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Non-(k)-monogenic points of functions of a quaternion variable

Part of the Lecture Notes in Mathematics book series (LNM,volume 561)

Abstract

The concept of (k)-monogeneity for functions with values in the algebra ℜ of real quaternions has been introduced in [1]. Its definition and some results concerning Cauchy's Formula, the Taylor expansion and Weierstrass's convergence theorem are recalled in a first paragraph; for the proofs we refer the reader to [1] and [2].

We then continue the study of such functions by examining their behaviour in the neighbourhood of singular points. This gives rise to Laurent's expansion (§ 3), the notions of (k)-pole, essential-non-(k)-monogenic point and removable singularity (§§ 4 and 6), Mittag-Leffler's theorem on (k)-meromorphic functions (§ 5) and a finite residue theory (§ 7).

Keywords

  • Entire Function
  • Meromorphic Function
  • Principal Part
  • Laurent Series
  • Monogenic Function

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. F. BRACKX, On (k)-monogenic functions of a quaternion variable (to appear)

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  2. F. BRACKX, On the space of left-(k)-monogenic functions of a quaternion variable and an associated quaternion Hilbert space with reproducing kernel (to appear)

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  3. R. FUETER, Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen, Comm. Math. Helv., 8 (1935), 371–378

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  4. H.G. GARNIR, Fonctions de variables réelles, II, Gauthier-Villars, Paris, 1965

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  5. O. KELLOG, Foundations of potential theory, Springer, Berlin, 1929

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© 1976 Springer-Verlag

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Brackx, F.F. (1976). Non-(k)-monogenic points of functions of a quaternion variable. In: Meister, V.E., Wendland, W.L., Weck, N. (eds) Function Theoretic Methods for Partial Differential Equations. Lecture Notes in Mathematics, vol 561. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087632

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  • DOI: https://doi.org/10.1007/BFb0087632

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08054-1

  • Online ISBN: 978-3-540-37536-4

  • eBook Packages: Springer Book Archive