Advertisement

Exponents and Logarithms

  • João Pedro Morais
  • Svetlin Georgiev
  • Wolfgang Sprößig
Chapter

Abstract

The real exponential and logarithmic functions play an important role in advanced mathematics, including applications to calculus, differential equations, and complex analysis. In this chapter we use the properties of quaternions described in the previous chapters to define and study the quaternionic analogues of these functions.

Keywords

Fundamental Region Vector Part Principal Branch Quaternion Space Real Quaternion 
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.

Bibliography

  1. 1.
    K. Abdel-Khalek, Quaternion analysis (1996, peprint). arXiv:hepth/9607152Google Scholar
  2. 2.
    G. Aeberli, Der Zusammenhang zwischen quaternären quadratischen formen und idealen in quaternionenringen. Comment. Math. Helv. 33, 212–239 (1959)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    V. Bargmann, Über den Zusammenhang zwischen Semivektoren und Spinoren und die Reduktion der Diracgleichungen für Semivektoren. Helv. Phys. Acta 7, 57–82 (1934)Google Scholar
  4. 4.
    I.Y. Baritzhack, R.R. Harman, Optimal fusion of a given quaternion with vector measurements. J. Guid. Control Dyn. 25(1), 188–190 (2002)CrossRefGoogle Scholar
  5. 5.
    W.E. Baylis, Why i? Am. J. Phys. 60, 788–797 (1992)Google Scholar
  6. 6.
    B. Beck, Sur les équations polynomiales dans les quaternions. Enseign. Math. 25(3–4), 193–201 (1979)zbMATHMathSciNetGoogle Scholar
  7. 7.
    J. Brenner, Matrices of quaternions. Pac. J. Math. 1, 329–335 (1951)CrossRefzbMATHGoogle Scholar
  8. 8.
    E. Cartan, Sur les groupes linéaires quaternioniens. Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich 85, 191–203 (1940)MathSciNetGoogle Scholar
  9. 9.
    G. Combebiac, Calcul des Triquaternions (GauthierVillars, Paris, 1902)zbMATHGoogle Scholar
  10. 10.
    S. De Leo, P. Rotelli, Representations of U(1, q) and constructive quaternion tensor products. Nuovo Cimento B 110, 33–51 (1995)CrossRefGoogle Scholar
  11. 11.
    P. Du Val, Homographies, Quaternions, and Rotations. Clarendon Press, Oxford (1964)zbMATHGoogle Scholar
  12. 12.
    S. Eilenberg, I. Niven, The “fundamental theorem of algebra” for quaternions. Bull. Am. Math. Soc. 50, 244–248 (1944)CrossRefMathSciNetGoogle Scholar
  13. 13.
    A. Einstein, W. Mayer, Die Diracgleichungen für Semivektoren. Proc. R. Acad. Amst. 56, 497–516 (1933)Google Scholar
  14. 14.
    I. Gelfand, V. Retakh, R.L. Wilson, Quaternionic quasideterminants and determinants, in Lie Groups and Symmetric Spaces, ed. by F.I. Karpelevich, S.G. Gindikin. American Mathematical Society Translations Series 2, vol. 210 (American Mathematical Society, Providence, 2003), pp. 111–123Google Scholar
  15. 15.
    K. Gürlebeck, W. Sprößig, Quaternionic Analysis and Elliptic Boundary Value Problems (Birkhäuser, Basel, 1990)CrossRefzbMATHGoogle Scholar
  16. 16.
    K. Gürlebeck, W. Sprößig, Quaternionic and Clifford Calculus for Physicists and Engineers (Wiley, New York, 1997)zbMATHGoogle Scholar
  17. 17.
    K. Gürlebeck, K. Habetha, W. Sprößig, Holomorphic Functions in the Plane and n-Dimensional Space (Birkhäuser, Basel, 2008)zbMATHGoogle Scholar
  18. 18.
    F. Gürsey, Applications of Quaternions to Field Equations. Ph.D. thesis, University of London, 1950Google Scholar
  19. 19.
    W.R. Hamilton, On quaternions and the rotation of a solid body. Proc. R. Ir. Acad. 4, 38–56 (1850)Google Scholar
  20. 20.
    W.R. Hamilton, Elements of Quaternions, Vol. I et II (First edition 1866; second edition edited and expanded by C.J. Joly 1899–1901; reprinted by Chelsea Publishing, New York, 1969)Google Scholar
  21. 21.
    T. Haslwanter, Mathematics of the three dimensional eye rotations. Vis. Res. 35, 1727–1739 (1995)CrossRefGoogle Scholar
  22. 22.
    A.S. Hathaway, Quaternion space. Trans. Am. Math. Soc. 3, 46–59 (1902)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    D. Hestenes, A unified language for mathematics and physics, in Clifford Algebras and Their Applications in Mathematical Physics, ed. by J.S.R. Chisholmand, A.K.Common (Reidel, Dordrecht, 1986), pp. 1–23CrossRefGoogle Scholar
  24. 24.
    D. Janovská, G. Opfer, The classification and the computation of the zeros of quaternionic, two-sided polynomials. Numer. Math. 115, 81–100 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    D. Janovská, G. Opfer, A note on the computation of all zeros of simple quaternionic polynomials. SIAM J. Numer. Anal. 48(1), 244–256 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    T. Jiang, Cramer rule for quaternionic linear equations in quaternionic quantum theory. Rep. Math. Phys. 57(3), 463–468 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    T. Jiang, M. Wei, On a solution of the quaternion matrix equation XAXB = C and its application. Acta Math. Sin. 21, 483–490 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    W.J. Johnston, A quaternion substitute for the theory of tensors. Proc. R. Ir. Acad. A 37, 13–27 (1926)Google Scholar
  29. 29.
    C.J. Joly, Quaternion invariants of linear vector functions and quaternions determinants. Proc. R. Ir. Acad. 4, 1–15 (1896)Google Scholar
  30. 30.
    V. Kravchenko, M. Shapiro, Integral Representations for Spatial Models of Mathematical Physics. Research Notes in Mathematics (Pitman Advanced Publishing Program, London, 1996)Google Scholar
  31. 31.
    T.Y. Lam, A First Course in Noncommutative Rings. Graduate Texts in Mathematics, vol. 131 (Springer, New York, 1991)Google Scholar
  32. 32.
    J. Lambek, If Hamilton had prevailed: quaternions in physics. Math. Intell. 17, 7–15 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    H.C. Lee, Eigenvalues and canonical forms of matrices with quaternion coefficients. Proc. R. Ir. Acad. Sect. A 52, 253–260 (1949)Google Scholar
  34. 34.
    M. Markic, Transformantes nouveau véhicule mathématique. Synth\(\grave{e}\) se des triquaternions de Combebiac et du syst\(\grave{e}\) me géométrique de Grassmann Calcul des quadriquaternions. Ann. Fac. Sci. Toulouse, 28, 103–148 (1936)Google Scholar
  35. 35.
    M. Markic, Transformantes nouveau véhicule mathématique. Synth\(\grave{e}\) se des triquaternions de Combebiac et du syst\(\grave{e}\) me géométrique de Grassmann Calcul des quadriquaternions. Ann. Fac. Sci. Toulouse 1, 201–248 (1937)Google Scholar
  36. 36.
    I. Niven, Equations in quaternions. Am. Math. Mon. 48, 654–661 (1941)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    R.E. O’Connor, G. Pall, The quaternion congruence \(\overline{t}at = b(mod\,g)\). Am. J. Math. 61(2), 487–508 (1939)CrossRefMathSciNetGoogle Scholar
  38. 38.
    S. O’Donnel, William Rowan Hamilton, Portrait of a Prodigy (Boole Press, Dublin, 1983)Google Scholar
  39. 39.
    H.L. Olson, Doubly divisible quaternions. Ann. Math. 31(3), 371–374 (1930)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    G.B. Price, An Introduction to Multicomplex Spaces and Functions (Marcel Dekker, New York, 1991)zbMATHGoogle Scholar
  41. 41.
    V. Retakh, R.L. Wilson, Advanced Course on Quasideterminants and Universal Localization. Notes of the Course (Centre de Recerca Matematica (CRM), Bellaterra, 2007)Google Scholar
  42. 42.
    M. Riesz, Clifford Numbers and Spinors. Lecture Series No. 38 (Institute for Fluid Dynamics and Applied Mathematics/University of Maryland, Baltimore, 1958). Reprinted in Clifford Numbers and Spinors, ed. by M. Riesz, E.F. Bolinder, P. Lounesto (Kluwer, Dordrecht, 1993)Google Scholar
  43. 43.
    W. Rindler, I. Robinson, A plainmans guide to bivectors, biquaternions, and the algebra and geometry of Lorentz transformations, in On Einsteins Path Essays in Honour of Engelbert Schucking, ed. by A. Harvey (Springer, New York, 1999), pp. 407–433Google Scholar
  44. 44.
    E. Sarrau, Notions sur la théorie des quaternions, Paris, Gauthier-Villars, 1889, 46 p.Google Scholar
  45. 45.
    R. Serôdio, L.-S. Siu, Zeros of quaternion polynomials. Appl. Math. Lett. 14, 237–239 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    A. Sudbery, Quaternion analysis. Math. Proc. Camb. Philos. Soc. 85, 199–225 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    P.G. Tait, On the linear and vector function. Proc. R. Soc. Edinb. 23, 424–426 (1899)Google Scholar
  48. 48.
    P.G. Tait, On the claim recently made for Gauss to the invention (not the discovery) of quaternions. Proc. R. Soc. Edinb. 23, 17–23 (1899/1900)Google Scholar
  49. 49.
    O. Teichmüller, Operatoren im Wachsschen Raum. J. Math. 174, 73–124 (1935)zbMATHGoogle Scholar
  50. 50.
    J.P. Ward, Quaternions and Cayley Numbers (Kluwer, Dordrecht, 1997)CrossRefzbMATHGoogle Scholar
  51. 51.
    W.H. Watson, On a system of functional dynamics and optics. Philos. Trans. R. Soc. A 236, 155–190 (1937)CrossRefGoogle Scholar
  52. 52.
    D. Weingarten, Complex symmetries of electrodynamics. Ann. Phys. 76(2), 510–548 (1973)CrossRefMathSciNetGoogle Scholar
  53. 53.
    P. Weiss, On some applications of quaternions to restricted relativity and classical radiation theory. Proc. R. Ir. Acad. A 46, 129–168 (1941)zbMATHGoogle Scholar
  54. 54.
    P. Weiss, An extension of Cauchy’s integral formula bymeans of a Maxwell’s stress tensor. J. Lond. Math. Soc. 21(3), 210–218 (1946)CrossRefzbMATHGoogle Scholar
  55. 55.
    H. Weyl, Quantenmechanik und Gruppentheorie. Weyl, H. Publication: Zeitschrift für Physik 46(1–2), 1–46 (1927)Google Scholar
  56. 56.
    N.A. Wiegmann, Some theorems on matrices with real quaternion elements. Can. J. Math. 7, 191–201 (1955)CrossRefzbMATHMathSciNetGoogle Scholar
  57. 57.
    D.R. Wilkins, William Rowan Hamilton: mathematical genius. Phys. World 18, 33–36 (2005)Google Scholar
  58. 58.
    J.G. Winans, Quaternion physical quantities. Found. Phys. 7, 341–349 (1977)CrossRefMathSciNetGoogle Scholar
  59. 59.
    L.A. Wolf, Similarity of matrices in which the elements are real quaternions. Bull. Am. Math. Soc. 42, 737–743 (1936)CrossRefGoogle Scholar
  60. 60.
    R.M.W. Wood, Quaternionic eigenvalues. Bull. Lond. Math. Soc. 17, 137–138 (1985)CrossRefzbMATHGoogle Scholar
  61. 61.
    F. Zhang, Permanent Inequalities and Quaternion Matrices. Ph.D. dissertation, University of California at Santa Barbara, 1993Google Scholar
  62. 62.
    F. Zhang, On numerical range of normal matrices of quaternions. J. Math. Phys. Sci. 29(6), 235–251 (1995)zbMATHMathSciNetGoogle Scholar
  63. 63.
    F. Zhang, Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • João Pedro Morais
    • 1
  • Svetlin Georgiev
    • 2
  • Wolfgang Sprößig
    • 3
  1. 1.CIDMAUniversity of AveiroAveiroPortugal
  2. 2.Department of Differential EquationsUniversity of Sofia St Kliment Ohridski Faculty of Mathematics and InformaticsSofiaBulgaria
  3. 3.Institut für Angewandte AnalysisTU Bergakademie FreibergFreibergGermany

Personalised recommendations