By G Mazzola; Gérard Milmeister; Jody Weissmann

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Deﬁnition 189 If f : U → Rm and g : V → Rm are two functions deﬁned in open neighborhoods U , V of 0 ∈ Rn , then they are called equivalent if there is a neighborhood N ⊂ U ∩ V of 0 such that f |N = g|N . This relation is an equivalence relation, and the equivalence class [f ] of a function f is called the germ of f (at 0). The set F0 of germs of functions f : U → Rm with f (0) = 0 is a real vector space as follows: (1) The sum of germs is [f ] + [g] = [f |W + g|W ], W = U ∩ V being the intersection of the domains U and V of f and g representing the germs [f ] and [g], and (2) the scalar multiplication is λ[f ] = [λf ].

An , bn is continuously diﬀerentiable such that there is a number L with |Dj fi (x)| ≤ L for all x ∈ K o . Then for all x, y ∈ K, we have f (x) − f (y) ≤ n2 L x − y . Proof For x = (x1 , . . xn ), y = (y1 , . . yn ) ∈ K, and for any index i, we have fi (y) − fi (x) = (fi (y1 , . . yj , xj+1 , . . xn ) − fi (y1 , . . yj−1 , xj , . . n and then, by the mean value theorem 267 on each coordinate, fi (y1 , . . yj , xj+1 , . . xn ) − fi (y1 , . . yj−1 , xj , . . xn ) = yj − xj · Dj fi (wij ) for some vector wij .

All this together proves that i · 2π Z is the kernel of exp. Let us ﬁnally concentrate on the real arguments in exp. Since e = exp(1) > 1, there are arbitrary large real numbers exp(n) = e n for real arguments, and by 1 also arbitrary small real values for real arguments. By propoexp(−n) = exp(n) sition 245, every positive real value is taken by exp(x) for x ∈ R. Now, every complex number z ≠ 0 can be written as z = z u, u ∈ U. Therefore there are x, θ ∈ R, such that exp(x) = z and exp(i · θ) = u.