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ON COMPLEX MULTIPLICATIVE DIFFERENTIATION

Year 2011, Volume 01, Issue 1, 75 - 85, 01.06.2011

Abstract

In the present paper we discuss multiplicative differentiation for complexvalued functions. Some drawbacks, arising with this concept in the real case, are explained satisfactorily. Some new difficulties, coming from the complex nature of variables, are discussed and they are outreached. Multiplicative Cauchy–Riemann conditions are established. Properties of complex multiplicative derivatives are studied.

References

  • L.V.Ahlfors, (1979), Complex Analysis, 3rd ed., McGraw-Hill, New York.
  • D.Aniszewska, (2007), Multiplicative Runge-Kutta method, Nonlinear Dynamics 50, 265-272.
  • A.E.Bashirov, E.Kurpınar, A. ¨Ozyapıcı, (2008), Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications, 337(1), 36–48.
  • A.E.Bashirov, E.Kurpınar, M.Riza, A. ¨Ozyapıcı, On modeling with multiplicative differential equa- tions, Applied Mathematics-A Journal of Chinese Universities, submitted for publication.
  • A.E.Bashirov, G.Bashirova, (2011), Dynamics of literary texts and diffusion, Communication and Media Technologies, Vol.1, Issue 3.
  • J.B.Conway, (1978), Functions of One Complex Variable, 2nd ed., Springer-Verlag, New York.
  • F.C´ordova-Lepe, M.Pinto, (2009), From quotient operation toward a proportional calculus, Interna- tional Journal of Mathematics, Game Theory and Algebra, 18(6), 527–536.
  • Yu.L.Daletskii, N.I.Teterina, (1972), Multiplicative stochastic integrals, Uspekhi Matematicheskikh Nauk, 27:2(164), 167–168.
  • W.R.Derrick, (1984), Complex Analysis and Applications, 2nd ed., Wadsworth International Group, Belmont, CA.
  • R.E.Greene, S.G.Krantz, (2006), Function Theory of One Complex Variable, 3rd ed., Amer. Math Society, Providence, RI.
  • M.Grossman, (1983), Bigeometric Calculus: A System with a Scale-Free Derivative, Archimedes Foun- dation, Rockport, MA.
  • M.Grossman and R.Katz, (1972), Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA.
  • R.L.Karandikar, (1982), Multiplicative decomposition of non-singular matrix valued continuous semi- martingales, The Annales of Probability, 10(4), 1088–1091.
  • W.Kasprzak, B.Lysik, M.Rybaczuk, (2004), Dimensions, Invariants Models and Fractals, Ukrainian Society on Fracture Mechanics, SPOLOM, Wroclaw-Lviv, Poland.
  • S.Lang, (2007), Complex Analysis, 3rd ed., Springer-Verlag, New York, NY.
  • B.P.Palka, (1991), An Introduction to Complex Function Theory, Springer-Verlag, New York, NY.
  • M.Riza, A. ¨Ozyapıcı, E.Kurpınar, Multiplicative finite difference methods, Quarterly of Applied Math- ematics, PII S0033-569X-09-01158-2, to appear in print.
  • M.Rybaczuk, A.Kedzia, W.Zielinski, (2001), The concepts of physical and fractional dimensions II. The differential calculus in dimensional spaces, Chaos Solutions Fractals 12, 2537–2552.
  • D.Sarason, (2007), Complex Functions Theory, Amer. Math Society, Providence, RI.
  • D.Slavik, (2007), Product Integration, Its History and Applications, Matfyz Press, Prague.
  • D.Stanley, (1999), A multiplicative calculus, Primus, IX(4), 310–326.
  • A.Uzer, (2010), Multiplicative type complex calculus as an alternative to the classical calculus, Com- puters and Mathematics with Applications, 60(10), 2725–2737.
  • V.Volterra, B.Hostinsky, (1938), Operations Infinitesimales Lineares, Herman, Paris.

Year 2011, Volume 01, Issue 1, 75 - 85, 01.06.2011

Abstract

References

  • L.V.Ahlfors, (1979), Complex Analysis, 3rd ed., McGraw-Hill, New York.
  • D.Aniszewska, (2007), Multiplicative Runge-Kutta method, Nonlinear Dynamics 50, 265-272.
  • A.E.Bashirov, E.Kurpınar, A. ¨Ozyapıcı, (2008), Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications, 337(1), 36–48.
  • A.E.Bashirov, E.Kurpınar, M.Riza, A. ¨Ozyapıcı, On modeling with multiplicative differential equa- tions, Applied Mathematics-A Journal of Chinese Universities, submitted for publication.
  • A.E.Bashirov, G.Bashirova, (2011), Dynamics of literary texts and diffusion, Communication and Media Technologies, Vol.1, Issue 3.
  • J.B.Conway, (1978), Functions of One Complex Variable, 2nd ed., Springer-Verlag, New York.
  • F.C´ordova-Lepe, M.Pinto, (2009), From quotient operation toward a proportional calculus, Interna- tional Journal of Mathematics, Game Theory and Algebra, 18(6), 527–536.
  • Yu.L.Daletskii, N.I.Teterina, (1972), Multiplicative stochastic integrals, Uspekhi Matematicheskikh Nauk, 27:2(164), 167–168.
  • W.R.Derrick, (1984), Complex Analysis and Applications, 2nd ed., Wadsworth International Group, Belmont, CA.
  • R.E.Greene, S.G.Krantz, (2006), Function Theory of One Complex Variable, 3rd ed., Amer. Math Society, Providence, RI.
  • M.Grossman, (1983), Bigeometric Calculus: A System with a Scale-Free Derivative, Archimedes Foun- dation, Rockport, MA.
  • M.Grossman and R.Katz, (1972), Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA.
  • R.L.Karandikar, (1982), Multiplicative decomposition of non-singular matrix valued continuous semi- martingales, The Annales of Probability, 10(4), 1088–1091.
  • W.Kasprzak, B.Lysik, M.Rybaczuk, (2004), Dimensions, Invariants Models and Fractals, Ukrainian Society on Fracture Mechanics, SPOLOM, Wroclaw-Lviv, Poland.
  • S.Lang, (2007), Complex Analysis, 3rd ed., Springer-Verlag, New York, NY.
  • B.P.Palka, (1991), An Introduction to Complex Function Theory, Springer-Verlag, New York, NY.
  • M.Riza, A. ¨Ozyapıcı, E.Kurpınar, Multiplicative finite difference methods, Quarterly of Applied Math- ematics, PII S0033-569X-09-01158-2, to appear in print.
  • M.Rybaczuk, A.Kedzia, W.Zielinski, (2001), The concepts of physical and fractional dimensions II. The differential calculus in dimensional spaces, Chaos Solutions Fractals 12, 2537–2552.
  • D.Sarason, (2007), Complex Functions Theory, Amer. Math Society, Providence, RI.
  • D.Slavik, (2007), Product Integration, Its History and Applications, Matfyz Press, Prague.
  • D.Stanley, (1999), A multiplicative calculus, Primus, IX(4), 310–326.
  • A.Uzer, (2010), Multiplicative type complex calculus as an alternative to the classical calculus, Com- puters and Mathematics with Applications, 60(10), 2725–2737.
  • V.Volterra, B.Hostinsky, (1938), Operations Infinitesimales Lineares, Herman, Paris.

Details

Primary Language English
Journal Section Research Article
Authors

Agamirza E. BASHİROV This is me
İnstitute of Cybernetics, National Academy of Sciences, Baku, Azerbaijan


Mustafa RİZA This is me
Department of Mathematics, Eastern Mediterranean University, Gazimagusa - North Cyprus, via Mersin 10 - Turkey

Publication Date June 1, 2011
Published in Issue Year 2011, Volume 01, Issue 1

Cite

Bibtex @ { twmsjaem761811, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2011}, volume = {01}, number = {1}, pages = {75 - 85}, title = {ON COMPLEX MULTIPLICATIVE DIFFERENTIATION}, key = {cite}, author = {Bashirov, Agamirza E. and Riza, Mustafa} }