On the relation between the growth and the Taylor coefficients of entire solutions to the higher-dimensional Cauchy–Riemann system in \(\mathbb R^{n+1}\). (English) Zbl 1108.30041

The authors establish an interesting relation between the coefficients of the Taylor expansion of entire holomorphic functions in Clifford analysis and the growth order of the maximum modulus which is known from classical function theory. This allows to determine the growth order without knowing the maximum modulus. Examples of any finite order are given, the generalized trigonometric functions have the order 1.


30G35 Functions of hypercomplex variables and generalized variables
30D15 Special classes of entire functions of one complex variable and growth estimates
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[1] Abul-Ez, M.A.; Constales, D., Basic sets of polynomials in Clifford analysis, Complex var. theory appl., 14, 1-4, 177-185, (1990) · Zbl 0663.41009
[2] Abul-Ez, M.A.; Constales, D., On the order of basic series representing Clifford-valued functions, Appl. math. comput., 142, 2-3, 575-584, (2003) · Zbl 1058.30040
[3] Abul-Ez, M.A.; Constales, D., On the convergence properties of basic series representing special monogenic functions, Arch. math., 81, 1, 62-71, (2003) · Zbl 1052.30047
[4] Brackx, F.; Delanghe, R.; Sommen, F., Clifford analysis, Res. notes math., vol. 76, (1982), Pitman London · Zbl 0529.30001
[5] Constales, D.; Kraußhar, R.S., Representation formulas for the general derivatives of the fundamental solution of the cauchy – riemann operator in Clifford analysis and applications, Z. anal. anwendungen, 21, 3, 579-597, (2002) · Zbl 1018.30037
[6] De Almeida, R.; Kraußhar, R.S., On the asymptotic growth of monogenic functions, Z. anal. anwendungen, 24, 4, 791-813, (2005) · Zbl 1095.30044
[7] Delanghe, R., On regular points and Liouville’s theorem for functions with values in a Clifford algebra, Simon stevin, 44, 55-66, (1970-1971) · Zbl 0204.09402
[8] Delanghe, R.; Sommen, F.; Souček, V., Clifford algebra and spinor valued functions, (1992), Kluwer Dordrecht · Zbl 0747.53001
[9] Gürlebeck, K.; Sprössig, W., Quaternionic and Clifford calculus for physicists and engineers, (1997), John Wiley & Sons Chichester · Zbl 0897.30023
[10] Jank, G.; Volkmann, L., Meromorphe funktionen und differentialgleichungen, (1985), Birkhäuser Basel
[11] Hayman, W.K., The local growth of power series: A survey of the wiman – valiron method, Canad. math. bull., 17, 317-358, (1974) · Zbl 0314.30021
[12] Hempfling, T.; Kraußhar, R.S., Order theory for isolated points of monogenic functions, Arch. math., 80, 4, 406-423, (2003) · Zbl 1043.30031
[13] Kravchenko, K., Applied quaternionic analysis, Res. exp. math., vol. 28, (2003), Heldermann Lemgo · Zbl 1014.78003
[14] Lindelöf, E., Sur la détermination de la croissance des fonctions entières définies par un développement de Taylor, Darb. bull., 27, 2, 213-226, (1903) · JFM 34.0440.01
[15] Nevanlinna, R., Zur theorie der meromorphen funktionen, Acta math., 46, 1-99, (1925) · JFM 51.0254.05
[16] Pringsheim, A., Elementare theorie der ganzen transzendenten funktionen von endlicher ordnung, Math. ann., 58, 257-342, (1904) · JFM 35.0405.01
[17] Shah, S.M., On the lower order of integral functions, Bull. amer. math. soc., 52, 1046-1052, (1946) · Zbl 0061.15109
[18] Sprössig, W., Clifford analysis and its applications in mathematical physics, Cubo matemática educacional, 4, 2, 253-314, (2002)
[19] Valiron, G., Lectures on the general theory of integral functions, (1949), Chelsea New York
[20] Wiman, A., Über den zusammenhang zwischen dem maximalbetrage einer analytischen funktion und dem größten gliede der zugehörigen taylorschen reihe, Acta math., 37, 305-326, (1914) · JFM 45.0641.02
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