Discovering The Relationships between Fractional Order Derivatives and Complex Numbers

Abstract

The derivative concept was defined by Newton and Leipzig. After these scientists, there are many approaches about the order of derivative, since derivative defined by Newton and Leipzig considered as order of 1. So, imaginary axis vanishes and the result of derivation is a real number / function. However, in case of other orders of derivations, the obtained results have real and imaginary axises, since complex numbers and derivative have directions and magnitudes. This paper includes these relationships by using fractional order derivative (_α^∂)Kf(t) defined by Karcı.

Keywords

• Fractional order derivatives

References

1. Das, S., “Functional Fractional Calculus”, Springer-Verlag Berlin Heidelberg, 2011.
2. Karcı, A.,”Kesirli Türev için Yapılan Tanımlamaların Eksiklikleri ve Yeni Yaklaşım”, TOK-2013 Turkish Automatic Control National Meeting and Exhibition, 2013a.
3. Karcı,A., “A New Approach for Fractional Order Derivative and Its Applications”, Universal Journal of Engineering Sciences, Vol:1, pp: 110-117, 2013b.
4. Karcı, A., “Properties of Fractional Order Derivatives for Groups of Relations/Functions”, Universal Journal of Engineering Sciences, vol:3, pp:39-45, 2015a.
5. Karcı, A.,”The Linear, Nonlinear and Partial Differential Equations are not Fractional Order Differential Equations”, Universal Journal of Engineering Sciences, vol:3, pp:46-51, 2015b.
6. Karcı, A.,“Generalized Fractional Order Derivatives for Products and Quotients”, Science Innovation, vol:3, pp:58-62, 2015c.
7. Karcı, A.,“Chain Rule for Fractional Order Derivatives”, Science Innovation, vol:3, 63-67, 2015d.
8. Karcı, A.“The Properties of New Approach of Fractional Order Derivative”, Journal of the Faculty of Engineering and Architecture of Gazi University, Vol.30, pp:487-501, 2015e.

9. Karcı, A.,“Fractional order entropy New perspectives”, Optik - International Journal for Light and Electron Optics, vol:127, no:20, pp:9172-9177, 2016a.
10. Karcı, A.,“New Kinds of Entropy Fractional Entropy”, International Conference on Natural Science and Engineering, 1-4, 2016b.
11. Karcı, A.,“Malatya Functions: Symmetric Functions Obtained by Applying Fractional Order Derivative to Karcı Entropy”, Anatolian Science-Journal of Computer Sciences, Vol:2, Issue: 2, pp:1-8, 2017.