reversion of series | series inversion

Series reversion is the computation of the coefficients of the inverse function given those of the forward function. For a function expressed in a series with no constant term (i.e., a_0 = 0) as y = a_1 x + a_2 x^2 + a_3 x^3 + ..., the series expansion of the inverse series is given by x = A_1 y + A_2 y^2 + A_3 y^3 + .... By plugging (-1) into (-2), the following equation is obtained y = a_1 A_1 y + (a_2 A_1^2 + a_1 A_2) y^2 + (a_3 A_1^3 + 2a_2 A_1 A_2 + a_1 A_3) y^3 + (3a_3 A_1^2 A_2 + a_2 A_2^2 + a_2 A_1 A_3) + ....

power series | series