#lagrangeburmanntheorem — Public Fediverse posts

LAGRANGE-BÜRMANN THEOREM
Have you heard about the Lagrange-Bürmann formula? It gives the Taylor series expansion for the inverse of a function.

If \(z=f(\omega)\) with \(f\) analytic at a point \(a\) and \(f(a)\neq0\), then
\[\omega=g(z)=a+\displaystyle\sum_{n=1}^\infty g_n\dfrac{(z-f(a))^n}{n!}\]
\[\text{where }g_n=\displaystyle\lim_{\omega\to a}\left[\dfrac{\mathrm{d}^{n-1}}{d\omega^{n-1}}\left(\dfrac{\omega-a}{f(\omega)-f(a)}\right)^n\right]\]

#Lagrange #Burmann #LagrangeBurmannTheorem #TaylorSeries #InverseFunction #Mathematics #Math #Maths