ON COMPLEX MULTIPLICATIVE INTEGRATION
In the present paper, we extend the multiplicative integral to complex-valued functions of complex variable. The main difficulty in this way, that is, the multi-valued nature of the complex logarithm is avoided by division of the interval of integration to a finite number of local intervals, in each of which the complex logarithm can be localized in one of its branches. Interestingly, the complex multiplicative integral became a multivalued function. Some basic properties of this integral are considered. In particular, it is proved that this integral and the complex multiplicative derivative are bonded in a kind of fundamental theorem.
Keywords:
complex calculus, complex integral multiplicative calculus, fundamental theorem of calculus.,
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- [1] Ahlfors,L.V., (1979), Complex Analysis, 3rd ed., McGraw-Hill, New York.
- [2] Bashirov,A.E., (2013), On line and double multiplicative integrals, TWMS Journal of Applied and Engineering Mathematics, 3(1), 103–107.
- [3] Bashirov,A.E. and Bashirova,G., (2015), Literature as a diffusion process, in: Brownian Motion: Elements, Dynamics and Applications, Editors: M.A. McKibben and M. Webster, Nova Science Publications, pp. 201-218.
- [4] Bashirov,A.E., Kurpınar,E., and Ozyapıcı,A., (2008), Multiplicative calculus and its applications, ¨ Journal of Mathematical Analysis and Applications, 337(1), 36–48.
- [5] Bashirov,A.E., Mısırlı,E., Tando˘gdu,Y., and Ozyapıcı,A., (2011), On modeling with multiplicative ¨ differential equations, Applied Mathematics – Journal of Chinese Universities, 26(4), 425–438.
- [6] Bashirov,A.E. and Riza,M., (2011), On complex multiplicative differentiation, TWMS Journal of Applied and Engineering Mathematics, 1(1), 51–61.
- [7] C´ordova-Lepe,F., (2006), The multiplicative derivative as a measure of elasticity in economics, TEMAT-Theaeteto Atheniensi Mathematica, 2(3), online.
- [8] C´ordova-Lepe,F. and Pinto,M., (2009), From quotient operation toward a proportional calculus, International Journal of Mathematics, Game Theory and Algebra, 18(6), 527–536.
- [9] Daletskii,Yu.L. and Teterina,N.I., (1972), Multiplicative stochastic integrals, Uspekhi Matematicheskikh Nauk, 27(2:164), 167–168.
- [10] Florack,L. and Van Assen,H., (2011), Multiplicative calculus in biomedical image analysis, Journal of Mathematical Imaging and Vision, doi: 10.1007/s10851-011-0275-1.
- [11] Greene,R.E. and Krantz,S.G., (2006), Function Theory of One Complex Variable, 3rd ed., AMS, Providence, RI.
- [12] Grossman,M., (1983), Bigeometric Calculus: A System with a Scale-Free Derivative, Archimedes Foundation, Rockport, MA.
- [13] Grossman,M. and Katz,R., (1972), Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA.
- [14] Karandikar,R.L., (1982), Multiplicative decomposition of non-singular matrix valued continuous semimartingales, The Annals of Probability, 10(4), 1088–1091.
- [15] Mısırlı,E. and G¨urefe,Y., (2011), Multiplicative Adams Bashforth–Moulton methods, Numerical Algorithms, 57(4), 425–439
- [16] Ozyapıcı,A., Riza,M., Bilgehan,B., and Bashirov,A.E., (2014), On multiplicative and Volterra mini- ¨ mization method, Numerical Algorithms, 67(3), 623–636.
- [17] Ozyapıcı,A. and Bilgehan,E., (2016), Finite product representation via multiplicative calculus and its ¨ applications to exponential signal processing, Numerical Algorithms, 71(2), 475–489.
- [18] Riza,M., Ozyapıcı,A., and Kurpınar,E., (2009), Multiplicative finite difference methods, ¨ Quarterly of Applied Mathematics, 67(4), 745–754.
- [19] Sarason,D., (2007), Complex Functions Theory, AMS, Providence, RI.
- [20] Stanley,D., (1999), A multiplicative calculus, Primus, IX(4), 310–326.
- ISSN: 2146-1147
- Başlangıç: 2010
- Yayıncı: Turkic World Mathematical Society
Sayıdaki Diğer Makaleler
H. JAMALİ, K.h. Shokri TERNONİZ
Manısh JAIN, Neetu GUPTA, S. KUMAR
Bhavana DESHPANDE, A. HANDA, L. Narayan MİSHRA