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By Constantin Caratheodory

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Let UD be the axially symmetric s-domain defined by UD = (u + Iv). 14) 2 2 is the unique s-monogenic extension of f to UD . Similarly, let J2 , . . , Jn be a completion of J to an orthonormal basis of Rn and let f : D → Rn n−1 defined by f = |A|=0 FA JA , A ⊆ {2, . . , n}, FA : D → CJ holomorphic. Then, ∂ J f (u + Jv) = 0 and the function obtained by extending each of its holomorphic components FA is the unique s-monogenic extension of f to UD . Proof. 18. 14) we have that ext(f )(u + Jv) = f (u + Jv), and hence ext(f ) is the unique extension of f by the Identity Principle.

3) to the case of a general Clifford number. 14. The modulus of Clifford numbers satisfies: (1) |λa| = |λ| |a| for all λ ∈ R, a ∈ Rn ; (2) ||x| − |y|| ≤ |x − y| ≤ |x| + |y|; However, the modulus is not multiplicative, as shown in the next result. 15. For any two elements a, b ∈ Rn we have |ab| ≤ Cn |a| |b| where Cn is a constant depending only on the dimension of the Clifford algebra Rn . Moreover, we have Cn ≤ 2n/2 . 16. The modulus is multiplicative in the case of complex numbers and quaternions.

8. Let p be a polynomial in the paravector variable x with real coefficients. Then the zero set of p is the union of isolated points (belonging to R) and isolated (n − 1)-spheres. 9. As we have already pointed out, in the case n = 1 the set of s-monogenic functions coincide with the set of holomorphic functions in one complex variable (by identifying R2 with C). 7 corresponds to the well-known result saying that the zeros of a holomorphic function whose series expansion has real coefficients has isolated zeros which are either real or complex conjugates.

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