ABSTRACT: ON MULTIPLICATIVE COMPLEX INTEGRAL Agamirza E. Bashirov and Sajedeh Norozpour Eastern Mediterranean University Gazimagusa, North Cyprus
[email protected] [email protected] In 1972 Grossman and Katz [1] proposed alternative calculi to the calculus of Newton and Leibnitz. Among them multiplicative calculus is most popular. This pioneering work initiated numerous studies. What is the difference between Newtonian and multiplicative calculi? Both them describe the system of knowledges which is called calculus. Newtonian calculus describes it with reference to the linear function while multiplicative calculus with reference to the exponential function. Therefore, every theorem in one of them has an analog in the other one. Is it reasonable a consideration of a new calculus while Newtonian calculus is already well established? It is reasonable because different presentations of calculus provide different views. If proving some theorem is difficult in one of them, it may be more visible and easy in another one. In such a way, in [2] it is proved the non-analyticity of some infinitely many times differentiable function in a rather compact form by means of multiplicative calculus while its prove by means of Newtonian calculus is rather complicated. Motivated from the exponential nature of complex numbers, in [3] complex differentiation was revised by means of multiplicative calculus. In the present presentation we are aiming to present the results related to multiplicative complex integration from [4]. Since the multivalued nature of complex logarithm, which has an underlying role in multiplicative calculus, complex multiplicative calculus is not one-to-one transformation of ordinary complex calculus. In particular, unlike ordinary complex integral, multiplicative complex integral has a multivalued nature as well. This makes its properties to be in the form of inclusion rather than in the form of equality. The Cauchy integral formula is not affected in the multiplicative case, that is, the multiplicative complex integral does not count residues, etc. All these demonstrate that we should expect nontrivial transformation of complex calculus to multiplicative case, that is not yet completely formalized. We are going to discuss these issues in our presentation. 1
References [1] Grossman, M., and Katz, R., Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA, 1972. ¨ [2] Bashirov, A. E., Kurpınar, E., and Ozyapıcı, A., Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications, 337(1), 36–48, 2008. [3] Bashirov, A. E., Riza, M., On complex multiplicative differentiation, TWMS Journal of Applied and Engineering Mathematics, 1(1), 75–85, 2011. [4] Bashirov, A. E., Norozpour S., (submitted), On complex multiplicative integration, TWMS Journal of Applied and Engineering Mathematics, (to appear).
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