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The Lagrange inversion theorem in the smooth case. (English) Zbl 1194.26040
Summary: The classical Lagrange inversion theorem is a concrete, explicit form of the implicit function theorem for real analytic functions. An explicit construction shows that the formula is not true for all merely smooth functions. The authors modify the Lagrange formula by replacing the smooth function by its Maclaurin polynomials. The resulting modified Lagrange series is, in analogy to the Maclaurin polynomials, an approximation to the solution function accurate to $o(x^N)$ as $x\rightarrow 0$.

26E05Real-analytic functions
26A24Differentiation of functions of one real variable
58C15Implicit function theorems and global Newton methods on manifolds
Full Text: DOI arXiv
[1] Borel, Émile: Sur quelques points de la théorie des fonctions, Ann. sci. École norm. Sup. (3) 12, 9-55 (1895) · Zbl 26.0429.03 · numdam:ASENS_1895_3_12__9_0
[2] Di Bruno, Francesco Faá: Note sur une nouvelle formule de calcul différentiel, Q. J. Pure appl. Math. 1, 359-360 (1857)
[3] Grossman, Nathaniel: A C$\infty $ Lagrange inversion theorem, Amer. math. Monthly 112, 512-514 (2005) · Zbl 1128.26007 · doi:10.2307/30037521
[4] Krantz, Steven G.; Parks, Harold R.: A primer of real analytic functions, (2002) · Zbl 1015.26030
[5] Krantz, Steven G.; Parks, Harold R.: The implicit function theorem, (2002) · Zbl 1012.58003
[6] Lagrange, Joseph Louis: Nouvelle méthode pour résoudre LES équations littérales par le moyen des séries, Mém. acad. Roy. sci. Belles-lettres Berlin 24 (1869)