In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. Lagrange inversion is a special case of the inverse function theorem.
Statement Suppose z is defined as a function of w by an equation of the form
z = f ( w ) {\displaystyle z=f(w)}
where f is analytic at a point a and f ′ ( a ) ≠ 0. {\displaystyle f'(a)\neq 0.} Then it is possible to invert or solve the equation for w, expressing it in the form w = g ( z ) {\displaystyle w=g(z)} given by a power series
g ( z ) = a + ∑ n = 1 ∞ g n ( z − f ( a ) ) n n ! , {\displaystyle g(z)=a+\sum _{n=1}^{\infty }g_{n}{\frac {(z-f(a))^{n}}{n!}},}
where
g n = lim w → a d n − 1 d w n − 1 [ ( w − a f ( w ) − f ( a ) ) n ] . {\displaystyle g_{n}=\lim _{w\to a}{\frac {d^{n-1}}{dw^{n-1}}}\left[\left({\frac {w-a}{f(w)-f(a)}}\right)^{n}\right].}
The theorem further states that this series has a non-zero radius of convergence, i.e., g ( z ) {\displaystyle g(z)} represents an analytic function of z in a neighbourhood of z = f ( a ) . {\displaystyle z=f(a).} This is also called reversion of series. If the assertions about analyticity are omitted, the formula is also valid for formal power series and can be generalized in various ways: It can be formulated for functions of several variables; it can be extended to provide a ready formula for F(g(z)) for any analytic function F; and it can be generalized to the case f ′ ( a ) = 0 , {\displaystyle f'(a)=0,} where the inverse g is a multivalued function. The theorem was proved by Lagrange and generalized by Hans Heinrich Bürmann, both in the late 18th century. There is a straightforward derivation using complex analysis and contour integration; the complex formal power series version is a consequence of knowing the formula for polynomials, so the theory of analytic functions may be applied. Actually, the machinery from analytic function theory enters only in a formal way in this proof, in that what is really needed is some property of the formal residue, and a more direct formal proof is available. In fact, the Lagrange inversion theorem has a number of additional rather different proofs, including ones using tree-counting arguments or induction. If f is a formal power series, then the above formula does not give the coefficients of the compositional inverse series g directly in terms for the coefficients of the series f. If one can express the functions f and g in formal power series as
f ( w ) = ∑ k = 0 ∞ f k w k k ! and g ( z ) = ∑ k = 0 ∞ g k z k k ! {\displaystyle f(w)=\sum _{k=0}^{\infty }f_{k}{\frac {w^{k}}{k!}}\qquad {\text{and}}\qquad g(z)=\sum _{k=0}^{\infty }g_{k}{\frac {z^{k}}{k!}}}
with f0 = 0 and f1 ≠ 0, then an explicit form of inverse coefficients can be given in term of Bell polynomials:
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