Mathematica

P _ A_[x_] := (# [[ 1, 1 ]]/# [[ 2, 1 ]] &)[A . (x)] 1

P _ ( α β )[x] γ δ

(x α + β)/(x γ + δ)

Composition[Π _ A^', Π _ A][x] ^= Composition[P _ ( ' ' ), P _ ( α β )][x]      // Simplify // Map[Collect[#, x] &, #, 2] & α β γ δ ' ' γ δ

(β α^' + δ β^' + x (α α^' + γ β^'))/(β γ^' + δ δ^' + x (α γ^' + γ δ^'))

( ' ' ) . ( α β ) // MatrixForm α β γ δ ' ' γ δ

( ' ' ' ' ) α α + γ β β α + δ β ' ' ' ' α γ + γ δ β γ + δ δ

P _ (( ' ' ) . ( α β ))[x] α β γ δ ' ' γ δ

(β α^' + δ β^' + x (α α^' + γ β^'))/(β γ^' + δ δ^' + x (α γ^' + γ δ^'))

Composition[Π _ A^', Π _ A][x] == P _ (( ' ' ) . ( α β ))[x] // Simplify α β γ δ ' ' γ δ

True

P _ Inverse[( α β )][x] // Simplify γ δ

(β - x δ)/(-α + x γ)

Converted by Mathematica  (October 16, 2002)