Regular Functions of a Quaternionic Variable by Graziano Gentili Caterina Stoppato & Daniele C. Struppa

Author:Graziano Gentili, Caterina Stoppato & Daniele C. Struppa
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

Proof.

Suppose | f | to have a minimum point with f(p) ≠ 0 and let . If f vanished at some point p′ ∈ S, then, according to Formula (1.9), f(S) would be a 2-sphere passing through the origin of . The modulus would then have a global minimum at p′, a global maximum at some other point, and no other extremal point, in contradiction with the hypothesis on p. Hence f does not have zeros in S nor does f s . As a consequence, the domain of the regular reciprocal f  − ∗  includes S. By Proposition 5.32,

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