^4) r0+12332 (M^3) (r0^2)-3983 (M^2) (r0^3)+596 M (r0^4)-32 (r0^5)) EllipticE[(M/(-2 M+r0))]+(-12453 (M^5)+21071 (M^4) r0-13683 (M^3) (r0^2)+4229 (M^2) (r0^3)-612 M (r0^4)+32 (r0^5)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π r0 (((-3 M+r0))^(5/2)) (Sqrt[-2 M+r0]))) (WignerD[{l,m,-2},π,(π/2),(π/2)]+WignerD[{l,m,2},π,(π/2),(π/2)])]+(1/((Sqrt[1+2 l]) (r0+Δr)))2 (Sqrt[π]) (-((2 (1+2 l) M (Sqrt[r0]) (Sqrt[((Δr)^2)]))/((Sqrt[-3 M+r0]) (-2 M+r0)))-((2 (1+2 l) M (Sqrt[-3 M+r0]) Δr (Sqrt[((Δr)^2)]))/((Sqrt[r0]) (((-2 M+r0))^2)))+(((((1+2 l))^2) M ((Δr)^2) EllipticE[(M/(-2 M+r0))])/(π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2))))+((8 M r0 EllipticK[(M/(-2 M+r0))])/(π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+((4 M Δr (EllipticE[(M/(-2 M+r0))]+2 EllipticK[(M/(-2 M+r0))]))/(π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+(((Δr)^2) ((158 (M^3)-173 (M^2) r0+65 M (r0^2)-8 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-3 M+r0) (61 (M^2)-51 M r0+8 (r0^2)) EllipticK[(M/(-2 M+r0))]))/(2 π r0 (((-3 M+r0))^(3/2)) (((-2 M+r0))^(3/2)))-(4 ((2 M-r0) (17 (M^2)-15 M r0+8 (r0^2)) EllipticE[(M/(-2 M+r0))]+(-3 M+r0) (32 (M^2)-31 M r0+8 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) π (((-3 M+r0))^(3/2)) (Sqrt[-2 M+r0]))+(Δr ((-596 (M^4)+917 (M^3) r0-412 (M^2) (r0^2)+35 M (r0^3)+8 (r0^4)) EllipticE[(M/(-2 M+r0))]-(-3 M+r0) (197 (M^3)-164 (M^2) r0+11 M (r0^2)+8 (r0^3)) EllipticK[(M/(-2 M+r0))]))/(2 (-1+2 l) (3+2 l) π r0 (((-3 M+r0))^(5/2)) (Sqrt[-2 M+r0]))) WignerD[{l,m,0},π,(π/2),(π/2)];

hP[7]=If[l<2,0,(1/((Sqrt[1+2 l]) (r0+Δr)))2 (Sqrt[(-1+l) l (1+l) (2+l)]) (Sqrt[π]) (-(((1+2 l) M (Sqrt[r0]) (Sqrt[((Δr)^2)]))/((Sqrt[-3 M+r0]) (-2 M+r0)))-(((1+2 l) M (Sqrt[-3 M+r0]) Δr (Sqrt[((Δr)^2)]))/((Sqrt[r0]) (((-2 M+r0))^2)))+((32 M r0 EllipticK[(M/(-2 M+r0))])/((-1+2 l) (3+2 l) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+(168 r0 (20 (10 (M^2)-9 M r0+2 (r0^2)) EllipticE[(M/(-2 M+r0))]+(-247 (M^2)+200 M r0-40 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))-(8 r0 (4 (-2 M+r0) (-5 M+2 r0) EllipticE[(M/(-2 M+r0))]-(51 (M^2)-40 M r0+8 (r0^2)) EllipticK[(M/(-2 M+r0))]))/(3 M π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))+(5040 r0 (4 (-2 M+r0) (-5 M+2 r0) EllipticE[(M/(-2 M+r0))]-(51 (M^2)-40 M r0+8 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) M π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))+(48 ((34 (M^3)-79 (M^2) r0+47 M (r0^2)-8 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-96 (M^3)+125 (M^2) r0-55 M (r0^2)+8 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(4 (3 (154 (M^3)-155 (M^2) r0+47 M (r0^2)-4 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-552 (M^3)+511 (M^2) r0-145 M (r0^2)+12 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(16632 ((734 (M^3)-1097 (M^2) r0+517 M (r0^2)-76 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-1320 (M^3)+1511 (M^2) r0-585 M (r0^2)+76 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(108 ((3778 (M^3)-4519 (M^2) r0+1739 M (r0^2)-212 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-5400 (M^3)+5577 (M^2) r0-1895 M (r0^2)+212 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))+((Δr)^2) (-((4 M EllipticE[(M/(-2 M+r0))])/(π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2))))-(210 ((49 (M^2)-40 M r0+8 (r0^2)) EllipticE[(M/(-2 M+r0))]-4 (-3 M+r0) (-5 M+2 r0) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2)))+(5 ((433 (M^2)-360 M r0+72 (r0^2)) EllipticE[(M/(-2 M+r0))]-36 (15 (M^2)-11 M r0+2 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) M π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2)))-((((1+2 l))^2) ((41 (M^2)-40 M r0+8 (r0^2)) EllipticE[(M/(-2 M+r0))]-4 (15 (M^2)-11 M r0+2 (r0^2)) EllipticK[(M/(-2 M+r0))]))/(15 M π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2))))+Δr (((16 M (EllipticE[(M/(-2 M+r0))]+2 EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+(4 ((67 (M^2)-48 M r0+8 (r0^2)) EllipticE[(M/(-2 M+r0))]-2 (39 (M^2)-26 M r0+4 (r0^2)) EllipticK[(M/(-2 M+r0))]))/(3 M π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))-(2520 ((67 (M^2)-48 M r0+8 (r0^2)) EllipticE[(M/(-2 M+r0))]-2 (39 (M^2)-26 M r0+4 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) M π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))-(84 ((327 (M^2)-240 M r0+40 (r0^2)) EllipticE[(M/(-2 M+r0))]-2 (203 (M^2)-130 M r0+20 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+((Δr)^2) ((2 ((158 (M^3)-173 (M^2) r0+65 M (r0^2)-8 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-183 (M^3)+166 (M^2) r0-59 M (r0^2)+8 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) π r0 (((6 (M^2)-5 M r0+(r0^2)))^(3/2)))+(21 ((11734 (M^4)-20169 (M^3) r0+12025 (M^2) (r0^2)-2924 M (r0^3)+240 (r0^4)) EllipticE[(M/(-2 M+r0))]+(-14559 (M^4)+23618 (M^3) r0-13227 (M^2) (r0^2)+3044 M (r0^3)-240 (r0^4)) EllipticK[(M/(-2 M+r0))]))/(2 (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π (2 M-r0) (3 M-r0) r0 (Sqrt[6 (M^2)-5 M r0+(r0^2)]))+(315 ((2094 (M^4)-3757 (M^3) r0+2301 (M^2) (r0^2)-572 M (r0^3)+48 (r0^4)) EllipticE[(M/(-2 M+r0))]+(-2619 (M^4)+4458 (M^3) r0-2551 (M^2) (r0^2)+596 M (r0^3)-48 (r0^4)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) M π r0 (((6 (M^2)-5 M r0+(r0^2)))^(3/2)))+((-2094 (M^4)+3757 (M^3) r0-2301 (M^2) (r0^2)+572 M (r0^3)-48 (r0^4)) EllipticE[(M/(-2 M+r0))]+(2619 (M^4)-4458 (M^3) r0+2551 (M^2) (r0^2)-596 M (r0^3)+48 (r0^4)) EllipticK[(M/(-2 M+r0))])/(6 M π (2 M-r0) (3 M-r0) r0 (Sqrt[6 (M^2)-5 M r0+(r0^2)])))) (WignerD[{l,m,-2},π,(π/2),(π/2)]+WignerD[{l,m,2},π,(π/2),(π/2)])]+If[l<4,0,(1/((Sqrt[1+2 l]) (r0+Δr)))2 (Sqrt[(-3+l) (-2+l) (-1+l) l (1+l) (2+l) (3+l) (4+l)]) (Sqrt[π]) ((32 r0 (2 (-2 M+r0) (403 (M^2)-320 M r0+64 (r0^2)) EllipticE[(M/(-2 M+r0))]+(5 M-2 r0) (-21 M+8 r0) (-19 M+8 r0) EllipticK[(M/(-2 M+r0))]))/(15 (-1+2 l) (3+2 l) (M^2) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))-(1729728 r0 (2 (-2 M+r0) (403 (M^2)-320 M r0+64 (r0^2)) EllipticE[(M/(-2 M+r0))]+(5 M-2 r0) (-21 M+8 r0) (-19 M+8 r0) EllipticK[(M/(-2 M+r0))]))/((-9+2 l) (-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (11+2 l) (M^2) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))+(288 r0 (-2 (5722 (M^3)-7341 (M^2) r0+3136 M (r0^2)-448 (r0^3)) EllipticE[(M/(-2 M+r0))]+(14165 (M^3)-16866 (M^2) r0+6720 M (r0^2)-896 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (M^2) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))+(32 r0 (-2 (5690 (M^3)-7325 (M^2) r0+3136 M (r0^2)-448 (r0^3)) EllipticE[(M/(-2 M+r0))]+(14085 (M^3)-16834 (M^2) r0+6720 M (r0^2)-896 (r0^3)) EllipticK[(M/(-2 M+r0))]))/(3 (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (M^2) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))+(22176 r0 (2 (7094 (M^3)-9307 (M^2) r0+4032 M (r0^2)-576 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-17555 (M^3)+21422 (M^2) r0-8640 M (r0^2)+1152 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (M^2) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))-(48 ((1451242 (M^5)-379027 (M^4) r0-1880025 (M^3) (r0^2)+1683316 (M^2) (r0^3)-533248 M (r0^4)+59136 (r0^5)) EllipticE[(M/(-2 M+r0))]+(-1790940 (M^5)+304903 (M^4) r0+2290011 (M^3) (r0^2)-1887108 (M^2) (r0^3)+562816 M (r0^4)-59136 (r0^5)) EllipticK[(M/(-2 M+r0))]))/((-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (M^2) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(5559840 ((60794 (M^5)-37899 (M^4) r0-40705 (M^3) (r0^2)+46292 (M^2) (r0^3)-15616 M (r0^4)+1792 (r0^5)) EllipticE[(M/(-2 M+r0))]+(-74940 (M^5)+40031 (M^4) r0+51347 (M^3) (r0^2)-52196 (M^2) (r0^3)+16512 M (r0^4)-1792 (r0^5)) EllipticK[(M/(-2 M+r0))]))/((-11+2 l) (-9+2 l) (-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (11+2 l) (13+2 l) (M^2) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(16 ((29082 (M^5)+11909 (M^4) r0-71377 (M^3) (r0^2)+55188 (M^2) (r0^3)-16640 M (r0^4)+1792 (r0^5)) EllipticE[(M/(-2 M+r0))]-(35964 (M^5)+18033 (M^4) r0-85379 (M^3) (r0^2)+61604 (M^2) (r0^3)-17536 M (r0^4)+1792 (r0^5)) EllipticK[(M/(-2 M+r0))]))/(3 (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (M^2) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(48 ((103102 (M^5)+10823 (M^4) r0-198795 (M^3) (r0^2)+161116 (M^2) (r0^3)-49408 M (r0^4)+5376 (r0^5)) EllipticE[(M/(-2 M+r0))]-(127380 (M^5)+25067 (M^4) r0-239121 (M^3) (r0^2)+180108 (M^2) (r0^3)-52096 M (r0^4)+5376 (r0^5)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (M^2) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(5616 ((896282 (M^5)+492613 (M^4) r0-2416785 (M^3) (r0^2)+1838996 (M^2) (r0^3)-551168 M (r0^4)+59136 (r0^5)) EllipticE[(M/(-2 M+r0))]-(1108860 (M^5)+711217 (M^4) r0-2885571 (M^3) (r0^2)+2051748 (M^2) (r0^3)-580736 M (r0^4)+59136 (r0^5)) EllipticK[(M/(-2 M+r0))]))/((-9+2 l) (-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (11+2 l) (M^2) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))+Δr ((16 ((6451 (M^3)-7698 (M^2) r0+3008 M (r0^2)-384 (r0^3)) EllipticE[(M/(-2 M+r0))]-2 (19 M-8 r0) (21 M-8 r0) (10 M-3 r0) EllipticK[(M/(-2 M+r0))]))/(15 (-1+2 l) (3+2 l) (M^2) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))-(864864 ((6451 (M^3)-7698 (M^2) r0+3008 M (r0^2)-384 (r0^3)) EllipticE[(M/(-2 M+r0))]-2 (19 M-8 r0) (21 M-8 r0) (10 M-3 r0) EllipticK[(M/(-2 M+r0))]))/((-9+2 l) (-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (11+2 l) (M^2) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))+(144 ((45037 (M^3)-53806 (M^2) r0+21056 M (r0^2)-2688 (r0^3)) EllipticE[(M/(-2 M+r0))]+2 (-27850 (M^3)+30739 (M^2) r0-11200 M (r0^2)+1344 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (M^2) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))+(16 ((45085 (M^3)-53838 (M^2) r0+21056 M (r0^2)-2688 (r0^3)) EllipticE[(M/(-2 M+r0))]+2 (-27882 (M^3)+30755 (M^2) r0-11200 M (r0^2)+1344 (r0^3)) EllipticK[(M/(-2 M+r0))]))/(3 (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (M^2) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))-(11088 ((58299 (M^3)-69442 (M^2) r0+27072 M (r0^2)-3456 (r0^3)) EllipticE[(M/(-2 M+r0))]+2 (-36070 (M^3)+39653 (M^2) r0-14400 M (r0^2)+1728 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (M^2) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+((Δr)^2) (-((83160 ((5 M-2 r0) (387 (M^2)-320 M r0+64 (r0^2)) EllipticE[(M/(-2 M+r0))]+2 (-3 M+r0) (-21 M+8 r0) (-19 M+8 r0) EllipticK[(M/(-2 M+r0))]))/((-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (M^2) π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2))))-(4 ((1975 (M^3)-2390 (M^2) r0+960 M (r0^2)-128 (r0^3)) EllipticE[(M/(-2 M+r0))]+2 (-1221 (M^3)+1367 (M^2) r0-512 M (r0^2)+64 (r0^3)) EllipticK[(M/(-2 M+r0))]))/(3 (-1+2 l) (3+2 l) (M^2) π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2)))-(4 ((2015 (M^3)-2406 (M^2) r0+960 M (r0^2)-128 (r0^3)) EllipticE[(M/(-2 M+r0))]+2 (-1245 (M^3)+1375 (M^2) r0-512 M (r0^2)+64 (r0^3)) EllipticK[(M/(-2 M+r0))]))/(105 (M^2) π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2)))+(1620 ((13465 (M^3)-16586 (M^2) r0+6720 M (r0^2)-896 (r0^3)) EllipticE[(M/(-2 M+r0))]+2 (-8331 (M^3)+9497 (M^2) r0-3584 M (r0^2)+448 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (M^2) π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2)))-(4 ((68805 (M^3)-83522 (M^2) r0+33600 M (r0^2)-4480 (r0^3)) EllipticE[(M/(-2 M+r0))]+2 (-42543 (M^3)+47781 (M^2) r0-17920 M (r0^2)+2240 (r0^3)) EllipticK[(M/(-2 M+r0))]))/(3 (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (M^2) π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2))))+((Δr)^2) ((108108 ((390642 (M^5)-862915 (M^4) r0+740903 (M^3) (r0^2)-310320 (M^2) (r0^3)+63616 M (r0^4)-5120 (r0^5)) EllipticE[(M/(-2 M+r0))]+(-483435 (M^5)+1023438 (M^4) r0-840151 (M^3) (r0^2)+336688 (M^2) (r0^3)-66176 M (r0^4)+5120 (r0^5)) EllipticK[(M/(-2 M+r0))]))/((-9+2 l) (-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (11+2 l) (M^2) π r0 (((6 (M^2)-5 M r0+(r0^2)))^(3/2)))-(2 ((390642 (M^5)-862915 (M^4) r0+740903 (M^3) (r0^2)-310320 (M^2) (r0^3)+63616 M (r0^4)-5120 (r0^5)) EllipticE[(M/(-2 M+r0))]+(-483435 (M^5)+1023438 (M^4) r0-840151 (M^3) (r0^2)+336688 (M^2) (r0^3)-66176 M (r0^4)+5120 (r0^5)) EllipticK[(M/(-2 M+r0))]))/(15 (-1+2 l) (3+2 l) (M^2) π (2 M-r0) (3 M-r0) r0 (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(2 ((2676078 (M^5)-5954125 (M^4) r0+5135849 (M^3) (r0^2)-2158224 (M^2) (r0^3)+443776 M (r0^4)-35840 (r0^5)) EllipticE[(M/(-2 M+r0))]+(-3311973 (M^5)+7064082 (M^4) r0-5825209 (M^3) (r0^2)+2342032 (M^2) (r0^3)-461696 M (r0^4)+35840 (r0^5)) EllipticK[(M/(-2 M+r0))]))/(3 (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (M^2) π r0 (((6 (M^2)-5 M r0+(r0^2)))^(3/2)))-(18 ((2637134 (M^5)-5896605 (M^4) r0+5102201 (M^3) (r0^2)-2148880 (M^2) (r0^3)+442752 M (r0^4)-35840 (r0^5)) EllipticE[(M/(-2 M+r0))]+(-3263925 (M^5)+6997426 (M^4) r0-5787977 (M^3) (r0^2)+2332176 (M^2) (r0^3)-460672 M (r0^4)+35840 (r0^5)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (M^2) π r0 (((6 (M^2)-5 M r0+(r0^2)))^(3/2)))+(1386 ((3710498 (M^5)-8053835 (M^4) r0+6836367 (M^3) (r0^2)-2839600 (M^2) (r0^3)+577664 M (r0^4)-46080 (r0^5)) EllipticE[(M/(-2 M+r0))]+(-4591155 (M^5)+9544222 (M^4) r0-7747519 (M^3) (r0^2)+3079472 (M^2) (r0^3)-600704 M (r0^4)+46080 (r0^5)) EllipticK[(M/(-2 M+r0))]))/((-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (M^2) π r0 (((6 (M^2)-5 M r0+(r0^2)))^(3/2))))) (WignerD[{l,m,-4},π,(π/2),(π/2)]+WignerD[{l,m,4},π,(π/2),(π/2)])]+(1/((Sqrt[1+2 l]) (r0+Δr)))2 (-1+l) l (1+l) (2+l) (Sqrt[π]) ((4 (1+(3/((-1+2 l) (3+2 l)))-(18/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l)))) ((Δr)^2) ((-5 M+2 r0) EllipticE[(M/(-2 M+r0))]-2 (-3 M+r0) EllipticK[(M/(-2 M+r0))]))/(3 π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2)))+(1/(π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))16 ((1/((-1+2 l) (3+2 l)))+(3/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l)))+(30/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l)))) Δr ((-3 M+2 r0) EllipticE[(M/(-2 M+r0))]-2 (-2 M+r0) EllipticK[(M/(-2 M+r0))])+(1/(π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))32 ((1/((-1+2 l) (3+2 l)))+(3/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l)))+(30/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l)))) r0 (2 (-2 M+r0) EllipticE[(M/(-2 M+r0))]-(-5 M+2 r0) EllipticK[(M/(-2 M+r0))])+(16 (-(2 M-r0) (31 (M^2)-37 M r0+12 (r0^2)) EllipticE[(M/(-2 M+r0))]+(3 M-r0) (44 (M^2)-43 M r0+12 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(80 (-(2 M-r0) (31 (M^2)-37 M r0+12 (r0^2)) EllipticE[(M/(-2 M+r0))]+(3 M-r0) (44 (M^2)-43 M r0+12 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(3360 (-(2 M-r0) (31 (M^2)-37 M r0+12 (r0^2)) EllipticE[(M/(-2 M+r0))]+(3 M-r0) (44 (M^2)-43 M r0+12 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))+((Δr)^2) ((2 ((130 (M^3)-139 (M^2) r0+55 M (r0^2)-8 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-123 (M^3)+134 (M^2) r0-55 M (r0^2)+8 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) π r0 (((6 (M^2)-5 M r0+(r0^2)))^(3/2)))+(6 ((130 (M^3)-139 (M^2) r0+55 M (r0^2)-8 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-123 (M^3)+134 (M^2) r0-55 M (r0^2)+8 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) π r0 (((6 (M^2)-5 M r0+(r0^2)))^(3/2)))+(60 ((130 (M^3)-139 (M^2) r0+55 M (r0^2)-8 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-123 (M^3)+134 (M^2) r0-55 M (r0^2)+8 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) π r0 (((6 (M^2)-5 M r0+(r0^2)))^(3/2))))) WignerD[{l,-m,0},0,(π/2),(π/2)];

hP[8]=If[l<1,0,(1/(Sqrt[1+2 l]))2 I (Sqrt[l (1+l)]) (Sqrt[π]) (((2 (1+2 l) (Sqrt[M]) ( (Sqrt[((Δr)^2)])))/(r0 (Sqrt[-3 M+r0])))-((16 (Sqrt[-2 M+r0]) ((-2 M+r0) EllipticE[(M/(-2 M+r0))]+(3 M-r0) EllipticK[(M/(-2 M+r0))]))/((Sqrt[M]) π (Sqrt[r0]) (Sqrt[-3 M+r0])))-(40 (((-2 M+r0))^(3/2)) (2 (23 (M^2)-23 M r0+6 (r0^2)) EllipticE[(M/(-2 M+r0))]+(-39 (M^2)+49 M r0-12 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (Sqrt[M]) π (r0^(3/2)) (((-3 M+r0))^(3/2)))-(8 (Sqrt[-2 M+r0]) ((34 (M^3)-43 (M^2) r0+17 M (r0^2)-2 (r0^3)) EllipticE[(M/(-2 M+r0))]+2 (-9 (M^3)+21 (M^2) r0-9 M (r0^2)+(r0^3)) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) (Sqrt[M]) π (3 M-r0) (r0^(3/2)) (Sqrt[-3 M+r0]))+Δr (-((8 (Sqrt[M]) ((-2 M+r0) EllipticE[(M/(-2 M+r0))]+2 M EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) π (r0^(3/2)) (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+((120 (Sqrt[-2 M+r0]) ((-4 M+r0) EllipticE[(M/(-2 M+r0))]-(-5 M+r0) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (Sqrt[M]) π (r0^(3/2)) (Sqrt[-3 M+r0])))+((8 ((10 (M^2)-7 M r0+(r0^2)) EllipticE[(M/(-2 M+r0))]-(12 (M^2)-7 M r0+(r0^2)) EllipticK[(M/(-2 M+r0))]))/((Sqrt[M]) π (r0^(3/2)) (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))))+((Δr)^2) (-((2 (((1+2 l))^2) ((-M+r0) EllipticE[(M/(-2 M+r0))]-(-3 M+r0) EllipticK[(M/(-2 M+r0))]))/(3 (Sqrt[M]) π (r0^(3/2)) (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+(2 (60 (M^3)-76 (M^2) r0+27 M (r0^2)-2 (r0^3)) EllipticE[(M/(-2 M+r0))]-(162 (M^3)-179 (M^2) r0+55 M (r0^2)-4 (r0^3)) EllipticK[(M/(-2 M+r0))])/((Sqrt[M]) π (r0^(5/2)) (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2)))-(3 ((230 (M^3)-269 (M^2) r0+95 M (r0^2)-10 (r0^3)) EllipticE[(M/(-2 M+r0))]+2 (-141 (M^3)+155 (M^2) r0-51 M (r0^2)+5 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) (Sqrt[M]) π (r0^(5/2)) (((-3 M+r0))^(3/2)) (Sqrt[-2 M+r0])))) (WignerD[{l,m,-1},π,(π/2),(π/2)]+WignerD[{l,m,1},π,(π/2),(π/2)])]+If[l<3,0,(1/(Sqrt[1+2 l]))2 I (Sqrt[l (1+l)]) (Sqrt[(-2+l) (-1+l) (2+l) (3+l)]) (Sqrt[π]) ((64 (Sqrt[-2 M+r0]) ((46 (M^2)-39 M r0+8 (r0^2)) EllipticE[(M/(-2 M+r0))]+(-57 (M^2)+43 M r0-8 (r0^2)) EllipticK[(M/(-2 M+r0))]))/(3 (-1+2 l) (3+2 l) (M^(3/2)) π (Sqrt[r0]) (Sqrt[-3 M+r0]))-(4032 (((-2 M+r0))^(3/2)) (2 (237 (M^3)-660 (M^2) r0+431 M (r0^2)-80 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-585 (M^3)+1581 (M^2) r0-942 M (r0^2)+160 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) π (((M r0))^(3/2)) (((-3 M+r0))^(3/2)))-(96 (Sqrt[-2 M+r0]) (2 (3644 (M^4)-6132 (M^3) r0+4137 (M^2) (r0^2)-1311 M (r0^3)+160 (r0^4)) EllipticE[(M/(-2 M+r0))]+(-9030 (M^4)+14347 (M^3) r0-9245 (M^2) (r0^2)+2782 M (r0^3)-320 (r0^4)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) π (((M r0 (-3 M+r0)))^(3/2)))-(96 (Sqrt[-2 M+r0]) ((634 (M^4)+45 (M^3) r0-661 (M^2) (r0^2)+336 M (r0^3)-48 (r0^4)) EllipticE[(M/(-2 M+r0))]-2 (393 (M^4)+64 (M^3) r0-389 (M^2) (r0^2)+180 M (r0^3)-24 (r0^4)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) π (((M r0 (-3 M+r0)))^(3/2)))+(1/(3 (M^(3/2)) π (r0^(3/2)) (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))32 Δr ((210 ((-344 (M^3)+438 (M^2) r0-181 M (r0^2)+24 (r0^3)) EllipticE[(M/(-2 M+r0))]+(426 (M^3)-503 (M^2) r0+193 M (r0^2)-24 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l))+(1/((-1+2 l) (3+2 l)))((398 (M^3)-477 (M^2) r0+187 M (r0^2)-24 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-492 (M^3)+545 (M^2) r0-199 M (r0^2)+24 (r0^3)) EllipticK[(M/(-2 M+r0))])+(3 ((2044 (M^3)-2424 (M^2) r0+941 M (r0^2)-120 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-2526 (M^3)+2767 (M^2) r0-1001 M (r0^2)+120 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l)))+((Δr)^2) ((8 ((61 (M^2)-45 M r0+8 (r0^2)) EllipticE[(M/(-2 M+r0))]+(-75 (M^2)+49 M r0-8 (r0^2)) EllipticK[(M/(-2 M+r0))]))/(15 (M^(3/2)) π (r0^(3/2)) (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))-(4 (2 (3724 (M^4)-7006 (M^3) r0+4766 (M^2) (r0^2)-1385 M (r0^3)+144 (r0^4)) EllipticE[(M/(-2 M+r0))]+(-9222 (M^4)+16493 (M^3) r0-10611 (M^2) (r0^2)+2914 M (r0^3)-288 (r0^4)) EllipticK[(M/(-2 M+r0))]))/(3 (-1+2 l) (3+2 l) (M^(3/2)) π (2 M-r0) (r0^(5/2)) (Sqrt[(-3 M+r0) (-2 M+r0)]))-(280 ((13398 (M^4)-21985 (M^3) r0+13265 (M^2) (r0^2)-3484 M (r0^3)+336 (r0^4)) EllipticE[(M/(-2 M+r0))]-2 (8289 (M^4)-12840 (M^3) r0+7325 (M^2) (r0^2)-1826 M (r0^3)+168 (r0^4)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (M^(3/2)) π (r0^(5/2)) (((-3 M+r0))^(3/2)) (Sqrt[-2 M+r0]))-(4 ((107268 (M^5)-210880 (M^4) r0+159925 (M^3) (r0^2)-57995 (M^2) (r0^3)+9920 M (r0^4)-624 (r0^5)) EllipticE[(M/(-2 M+r0))]-4 (33210 (M^5)-62211 (M^4) r0+44956 (M^3) (r0^2)-15573 (M^2) (r0^3)+2558 M (r0^4)-156 (r0^5)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (M^(3/2)) π (2 M-r0) (3 M-r0) (r0^(5/2)) (Sqrt[(-3 M+r0) (-2 M+r0)])))) (WignerD[{l,m,-3},π,(π/2),(π/2)]+WignerD[{l,m,3},π,(π/2),(π/2)])];

hP[9]=If[l<2,0,(1/(Sqrt[1+2 l]))2 I (Sqrt[(-1+l) l (1+l) (2+l)]) (Sqrt[π]) (1-((2 M)/(r0+Δr))) (((4480 (Sqrt[-2 M+r0]) ((-5 M+2 r0) EllipticE[-(M/(2 M-r0))]-2 (-3 M+r0) EllipticK[-(M/(2 M-r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) M π (Sqrt[-3 M+r0])))-((384 ((-2 M+r0) EllipticE[-(M/(2 M-r0))]-(-3 M+r0) EllipticK[-(M/(2 M-r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+((128 (Sqrt[-3 M+r0]) (2 (-2 M+r0) EllipticE[-(M/(2 M-r0))]-(-5 M+2 r0) EllipticK[-(M/(2 M-r0))]))/((-1+2 l) (3+2 l) M π (Sqrt[-2 M+r0])))+((Δr)^2) (-((16 (Sqrt[-3 M+r0]) (EllipticE[-(M/(2 M-r0))]-EllipticK[-(M/(2 M-r0))]))/((-1+2 l) (3+2 l) π r0 (((-2 M+r0))^(3/2))))-((16 (Sqrt[-3 M+r0]) ((5 M-2 r0) EllipticE[-(M/(2 M-r0))]+2 (-3 M+r0) EllipticK[-(M/(2 M-r0))]))/(3 M π r0 (((-2 M+r0))^(3/2))))+((240 (Sqrt[-3 M+r0]) ((4 M-2 r0) EllipticE[-(M/(2 M-r0))]+(-5 M+2 r0) EllipticK[-(M/(2 M-r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π r0 (((-2 M+r0))^(3/2)))))+Δr (-((320 ((108 (M^3)-131 (M^2) r0+50 M (r0^2)-6 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-135 (M^3)+150 (M^2) r0-53 M (r0^2)+6 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π r0 (((-3 M+r0))^(3/2)) (Sqrt[-2 M+r0])))+((64 (3 (26 (M^3)-35 (M^2) r0+15 M (r0^2)-2 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-96 (M^3)+121 (M^2) r0-48 M (r0^2)+6 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) M π r0 (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2)))))) (-WignerD[{l,m,-2},π,(π/2),(π/2)]+WignerD[{l,m,2},π,(π/2),(π/2)])];

hP[10]=If[l<2,0,(1/((Sqrt[1+2 l]) (r0+Δr)))2 I (Sqrt[(-1+l) l (1+l) (2+l)]) (Sqrt[π]) ((((1+2 l) M (Sqrt[r0]) (Sqrt[((Δr)^2)]))/((Sqrt[-3 M+r0]) (-2 M+r0)))-((32 M r0 EllipticK[(M/(-2 M+r0))])/((-1+2 l) (3+2 l) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))-((32 r0 (Sqrt[-2 M+r0]) ((-5 M+2 r0) EllipticE[(M/(-2 M+r0))]-2 (-3 M+r0) EllipticK[(M/(-2 M+r0))]))/(3 M π (Sqrt[-3 M+r0])))+((20160 r0 (Sqrt[-2 M+r0]) ((-5 M+2 r0) EllipticE[(M/(-2 M+r0))]-2 (-3 M+r0) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) M π (Sqrt[-3 M+r0])))+(672 r0 (5 (-2 M+r0) (-5 M+2 r0) EllipticE[(M/(-2 M+r0))]-2 (31 (M^2)-25 M r0+5 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))-(66528 (2 M-r0) ((28 (M^2)+25 M r0-11 (r0^2)) EllipticE[(M/(-2 M+r0))]+(15 (M^2)-38 M r0+11 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(432 (3 (244 (M^3)-212 (M^2) r0+47 M (r0^2)-(r0^3)) EllipticE[(M/(-2 M+r0))]+(-750 (M^3)+613 (M^2) r0-130 M (r0^2)+3 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(16 ((124 (M^3)-136 (M^2) r0+47 M (r0^2)-5 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-162 (M^3)+159 (M^2) r0-50 M (r0^2)+5 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(48 ((34 (M^3)-79 (M^2) r0+47 M (r0^2)-8 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-96 (M^3)+125 (M^2) r0-55 M (r0^2)+8 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))+((Δr)^2) (((4 M EllipticE[(M/(-2 M+r0))])/(π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2))))-((840 (Sqrt[-3 M+r0]) (2 (-2 M+r0) EllipticE[(M/(-2 M+r0))]-(-5 M+2 r0) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π (((-2 M+r0))^(3/2))))+(20 (2 (55 (M^2)-45 M r0+9 (r0^2)) EllipticE[(M/(-2 M+r0))]-9 (-3 M+r0) (-5 M+2 r0) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) M π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2)))-(4 (((1+2 l))^2) (2 (7 (M^2)-5 M r0+(r0^2)) EllipticE[(M/(-2 M+r0))]-(-3 M+r0) (-5 M+2 r0) EllipticK[(M/(-2 M+r0))]))/(15 M π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2))))+Δr ((((1+2 l) M (Sqrt[-3 M+r0]) (Sqrt[((Δr)^2)]))/((Sqrt[r0]) (((-2 M+r0))^2)))-((16 M (EllipticE[(M/(-2 M+r0))]+2 EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+((16 (2 (8 (M^2)-6 M r0+(r0^2)) EllipticE[(M/(-2 M+r0))]-(7 M-2 r0) (3 M-r0) EllipticK[(M/(-2 M+r0))]))/(3 M π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))-(10080 (2 (8 (M^2)-6 M r0+(r0^2)) EllipticE[(M/(-2 M+r0))]-(7 M-2 r0) (3 M-r0) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) M π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))-(336 (2 (41 (M^2)-30 M r0+5 (r0^2)) EllipticE[(M/(-2 M+r0))]+(-101 (M^2)+65 M r0-10 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))+((Δr)^2) (((-428 (M^3)+570 (M^2) r0-226 M (r0^2)+24 (r0^3)) EllipticE[(M/(-2 M+r0))]+2 (264 (M^3)-325 (M^2) r0+119 M (r0^2)-12 (r0^3)) EllipticK[(M/(-2 M+r0))])/(3 M π (2 M-r0) r0 (Sqrt[6 (M^2)-5 M r0+(r0^2)]))+(2 ((-158 (M^3)+173 (M^2) r0-65 M (r0^2)+8 (r0^3)) EllipticE[(M/(-2 M+r0))]+(183 (M^3)-166 (M^2) r0+59 M (r0^2)-8 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-1+2 l) (3+2 l) π r0 (((6 (M^2)-5 M r0+(r0^2)))^(3/2)))+(1260 ((214 (M^3)-285 (M^2) r0+113 M (r0^2)-12 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-264 (M^3)+325 (M^2) r0-119 M (r0^2)+12 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) M π (2 M-r0) r0 (Sqrt[6 (M^2)-5 M r0+(r0^2)]))+(42 ((2894 (M^4)-4999 (M^3) r0+2990 (M^2) (r0^2)-729 M (r0^3)+60 (r0^4)) EllipticE[(M/(-2 M+r0))]+(-3594 (M^4)+5863 (M^3) r0-3292 (M^2) (r0^2)+759 M (r0^3)-60 (r0^4)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l) M π r0 (((6 (M^2)-5 M r0+(r0^2)))^(3/2))))) (-WignerD[{l,m,-2},π,(π/2),(π/2)]+WignerD[{l,m,2},π,(π/2),(π/2)])]+If[l<4,0,(1/((Sqrt[1+2 l]) (r0+Δr)))2 I (Sqrt[(-3+l) (-2+l) (-1+l) l (1+l) (2+l) (3+l) (4+l)]) (Sqrt[π]) ((256 r0 (Sqrt[-2 M+r0]) ((97 (M^2)-80 M r0+16 (r0^2)) EllipticE[(M/(-2 M+r0))]-8 (-3 M+r0) (-5 M+2 r0) EllipticK[(M/(-2 M+r0))]))/(15 (-1+2 l) (3+2 l) (M^2) π (Sqrt[-3 M+r0]))-(13837824 r0 (Sqrt[-2 M+r0]) ((97 (M^2)-80 M r0+16 (r0^2)) EllipticE[(M/(-2 M+r0))]-8 (-3 M+r0) (-5 M+2 r0) EllipticK[(M/(-2 M+r0))]))/((-9+2 l) (-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (11+2 l) (M^2) π (Sqrt[-3 M+r0]))+(256 r0 ((-1346 (M^3)+1793 (M^2) r0-784 M (r0^2)+112 (r0^3)) EllipticE[(M/(-2 M+r0))]+(1665 (M^3)-2066 (M^2) r0+840 M (r0^2)-112 (r0^3)) EllipticK[(M/(-2 M+r0))]))/(3 (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (M^2) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))+(2304 r0 ((-1338 (M^3)+1789 (M^2) r0-784 M (r0^2)+112 (r0^3)) EllipticE[(M/(-2 M+r0))]+(1655 (M^3)-2062 (M^2) r0+840 M (r0^2)-112 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (M^2) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))+(177408 r0 ((1786 (M^3)-2333 (M^2) r0+1008 M (r0^2)-144 (r0^3)) EllipticE[(M/(-2 M+r0))]+2 (-1105 (M^3)+1342 (M^2) r0-540 M (r0^2)+72 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (M^2) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0]))+(44478720 (2 M-r0) ((536 (M^4)-4173 (M^3) r0+4651 (M^2) (r0^2)-1784 M (r0^3)+224 (r0^4)) EllipticE[(M/(-2 M+r0))]+(-645 (M^4)+5114 (M^3) r0-5305 (M^2) (r0^2)+1896 M (r0^3)-224 (r0^4)) EllipticK[(M/(-2 M+r0))]))/((-11+2 l) (-9+2 l) (-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (11+2 l) (13+2 l) (M^2) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(128 ((2892 (M^5)+2656 (M^4) r0-9641 (M^3) (r0^2)+7107 (M^2) (r0^3)-2104 M (r0^4)+224 (r0^5)) EllipticE[(M/(-2 M+r0))]-(3582 (M^5)+3615 (M^4) r0-11470 (M^3) (r0^2)+7921 (M^2) (r0^3)-2216 M (r0^4)+224 (r0^5)) EllipticK[(M/(-2 M+r0))]))/(3 (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (M^2) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(384 ((6694 (M^5)+11081 (M^4) r0-30840 (M^3) (r0^2)+21877 (M^2) (r0^3)-6376 M (r0^4)+672 (r0^5)) EllipticE[(M/(-2 M+r0))]-(8310 (M^5)+14474 (M^4) r0-36537 (M^3) (r0^2)+24351 (M^2) (r0^3)-6712 M (r0^4)+672 (r0^5)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (M^2) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(384 ((33994 (M^5)+184151 (M^4) r0-377580 (M^3) (r0^2)+251767 (M^2) (r0^3)-71416 M (r0^4)+7392 (r0^5)) EllipticE[(M/(-2 M+r0))]-(42690 (M^5)+231794 (M^4) r0-444447 (M^3) (r0^2)+279621 (M^2) (r0^3)-75112 M (r0^4)+7392 (r0^5)) EllipticK[(M/(-2 M+r0))]))/((-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (M^2) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(44928 ((103364 (M^5)+75196 (M^4) r0-310485 (M^3) (r0^2)+232307 (M^2) (r0^3)-69176 M (r0^4)+7392 (r0^5)) EllipticE[(M/(-2 M+r0))]-(127950 (M^5)+104779 (M^4) r0-370002 (M^3) (r0^2)+259041 (M^2) (r0^3)-72872 M (r0^4)+7392 (r0^5)) EllipticK[(M/(-2 M+r0))]))/((-9+2 l) (-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (11+2 l) (M^2) π (3 M-r0) (Sqrt[6 (M^2)-5 M r0+(r0^2)]))+(1/(15 (M^2) π (Sqrt[-3 M+r0]) (Sqrt[-2 M+r0])))128 Δr (-(1/((-1+2 l) (3+2 l)))(2 (-2 M+r0) (203 (M^2)-140 M r0+24 (r0^2)) EllipticE[(M/(-2 M+r0))]-(-3 M+r0) (335 (M^2)-256 M r0+48 (r0^2)) EllipticK[(M/(-2 M+r0))])+(810810 (2 (-2 M+r0) (203 (M^2)-140 M r0+24 (r0^2)) EllipticE[(M/(-2 M+r0))]-(-3 M+r0) (335 (M^2)-256 M r0+48 (r0^2)) EllipticK[(M/(-2 M+r0))]))/((-9+2 l) (-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (11+2 l))+(5 ((5693 (M^3)-6768 (M^2) r0+2632 M (r0^2)-336 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-7047 (M^3)+7727 (M^2) r0-2800 M (r0^2)+336 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-3+2 l) (-1+2 l) (3+2 l) (5+2 l))+(135 ((5699 (M^3)-6772 (M^2) r0+2632 M (r0^2)-336 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-7055 (M^3)+7731 (M^2) r0-2800 M (r0^2)+336 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l))-(10395 ((7278 (M^3)-8674 (M^2) r0+3384 M (r0^2)-432 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-9005 (M^3)+9907 (M^2) r0-3600 M (r0^2)+432 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l)))+((Δr)^2) ((665280 (Sqrt[-3 M+r0]) (8 (-2 M+r0) (-5 M+2 r0) EllipticE[(M/(-2 M+r0))]-(-11 M+4 r0) (-9 M+4 r0) EllipticK[(M/(-2 M+r0))]))/((-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (M^2) π (((-2 M+r0))^(3/2)))-(32 (2 (115 (M^3)-146 (M^2) r0+60 M (r0^2)-8 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-285 (M^3)+335 (M^2) r0-128 M (r0^2)+16 (r0^3)) EllipticK[(M/(-2 M+r0))]))/(105 (M^2) π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2)))-(32 ((235 (M^3)-294 (M^2) r0+120 M (r0^2)-16 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-291 (M^3)+337 (M^2) r0-128 M (r0^2)+16 (r0^3)) EllipticK[(M/(-2 M+r0))]))/(3 (-1+2 l) (3+2 l) (M^2) π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2)))+(12960 (2 (845 (M^3)-1038 (M^2) r0+420 M (r0^2)-56 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-2091 (M^3)+2377 (M^2) r0-896 M (r0^2)+112 (r0^3)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (M^2) π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2)))-(32 ((8265 (M^3)-10306 (M^2) r0+4200 M (r0^2)-560 (r0^3)) EllipticE[(M/(-2 M+r0))]+(-10233 (M^3)+11811 (M^2) r0-4480 M (r0^2)+560 (r0^3)) EllipticK[(M/(-2 M+r0))]))/(3 (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (M^2) π (Sqrt[-3 M+r0]) (((-2 M+r0))^(3/2))))+((Δr)^2) (-((16 ((17798 (M^4)-32269 (M^3) r0+21429 (M^2) (r0^2)-6152 M (r0^3)+640 (r0^4)) EllipticE[(M/(-2 M+r0))]+(-22020 (M^4)+37907 (M^3) r0-23825 (M^2) (r0^2)+6472 M (r0^3)-640 (r0^4)) EllipticK[(M/(-2 M+r0))]))/(15 (-1+2 l) (3+2 l) (M^2) π (2 M-r0) r0 (Sqrt[6 (M^2)-5 M r0+(r0^2)])))+(864864 ((17798 (M^4)-32269 (M^3) r0+21429 (M^2) (r0^2)-6152 M (r0^3)+640 (r0^4)) EllipticE[(M/(-2 M+r0))]+(-22020 (M^4)+37907 (M^3) r0-23825 (M^2) (r0^2)+6472 M (r0^3)-640 (r0^4)) EllipticK[(M/(-2 M+r0))]))/((-9+2 l) (-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (11+2 l) (M^2) π (2 M-r0) r0 (Sqrt[6 (M^2)-5 M r0+(r0^2)]))-(144 ((385928 (M^5)-820210 (M^4) r0+686407 (M^3) (r0^2)-282115 (M^2) (r0^3)+56824 M (r0^4)-4480 (r0^5)) EllipticE[(M/(-2 M+r0))]+(-477435 (M^5)+971017 (M^4) r0-777309 (M^3) (r0^2)+305767 (M^2) (r0^3)-59064 M (r0^4)+4480 (r0^5)) EllipticK[(M/(-2 M+r0))]))/((-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (M^2) π r0 (((6 (M^2)-5 M r0+(r0^2)))^(3/2)))-(16 ((381060 (M^5)-813020 (M^4) r0+682201 (M^3) (r0^2)-280947 (M^2) (r0^3)+56696 M (r0^4)-4480 (r0^5)) EllipticE[(M/(-2 M+r0))]+(-471429 (M^5)+962685 (M^4) r0-772655 (M^3) (r0^2)+304535 (M^2) (r0^3)-58936 M (r0^4)+4480 (r0^5)) EllipticK[(M/(-2 M+r0))]))/(3 (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (M^2) π r0 (((6 (M^2)-5 M r0+(r0^2)))^(3/2)))+(11088 ((456206 (M^5)-995495 (M^4) r0+847974 (M^3) (r0^2)-353125 (M^2) (r0^3)+72008 M (r0^4)-5760 (r0^5)) EllipticE[(M/(-2 M+r0))]+(-564510 (M^5)+1180009 (M^4) r0-961168 (M^3) (r0^2)+383009 (M^2) (r0^3)-74888 M (r0^4)+5760 (r0^5)) EllipticK[(M/(-2 M+r0))]))/((-7+2 l) (-5+2 l) (-3+2 l) (-1+2 l) (3+2 l) (5+2 l) (7+2 l) (9+2 l) (M^2) π r0 (((6 (M^2)-5 M r0+(r0^2)))^(3/2))))) (-WignerD[{l,m,-4},π,(π/2),(π/2)]+WignerD[{l,m,4},π,(π/2),(π/2)])];