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Akn+i coefficients of lower conditions 32 conformal symmetries corresponding and lower corresponding expressions determinant for consideration direct periodic conformal discussion e2 are primitive eie2 elimination Equation 24 equations of 14 equivalent expressions expressions for Kt factored into transformations form kn form of 79 Functional Equations Furthermore given transformation Harry Langman independent irreducible period left member lower orders member of 95 method of Section necessary condition necessary relation observe order kn order t/2 period n periodic function periodic reverse conformal periodic transformations polynomial in lower polynomial in X's power series primitive root rational integral functions relations 32 resulting from 93 reverse conformal transformation root of unity satisfied series in xi Similarly substitution Suppose taken arbitrarily takes the form terms of X's Theorem transformation F(z transformations of order transformations of period uniquely vanish identically variables write X's and n's X's of corresponding X's of lower X's of order yield