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On some new properties of quaternion functions. (English. Russian original) Zbl 1408.30044
J. Math. Sci., New York 235, No. 5, 557-603 (2018); translation from Sovrem. Mat. Prilozh. 101 (2016).
Summary: Quaternions discovered by W. R. Hamilton made a great contribution to the progress in noncommutative algebra and vector analysis. However, the analysis of quaternion functions has not been duly developed. The matter is that the notion of a derivative of quaternion functions of a quaternion variable has not been known until recently. The author has succeeded in improving the situation. The present work contains an account of the results obtained by him in this direction. The notion of an \(\mathbb{H}\)-derivative is introduced for quaternion functions of a quaternion variable. The existence of an \(\mathbb{H}\)-derivative of elementary functions is established retaining the well-known formulas for the corresponding functions from complex (real) analysis. The rules on the \(\mathbb{H}\)-differentiation of a sum, a product, and an inverse function are formulated and proved. Necessary and sufficient conditions for the existence of an \(\mathbb{H}\)-derivative are established. The notions of \(\mathbb{C}^2\)-differentiation and \(\mathbb{C}^2\)-holomorphy are introduced for quaternion functions of a quaternion variable. Three equivalent conditions are found, each of them being a necessary and sufficient one for \(\mathbb{C}^2\)-differentiation. Representations by an integral and a power series are given for \(\mathbb{C}^2\)-holomorphic functions. It is proved that the harmonicity of functions \(f(z)\), \(z\cdot f(z)\), and \(f(z)\cdot z\) is the necessary and sufficient condition for a function \(f\) to be Fueter-regular.

MSC:
30G35 Functions of hypercomplex variables and generalized variables
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References:
[1] Aleksandrova, NV, Formation of the basic concepts of the vector calculus, Istor.-Mat. Issled., 26, 205-235, (1982) · Zbl 0531.01005
[2] A. Alle’gret, Essai sur la calcul des quaternions, Paris (1862).
[3] R. Argand, Essai sur une manière de représenter les quantit’s imaginaires dans les constructions géométriques, Gauthier-Villars, Paris (1874). · JFM 06.0234.04
[4] J. R. Argand, Imaginary Quantities: Their Geometrical Representation, New York (1881).
[5] Arias de Reyna Martinez, J., Differentiable quaternion functions, Gac. Mat. Madrid (1), 27, 127-129, (1975)
[6] Arnold, VI, The geometry of spherical curves and quaternion algebra, Russ. Math. Surv., 50, 1-68, (1995)
[7] Artin, E., Zur Theorie der hyperkomplexen Zahlen, Abh. Math. Sem. Univ. Hamburg, 5, 251-260, (1927) · JFM 53.0114.03
[8] Artin, E., Zur Arithmetik hyperkomplexer Zahlen, Abh. Math. Sem. Univ. Hamburg, 5, 261-289, (1927) · JFM 53.0115.01
[9] Bantsuri, L., On the relation between the differentiability condition and the condition of the existence of generalized gradient, Bull. Georgian Acad. Sci., 171, 241-242, (2005)
[10] L. Bantsuri, “On the relationship between the conditions of differentiability and existence of generalized gradient,” Proc. III Int. Conf. of the Georgian Math. Union Batumi (2012), pp. 65-66.
[11] Bantsuri, LD; Oniani, GG, On the differential properties of functions of bounded variation in Hardy sense, Proc. A. Razmadze Math. Inst., 139, 93-95, (2005) · Zbl 1096.26503
[12] Bantsuri, LD; Oniani, GG, On differential properties of functions of bounded variation, Anal. Math., 38, 1-17, (2012) · Zbl 1299.26036
[13] E. T. Bell, Men of Mathematics, New York (1937).
[14] E. T. Bell, The Development of Mathematics, McGraw-Hill, New York (1945). · Zbl 0061.00101
[15] Bezhko, AP; Branets, VN; Zakharov, YM; Shmiglevski, IP, Application of quaternions in the theory of finite rotation of a solid body, Izv. Akad. Nauk SSSR. Mekh. Tv. Tela, 1, 123-134, (1972)
[16] A. N. Bogolyubov, Mathematicians and Mechanicians [in Russian], Naukova Dumka, Kiev (1983).
[17] A. I. Borodin and A. S. Bugaǐ, Biographical Dictionary of Mathematicians [in Russian], Radyans’ka Shkola, Kiev (1979). · Zbl 0447.01008
[18] C. B. Boyer, History of Analytic Geometry, Script. Math., New York (1956). · Zbl 0073.00203
[19] Brackx, F. F., Non-(k)-monogenic points of functions of a quaternion variable, 138-149, (1976), Berlin, Heidelberg · Zbl 0346.30038
[20] Brackx, F. F., The exponential function of a quaternion variable, Applicable Analysis, 8, 265-276, (1979) · Zbl 0399.30039
[21] Brand, L., The roots of a quaternion, Am. Math. Monthly, 49, 519-520, (1942) · Zbl 0061.01408
[22] V. N. Branets and I. P. Shmyglevski, “Kinematic problems of orientation in a rotating coordinate system,” Izv. Akad. Nauk SSSR. Mekh. Tv. Tela, 6 (1972).
[23] V. N. Branets and I. P. Shmiglevski, “Application of quaternions in the control of the angular position of a rigid body,” Izv. Akad. Nauk SSSR. Mekh. Tv. Tela, 4 (1972).
[24] V. N. Branets and I. P. Shmyglevski, Applications of Quaternions in Problems on the Motion of Rigid Bodies [in Russian], Nauka, Moscow (1973).
[25] Buff, JJ, Characterization of an analytic function of a quaternion variable, Pi Mu Epsilon J., 5, 387-392, (1973)
[26] Bureš, J.; Souček, V., “Complexes of invariant operators in several quaternionic variables,” Complex Var, Elliptic Equ., 51, 463-485, (2006) · Zbl 1184.30044
[27] J. G. Coffin, Vector Analysis. An Introduction to Vector Methods and Their Various Applications to Physics and Mathematics, Wiley, New York-London (1909). · JFM 40.0141.01
[28] J. L. Coolidge, The Geometry of the Complex Domain, Oxford Univ. Press, New York (1924). · JFM 50.0394.01
[29] M. J. Crowe, A History of Vector Analysis. The Evolution of the Idea of a Vectorial System, Univ. Notre Dame Press, Notre Dame, Indiana-London (1967). · Zbl 0165.00303
[30] C. Crubin, “Derivation of the quaternion scheme via the Euler axis and agle,” Int. J. Control, 8, No. 3 (1968).
[31] Cullen, CG, An integral theorem for analytic intrinsic functions on quaternions, Duke Math. J., 32, 139-148, (1965) · Zbl 0173.09001
[32] Deavours, CA, The quaternion calculus, Am. Math. Monthly, 80, 995-1008, (1973) · Zbl 0282.30040
[33] Dickson, LE, Arithmetic of quaternions, Proc. London Math. Soc., 20, 225-232, (1921) · JFM 48.0130.06
[34] J. Dieudonné, Algèbre linéaire et gómétrie élémentaire, Hermann, Paris (1964). · Zbl 0185.48802
[35] Dirac, PAM, Application of quaternions to Lorentz transformations, Proc. Roy. Irish Acad. Sect. A., 50, 261-270, (1945) · Zbl 0060.44002
[36] P. Du Val, Homographies, Quaternions and Rotations, Oxford Math. Monogr., Clarendon Press, Oxford (1964). · Zbl 0128.15403
[37] Dzagnidze, OP, On the differentiability of functions of two variables and of indefinite double integrals, Proc. A. Razmadze Math. Inst., 106, 7-48, (1993) · Zbl 0836.26007
[38] O. P. Dzagnidze, “Separately continuous functions in a new sense are continuous,” Real Anal. Exchange, 24, No. 2, 695-702 (1998/99). · Zbl 0967.26010
[39] Dzagnidze, O., A necessary and sufficient condition for differentiability functions of several variables, Proc. A. Razmadze Math. Inst., 123, 23-29, (2000) · Zbl 0971.26007
[40] Dzagnidze, O., Some new results on the continuity and differentiability of functions of several real variables, Proc. A. Razmadze Math. Inst., 134, 1-138, (2004) · Zbl 1076.26009
[41] Dzagnidze, O., A criterion of joint ℂ-differentiability and a new proof of Hartogs’ main theorem, J. Appl. Anal., 13, 13-17, (2007) · Zbl 1147.58010
[42] O. Dzagnidze, “On the differentiability of quaternion functions,” Tbilis. Math. J., 5, 1-15 (2012); http://arxiv.org/pdf/1203.5619.pdf. · Zbl 1280.30022
[43] Dzagnidze, O., On the differentiability of real, complex and quaternion functions, Bull. TICMI, 18, 93-109, (2014) · Zbl 1325.30043
[44] Dzagnidze, O., Necessary and sufficient conditions for the ℍ-differentiability of quaternion functions, Georgian Math. J., 22, 215-218, (2015) · Zbl 1318.30079
[45] Dzagnidze, O., ℂ2-Differentiability of quaternion functions and their representation by integrals and series, Proc. A. Razmadze Math. Inst., 167, 19-27, (2015) · Zbl 1332.30074
[46] Eichler, M., Allgemeine Integration einiger partieller Differentialgleichungen der mathematischen Physik durch Quaternionenfunktionen, Comment. Math. Helv., 12, 212-224, (1940) · JFM 66.0440.01
[47] Faddeev, DK, Construction of algebraic domains whose Galois group is a quaternion group, Trans. Leningrad Univ., 3, 17-23, (1937)
[48] Faddeev, DK, Construction of fields of algebraic numbers whose Galois group is a group of quaternion units, Dokl. Akad. Nauk SSSR, 47, 404-407, (1945) · Zbl 0061.06006
[49] D. K. Faddeev and R. O. Kuzmin, Arithmetic and Algebra of Complex Numbers [in Russian], Moscow (1939).
[50] M. I. Falcao, I. F. Crus, and H. R. Malonek, “Remarks on the generation of monogenic functions,” in: Proc. 17 Int. Conf. “Application of Computer Sciences and Mathematics in Architecture and Civil Engineering (July 12-14, 2006), Weimar (2006).
[51] O. F. Fischer, Universal Mechanics and Hamilton’s Quaternions, Axion Institute, Stockholm (1951). · Zbl 0044.38501
[52] O. F. Fischer, Five Mathematical Structural Models, Axion Institute, Stockholm (1957). · Zbl 0078.37202
[53] Fueter, R., Analytische Funktionen einer Quaternionenvariablen, Comment. Math. Helv., 4, 9-20, (1932) · JFM 58.0144.05
[54] Fueter, R., Die Funktionentheorie der Differentialgleichungen Δ\(u\) = 0 und ΔΔ\(u\) = 0 mit vier reellen Variablen, Comment. Math. Helv., 7, 307-330, (1934) · Zbl 0012.01704
[55] Fueter, R., Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen, Comment. Math. Helv., 8, 371-378, (1935) · JFM 62.0120.04
[56] Fueter, R., Die Singularitäten der eindeutigen regulären Funktionen einer Quaternionenvariablen. I, Comment. Math. Helv., 9, 320-334, (1937) · JFM 63.0311.01
[57] Fueter, R., Integralsätze für reguläre Funktionen einer Quaternionen-Variablen, Comment. Math. Helv., 10, 306-315, (1938) · JFM 64.0326.01
[58] Fueter, R., Über einen Hartogs’schen Satz, Comment. Math. Helv., 12, 75-80, (1939) · JFM 65.0363.03
[59] Fueter, R., Die Funktionentheorie der Dirac’schen Differentialgleichungen, Comment. Math. Helv., 16, 19-28, (1944) · Zbl 0060.24506
[60] Gentili, G.; Struppa, DC, A new theory of regular functions of a quaternionic variable, Adv. Math., 216, 279-301, (2007) · Zbl 1124.30015
[61] Gormley, PG, Stereographic projection and the linear fractional group of transformations of quaternions, Proc. Roy. Irish Acad. Sect. A, 51, 67-85, (1947)
[62] R. P. Graves, Life of Sir William Rowan Hamilton, Dublin (1882-1891).
[63] K. Gürlebeck and W. Sprössig, Quaternionic Analysis and Elliptic Boundary-Value Problems, Math. Res., 56, Akademie-Verlag, Berlin (1989).
[64] Haddad, Z., Two remarks on the quaternions, Pi Mu Epsilon J., 7, 221-231, (1981) · Zbl 0465.30003
[65] W. R. Hamilton, Lectures of Quaternions, Hodges & Smith, Dublin (1853).
[66] W. R. Hamilton, Elements of Quaternions, Chelsea, New York (1866).
[67] W. R. Hamilton, Mathematical Papers, Vol. I. Geometrical Optics, Cambridge Univ. Press (1931). · JFM 57.0034.09
[68] W. R. Hamilton, Mathematical Papers, Vol. II. Dynamics, Cambridge Univ. Press (1940). · Zbl 0024.36304
[69] W. R. Hamilton, Mathematical Papers, Vol. III. Algebra, Cambridge Univ. Press (1967).
[70] W. R. Hamilton, Mathematical Papers, Vol. IV. Geometry, Analysis, Astronomy, Probability and Finite Differences, Miscellaneous, Cambridge Univ. Press (2000). · Zbl 0973.01105
[71] W. R. Hamilton, Selected Works. Optics. Dynamics. Quaternions [Russian translation], Nauka, Moscow (1994).
[72] H. Hankel, Theorie der complexen Zahlensysteme, Leipzig (1867).
[73] H. Hankel, Theory of Complex Numerical Systems, Mainly Ordinary Imaginary Numbers and Hamilton Quaternions Together with Their Geometric Interpretations [Russian translation], Kazan Univ., Kazan (1912).
[74] Hausdorff, F., Zur Theorie der Systeme comoplex Zahlen, Leipz. Ber., 52, 43-61, (1900)
[75] Häfeli, H., Quaternionengeometrie und das Abbildungsproblem der regulären Quaternionenfunktionen, Comment. Math. Helv., 17, 135-164, (1945) · Zbl 0060.24514
[76] S. Hoshi, “On some theories of quaternion functions,” Mem. Fac. Eng. Miyazaki Univ., 3 (1962).
[77] A. Hurwitz, “Über die Zahlentheorie der Quaternionen,” in: Nachrichten von der K. Gesellschaft der Wissenschaften zu Götingen, Mathematisch-Physikalische Klasse (1896), pp. 313-340; A. Hurwitz, Mathematische Werke, Band II. Zahlentheorie, Algebra and Geometrie, Basel (1933), pp. 303-330. · JFM 27.0162.01
[78] W. V. Ignatowski, Die Vektoranalysis und ihre Anwendung in der theoretischen Physik, B. G. Teubner, Leipzig (1909-1910). · JFM 52.0774.01
[79] C. J. Joly, A Manual of Quaternions, Macmillan, London (1905). · JFM 36.0144.04
[80] I. L. Kantor and A. S. Solodovnikov, Hypercomplex Numbers [in Russian], Nauka, Moscow (1973). · Zbl 0669.17002
[81] P. Kelland and P. G. Tait, Introduction to Quaternions, London (1873). · JFM 06.0229.02
[82] Kramar, FD, Vector analysis at the end of the 18th and beginning of the 19th centuries, Istor.-Mat. Issled., 15, 225-290, (1963)
[83] V. V. Kravchenko and M. V. Shapiro, Integral Representations for Spatial Models of Mathematical Physics, Pitman Res. Notes Math. Ser., 351, Longman, Harlow (1996). · Zbl 0872.35001
[84] Kryloff, NM, Sur les quaternions de W. R. Hamilton et la notion de la monogénéité, Dokl. Akad. Nauk SSSR, 55, 787-788, (1947) · Zbl 0029.03708
[85] Lawrence, WR, A particular application of extended dimensional complex numbers, Int. J. Math. Sci. Educ., 10, 569-574, (1979) · Zbl 0436.12017
[86] Linnik, YV, A centenary of the discovery of quaternions, Priroda, 2, 49, (1944)
[87] Linnik, YV, Quaternions and Cayley numbers; some applications of the arithmetic of quaternions, Usp. Mat. Nauk (N.S.), 4, 49-98, (1949)
[88] Linnik, YV, Application of the theory of Markov chains to the arithmetic of quaternions, Usp. Mat. Nauk (N.S.), 9, 203-210, (1954)
[89] Linnik, YV, Markov chains in the analytical arithmetic of quaternions and matrices, Vestn. Leningrad. Univ., 11, 63-68, (1956)
[90] Yu. V. Linnik and A. V. Malyshev, “Application of the arithmetic of quaternions to the theory of ternary quadratic forms and to the decomposition of numbers into cubes,” Usp. Mat. Nauk (N.S.), 8, No. 5 (57), 3-71 (1953); corrections: 10, No. 1(63), 243-244 (1955).
[91] H. Looman, “Über die Cauchy-Riemannschen Differentialgleichungen,” Gott. Nachr., 97-108 (1923). · JFM 49.0709.01
[92] Lugojan, S., Quaternionic derivability, Anal. Univ. Timi¸soara, 29, 175-190, (1991) · Zbl 0795.30041
[93] Lugojan, S., About the symmetry of the quaternionic derivatives, Bul. Ştiinṭ. Univ. Politeh. Timiş. Ser. Mat. Fiz., 42 (56), 7-11, (1997) · Zbl 1057.30504
[94] S. Lugojan, “Quaternionic derivability, II,” in: Quaternionic Structures in Mathematics and Physics, Int. Sch. Adv. Stud. SISSA, Trieste (1998), pp. 189-196. · Zbl 1058.30048
[95] A. Macfarlane, International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics, University Press, Dublin (1904). · JFM 35.0057.13
[96] A. Macfarlane, Vector Analysis and Quaternions, Wiley, New York (1906). · JFM 37.0115.07
[97] H. R. Malonek, M. I. Falcao, and A. M. Silva, “Maple tools for modified quaternionic analysis,” in: Proc. 17 Int. Conf. “Application of Computer Sciences and Mathematics in Architecture and Civil Engineering (July 12-14, 2006), Weimar (2006).
[98] J. E. Marsden, Elementary Classical Analysis, W. H. Freeman, San Francisco (1974). · Zbl 0285.26005
[99] Meǐlihzon, AS, On the assignment of monogeneity to quaternions, Dokl. Akad. Nauk SSSR (N.S.), 59, 431-434, (1948)
[100] Menchoff, D., Sur la generalisation des conditions de Cauchy-Riemann, Fundam. Math., 25, 59-97, (1935) · JFM 61.0305.01
[101] E. L. Mitchell and A. E. Rogers, Quaternion Parameters in the Simulation of a Spinning Rigid Body, Simulation, New York (1968).
[102] P. Molenbroek, Anwendung der Quaternionen anf die Geometrie, Leiden (1893). · JFM 25.1071.03
[103] P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York-Toronto-London (1953). · Zbl 0051.40603
[104] C. V. Mourey, La vrai théorie des quatities négatives et des quantités prétendues imaginaires, Paris (1828).
[105] Nef, W., Über die singulären Gebilde der regularen Funktionen einer Quaternionenvariabeln, Comment. Math. Helv., 15, 144-174, (1943) · Zbl 0027.30902
[106] Nef, W., Die unwesentlichen Singularitäten der regulären Funktionen einer Quaternionenvariabeln, Comment. Math. Helv., 16, 284-304, (1944) · Zbl 0060.24509
[107] Niven, I., Equations in quaternions, Am. Math. Mon., 48, 654-661, (1941) · Zbl 0060.08002
[108] Niven, I., The roots of a quaternion, Am. Math. Mon., 49, 386-388, (1942) · Zbl 0061.01407
[109] Oniani, G., On the relation between conditions of the differentiability and existence of the strong gradient, Proc. A. Razmadze Math. Inst., 132, 151-152, (2003) · Zbl 1054.26502
[110] Oniani, GG, On the interrelation between differentiability conditions and the existence of a strong gradient, Math. Notes, 77, 84-89, (2005) · Zbl 1101.26012
[111] Pall, G., On the arithmetic of quaternions, Trans. Am. Math. Soc., 47, 487-500, (1940) · Zbl 0023.19904
[112] Pall, G.; Taussky, O., Application of quaternions to the representations of a binary quadratic form as a sum of four squares, Proc. Roy. Irish Acad. Sect. A, 58, 23-28, (1957) · Zbl 0079.27103
[113] Peters, M., Ternäre und quaternäre quadratische Formen und Quaternionenalgebren, Acta Arithmetica, 15, 329-365, (1969) · Zbl 0188.11202
[114] Z. Piotrowski, “The genesis of separate versus joint continuity,” in: Real Functions ’94 (Liptovský Ján, 1994), Tatra Mt. Math. Publ., 8, 113-126 (1996).
[115] L. S. Polak, W. R. Hamilton and the Stationary Action Principle [in Russian], Akad. Nauk SSSR, Moscow (1936). · JFM 62.1036.06
[116] Polak, LS, William Rowan Hamilton (on the 150th anniversary of his birth), Tr. Inst. Istor. Estestv. Tekhnol., 15, 206-276, (1956)
[117] I. I. Privalov, Introduction to the Theory of Functions of a Complex Variable [in Russian], Fizmatgiz, Moscow (1960).
[118] Romoniro, A., Gli Elementi imaghinarii nela geometria, Battaglini’s Fiornale di Matematica, 35, 242-258, (1897)
[119] Schuler, B., Zur Theorie der regularen Funktionen einer Quaternionen-Variabeln, Comment. Math. Helv., 10, 327-342, (1937) · JFM 64.0325.04
[120] A. C. Robinson, “On the use of quaternions in simulation of rigid-body motuion,” WADD Tech. Rep., No. 58-67 (1957).
[121] Rose, A., On the use of a complex (quaternion) velocity potential in three dimensions, Comment. Math. Helv., 24, 135-148, (1950) · Zbl 0038.05703
[122] Salzer, HE, An elementary note on powers of quaternions, Am. Math. Mon., 59, 298-300, (1952) · Zbl 0046.01602
[123] B. V. Shabat, Introduction to Complex Analysis. Part II. Functions of Several Variables [in Russian], Nauka, Moscow (1976).
[124] G. E. Shilov. Mathematical Analysis. Functions of Several Real Variables [in Russian], Nauka, Moscow (1972).
[125] H. H. Snyder, A Hypercomplex Function Theory Associated with Laplace’s Equation, Berlin (1968). · Zbl 0185.34701
[126] Sommen, F., Plane wave decompositions of monogenic functions, Ann. Pol. Math., 49, 101-114, (1988) · Zbl 0673.30038
[127] P. O. Somov, Vector Analysis and Its Applications [in Russian], Saint Petersburg (1907).
[128] E. Study, Geometrie der Dynamen, Leipzig (1903). · JFM 34.0741.10
[129] Sudbery, A., Quaternionic analysis, Math. Proc. Cambridge Philos. Soc., 85, 199-224, (1979) · Zbl 0399.30038
[130] P. G. Tait, Elementary Treatise on Quaternions, Oxford (1867). · JFM 06.0229.01
[131] P.-G. Tait, Traité élémentaire des quaternions, Gauthier-Villars, Paris (1882).
[132] Taussky, O., Sums of squares, Am. Math. Mon., 77, 805-830, (1970) · Zbl 0208.05202
[133] Tolstov, G., Sur la fonctions bornées vérifiant les conditions de Cauchy-Riemann, Mat. Sb. (N.S.), 10, 79-85, (1942) · Zbl 0063.07812
[134] Tolstov, GP, On partial derivatives, Izv. Akad. Nauk SSSR. Ser. Mat., 13, 425-446, (1949)
[135] B. P. Uskes, “A new methiod of performing digital control system attitude computation using quaternions,” AIAA J., 8, No. 1 (1970).
[136] Vakhania, NN, Random vectors with values in quaternion Hilbert spaces, Theory Probab. Appl., 43, 99-115, (1999) · Zbl 0943.60004
[137] S. Valentiner, Vektoranalysis, Leipzig (1907).
[138] B. A. Venkov, “On the arithmetic of quaternions,” Izv. Akad. Nauk SSSR, Ser. VI. Fiz.-Mat. Nauki, 16, 205-220, 221-246 (1922).
[139] Venkov, BA, On the arithmetic of quaternions, Izv. Akad. Nauk SSSR, Ser. VII. Fiz.-Mat. Nauki, 5, 489-504, (1929)
[140] Venkov, BA, On the arithmetic of quaternions, Izv. Akad. Nauk SSSR, Ser. VII. Fiz.-Mat. Nauki, 6, 535-562, (1929)
[141] Venkov, BA, On the arithmetic of quaternions, Izv. Akad. Nauk SSSR, Ser. VII. Fiz.-Mat. Nauki, 7, 607-622, (1929)
[142] J. Warren, A Treatise on the Geometrical Representation of the Square Roots of Negative Quantities, Cambridge (1828).
[143] C. Wessel, Essai sur la représentation analytique de la direction, Copenhague (1897).
[144] Whittaker, ET, The Hamiltonian revival, Math. Gazette, 24, 153-158, (1940)
[145] Windred, G., History of the theory of imaginary and complex quantities, Math. Gazette, 14, 533-541, (1929) · JFM 55.0026.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.