On Factorization of Bicomplex Meromorphic Functions

  • K. S. Charak
  • D. Rochon
Part of the Trends in Mathematics book series (TM)


In this paper the factorization theory of meromorphic functions of one complex variable is promoted to bicomplex meromorphic functions. Many results of one complex variable case are seen to hold in bicomplex case, and it is found that there are results for meromorphic functions of one complex variable which are not true for bicomplex meromorphic functions. In particular, we show that for any bicomplex transcendental meromorphic function F, there exists a bicomplex meromorphic function G such that GF is prime even if the set:
$$ \{ a \in \mathbb{T}: F(w) + a\varphi (w) is not prime\} $$
is empty or of cardinality ie1 for any non-constant fractional linear bicomplex function Ø. Moreover, as specific application, we obtain six additional possible forms of factorization of the complex cosine cos z in the bicomplex space.


Bicomplex Numbers Factorization Meromorphic Functions 

Mathematics Subject Classification (2000)

30G 30D30 30G35 32A 32A30 


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  1. [1]
    L. Baoqin and S. Guodong, On Goldbach problem concerning factorization of meromorphic functions, Acta Math. Sinica 5(4) (1997), 337–344.CrossRefGoogle Scholar
  2. [2]
    W. Bergweiler, On factorization of certain entire functions, J. Math.Anal. Appl. 193 (1995), 1003–1007.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    C.T. Chuang and C.C. Yang, Fix-points and Factorization of meromorphic functions, World Scientific, 1990.Google Scholar
  4. [4]
    N. Fleury, M. Rausch de Traubenberg and R.M. Yamaleev, Commutative extended complex numbers and connected trigonometry, J. Math. Ann. and Appl. 180 (1993), 431–457.zbMATHCrossRefGoogle Scholar
  5. [5]
    F. Gross, Factorization of Meromorphic Function, U.S. Government Printing Office, Washington, D. C., 1972.Google Scholar
  6. [6]
    F. Gross, C.C. Yang and C. Osgoods, Primeable entire function, Nagoya Math J 51 (1973), 123–130.zbMATHMathSciNetGoogle Scholar
  7. [7]
    F. Gross, On factorization of meromorphic functions, Trans. Amer. Math. Soc. 131 (1968), 215–222.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Q. Jin and G. Yongxing, On Factorization of Meromorphic Function, Acta Math. Sinica 13(4) (1997), 509–512.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    T.W. Ng, Imprimitive parameterization of analytic curves and factorization of entire functions, J. London Math. Soc. 64 (2001), 385–394.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    T.W. Ng and C.C. Yang, On the common right factors of meromorphic functions, Bull. Austral. Math. Soc. 55 (1997), 395–403.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    T.W. Ng and C.C. Yang, Certain criteria on the existence of transcendental entire common right factor, Analysis 17 (1997), 387–393.zbMATHMathSciNetGoogle Scholar
  12. [12]
    T.W. Ng and C.C. Yang, On the composition of prime transcendental function and a prime polynomial, Pacific J. Math. 193 (2000), 131–141.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    Y. Noda, On factorization of entire function, Kodai Math J 4 (1981), 480–494.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    G.B. Price, An introduction to multicomplex spaces and functions, Marcel Dekker Inc., New York, 1991.zbMATHGoogle Scholar
  15. [15]
    D. Rochon, A bicomplex Riemann zeta function, Tokyo J. Math. 27 (2004), 357–369.zbMATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    D. Rochon, A Bloch Constant for Hyperholomorphic Functions, Complex Variables 44 (2001), 85–101.zbMATHMathSciNetGoogle Scholar
  17. [17]
    D. Rochon, A generalized Mandelbrot set for bicomplex numbers, Fractal 8 (2000), 355–368.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    D. Rochon, On a relation of bicomplex pseudoanalytic function theory to the complexified stationary Schrödinger equation, Complex Variables, 53(6) (2008), 501–521.zbMATHMathSciNetGoogle Scholar
  19. [19]
    D. Rochon, Sur une généralisation des nombres complexes: les tétranombres, M. Sc. Université de Montréal, 1997.Google Scholar
  20. [20]
    D. Rochon and M. Shapiro, On algebraic properties of bicomplex and hyperbolic numbers, Anal. Univ. Oradea, fasc. math., 11 (2004), 71–110.zbMATHMathSciNetGoogle Scholar
  21. [21]
    D. Rochon and S. Tremblay, Bicomplex Quantum Mechanics: I. The Generalized Schrödinger Equation, Adv. App. Cliff. Alg. 12(2) (2004), 231–248.CrossRefMathSciNetGoogle Scholar
  22. [22]
    D. Rochon and S. Tremblay, Bicomplex Quantum Mechanics: II. The Hilbert Space, Adv. App. Cliff. Alg. 16(2) (2006), 135–157.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    P.C. Rosenbloom, The fix-points of entire functions, Medd. Lunds Univ. Math. Sem. M.Riesz (1952), 187–192.Google Scholar
  24. [24]
    W. Rudin, Real and Complex Analysis 3rd ed., New York, McGraw-Hill, 1976.Google Scholar
  25. [25]
    B.V. Shabat, Introduction to Complex Analysis part II: Functions of Several Variables, American Mathematical Society, 1992.Google Scholar
  26. [26]
    G. Sobczyk, The hyperbolic number plane, Coll. Maths. Jour. 26(4) (1995), 268–280.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • K. S. Charak
    • 1
  • D. Rochon
    • 2
  1. 1.Department of MathematicsUniversity of JammuIndia
  2. 2.Département de mathématiques et d’informatiqueUniversité du Québec à Trois-RivièresTrois-Rivières QuébecCanada

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