Bürmann–Lagrange series

Remarks

Proof idea: since $g'(a) \ne 0$, the function $g$ has a holomorphic inverse $g^{-1}$ near $b$. The series is the Taylor series of $w \mapsto H(g^{-1}(w))$ at $w = b$, evaluated at $w = g(z)$. Its coefficients are found by Cauchy's integral formula for derivatives, followed by the change of variable $w = g(z)$. An integration by parts then turns the integral into a residue at $z = a$.

With $H(z) = z$ the series is the classical Lagrange inversion formula for $g^{-1}$. Example: $g(z) = z e^{-z}$ and $a = 0$, so $\varphi(z) = e^{z}$ and $c_n = \frac{d^{\,n-1}}{dz^{\,n-1}} e^{nz} \big|_{z=0} = n^{n-1}$. So the inverse of $z e^{-z}$ near $0$ is $\sum_{n \ge 1} \frac{n^{n-1}}{n!} w^n$, which is $-W(-w)$ for the Lambert W function.

The hypothesis $g'(a) \ne 0$ is needed: for $g(z) = z^2$ and $a = 0$ the function $H(z) = z$ is not a power series in $g(z) = z^2$, since $z$ is odd. The formal version for power series over a field of characteristic $0$ is the Lagrange inversion theorem.

Going deeper

For a student

The series is the analytic form of Lagrange inversion. It gives the Taylor coefficients of an inverse function, for example of the solution $w(z)$ of Kepler's equation or of the Lambert $W$ function. In asymptotic analysis it is used to change variables near a critical point. For instance, it reduces an integral in Laplace's method to a Gaussian form with an explicit series of corrections. The combinatorial version, for formal power series, is the Lagrange inversion theorem, which counts trees.