By S.I. Andersson, H.-D. Doebner
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5. o Now the implicit function theorem, formulated in a manner that lends itself to proof by the contraction mapping fixed point principle, will be our next major result. We first introduce some definitions and notation. 7 (Landau) Fix a in the extended reals, that is, a E R U {±oo}. neighborhood of a. For a real-valued function f defined in a punctured neighborhood of a, we say f is little "0" ofg as x ~ a and write f(x) = o(g(x)) as x ~ a in case lim f(x) = O. 8 Let X, Z be normed linear spaces.
Obversely, if we look at the image of F, then we see that it is a smooth 2dimensional manifold given by {(x, y, z) : Z = x - y}. The rank theorem formalizes the observations that we have made for this specific mapping F in the context of a class of mappings having constant rank. 1 (The Rank Theorem). Let r, p, q be nonnegative integers and let M = R'+P. N = R,+q. Let W 5; M be an open set and suppose that F : W ~ N is a continuously differentiable mapping. Assume that D F has rank r at each point ofW.
21). 1 (Lagrange Inversion Theorem [La 69)). Let tjr(z) and ¢(z) be anaLytic on the open disc D(a, r) C C and continuous on the closed disc D(a, r). 3 Lagrange 23 We will give two proofs of Lagrange's theorem. The first proof uses the Cauchy theory from complex analysis. The second is a proof that is due to Laplace (17491827), and depends heavily on the chain rule of calculus. We will need some classical results from complex analysis. The first of these classical results is the Cauchy integral formula (see Greene and Krantz [GK 97; page 48]).