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Mathematica Notebook for Chapter 17

Ce[m_,0][z_]:=1

CeS[m_,i_][z_]:=Sum[Binomial[i,j]*(D[rho[z],{z,i-j}]/rho[z])*Ce[m,j][z],{j,0,i}]

alpha[0,j_][zeta_]:=1

alpha[i_,j_][zeta_]:=(Sum[alpha[i-1,k]'[zeta]-(i-1+k)(f''[zeta]/f'[zeta])*alpha[i-1,k][zeta],{k,1,j}])/;i≥1

beta[m_,r_,s_][zeta_]:=(Product[(m-s-k+1),{k,1,r}]/Product[(s+k),{k,1,r}])*alpha[r,s][zeta]

CeSS[m_,i_][zeta_]:=Sum[beta[m,i-j,m-i][zeta]*(f'[zeta])^j*Ce[m,j][f[zeta]],{j,0,i}]

Expand[(CeS[m,2][z]-CeS[m,1][z]^2-CeS[m,1]'[z])-(Ce[m,2][z]-Ce[m,1][z]^2-Ce[m,1]'[z])]

Together[(CeSS[m,2][zeta]-((m-2)(3m-1)/(3(m-1)^2))CeSS[m,1][zeta]^2-(2(m-2)/(3(m-1)))CeSS[m,1]'[zeta])-f'[zeta]^2 (Ce[m,2][f[zeta]]-((m-2)(3m-1)/(3(m-1)^2))Ce[m,1][f[zeta]]^2-(2(m-2)/(3(m-1)))Ce[m,1]'[f[zeta]])]

Together[(CeS[m,3][z]-3 CeS[m,2][z]*CeS[m,1][z]+2 CeS[m,1][z]^3-(3/2)CeS[m,2]'[z]+3 CeS[m,1][z]*CeS[m,1]'[z]+(1/2)CeS[m,1]''[z])-(Ce[m,3][z]-3 Ce[m,2][z]*Ce[m,1][z]+2 Ce[m,1][z]^3-(3/2)Ce[m,2]'[z]+3 Ce[m,1][z]*Ce[m,1]'[z]+(1/2)Ce[m,1]''[z])]

Together[(CeSS[m,3][zeta]-3CeSS[m,2][zeta]*CeSS[m,1][zeta]+2 CeSS[m,1][zeta]^3-(3/2)CeSS[m,2]'[zeta]+3 CeSS[m,1][zeta]*CeSS[m,1]'[zeta]+(1/2)CeSS[m,1]''[zeta])-(f'[zeta]^3)(Ce[m,3][f[zeta]]-3 Ce[m,2][f[zeta]]*Ce[m,1][f[zeta]]+2 Ce[m,1][f[zeta]]^3-(3/2)Ce[m,2]'[f[zeta]]+3 Ce[m,1][f[zeta]]*Ce[m,1]'[f[zeta]]+(1/2)Ce[m,1]''[f[zeta]])]