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  "body": "\n## Statement\n\nLet $g$ be a [[holomorphic function|0NAHXR1FRH]] on a neighbourhood of a point $a \\in \\C$ with $g'(a) \\ne 0$, and let $b = g(a)$.\nLet $H$ be holomorphic on a neighbourhood of $a$.\nPut $\\varphi(z) = \\dfrac{z - a}{g(z) - b}$ for $z \\ne a$ and $\\varphi(a) = 1/g'(a)$, which is holomorphic near $a$.\nThen there is a neighbourhood $U$ of $a$ such that for all $z \\in U$\n$$H(z) = H(a) + \\sum_{n=1}^{\\infty} \\frac{c_n}{n!} \\big( g(z) - b \\big)^n, \\qquad c_n = \\frac{d^{\\,n-1}}{dz^{\\,n-1}} \\Big[ H'(z)\\, \\varphi(z)^n \\Big]_{z = a},$$\nand the series converges absolutely on $U$.\n\n## Remarks\n\nProof idea: since $g'(a) \\ne 0$, the function $g$ has a holomorphic inverse $g^{-1}$ near $b$.\nThe series is the [[Taylor series|0Z49YH6ZWY]] of $w \\mapsto H(g^{-1}(w))$ at $w = b$, evaluated at $w = g(z)$.\nIts coefficients are found by Cauchy's integral formula for derivatives, followed by the change of variable $w = g(z)$.\nAn integration by parts then turns the integral into a residue at $z = a$.\n\nWith $H(z) = z$ the series is the classical Lagrange inversion formula for $g^{-1}$.\nExample: $g(z) = z e^{-z}$ and $a = 0$, so $\\varphi(z) = e^{z}$ and $c_n = \\frac{d^{\\,n-1}}{dz^{\\,n-1}} e^{nz} \\big|_{z=0} = n^{n-1}$.\nSo 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|FW2JKAN14B]].\n\nThe 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.\nThe formal version for power series over a field of characteristic $0$ is the [[Lagrange inversion theorem|BANY9YR82A]].\n\n## Explanations\n\n### L1 human\n\nA Taylor series writes a function as a sum of powers of $z - a$.\nThe Bürmann–Lagrange series instead writes it as a sum of powers of another function $g(z) - g(a)$.\nThis works whenever $g$ is locally one-to-one at $a$.\nThe coefficients come from derivatives at the single point $a$, so one never has to invert $g$ explicitly.\n\n### L2 human\n\nThe series is the analytic form of Lagrange inversion.\nIt 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.\nIn asymptotic analysis it is used to change variables near a critical point.\nFor instance, it reduces an integral in [[Laplace's method|9G7K8DBT9Y]] to a Gaussian form with an explicit series of corrections.\nThe combinatorial version, for formal power series, is the [[Lagrange inversion theorem|BANY9YR82A]], which counts trees.\n",
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