On the Leibniz rule and Laplace transform for fractional derivatives

Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. This paper investigates two typical applications: Lebiniz rule and Laplace transform. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann-Liouville derivative is doubtful for n-th continuously differentiable function. By the aid of this series representation, the exact formula of Caputo Leibniz rule and the explanation of Riemann-Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results

Orbital density wave induced by electron-lattice coupling in orthorhombic iron pnictides

In this paper we explore the magnetic and orbital properties closely related to a tetragonal-orthorhombic structural phase transition in iron pnictides based on both two- and five-orbital Hubbard models. The electron-lattice coupling, which interplays with electronic interaction, is self-consistently treated. Our results reveal that the orbital polarization stabilizes the spin density wave (SDW) order in both tetragonal and orthorhombic phases. However, the ferro-orbital density wave (F-ODW) only occurs in the orthorhombic phase rather than in the tetragonal one. Magnetic moments of Fe are small in the intermediate Coulomb interaction region for the striped antiferromangnetic phase in the realistic five orbital model. The anisotropic Fermi surface in the SDW/ODW orthorhombic phase is well in agreement with the recent angle-resolved photoemission spectroscopy experiments. These results suggest a scenario that the magnetic phase transition is driven by the ODW order mainly arising from the electron-lattice coupling.Comment: 21 pages, 10 figure