Fractional Calculus - Gauge-institute.org

GAUGE-INSTITUTEGauge Institute JournalFractional CalculusThe Fundamental Theorem of the Fractional Calculus, andthe Meaning of Fractional DerivativesH. Vic DannonAbstract: Since the beginning of the 18 th century, attempts weremade to generalize the Arithmetic Means Derivative, and obtain aGeneralized Arithmetic Means Calculus.As early as 1812, Lacroix [Lac] observed that for n = 1... m,nd m m! m−nΓ ( m + 1) m−nx = x =x ,ndx ( m −n)! Γ( m − n + 1)and by formal substitution ofn =12, obtained [Ross]1/2dΓ (1 + 1) 1−1/21x = x =( dx) Γ(1 − + 1) π /21/2 12x.In 1832, Liouville [Liou] used the Euler Gamma Integral formulau=∞−aa−1−xuΓ () ax = ∫ u e duu=0Differentiating both sides to an order ν , he obtained

Assuminghe hadνu=∞ν−aa−1d −νdxu=0dΓ () a x = u e( dx )ν ∫ ( )dνν( dx)νe−xuν−xu= ( −u)e , (3)u=∞d − a ν a+ ν−1Γ () a x = ( −1)u eν ∫( dx)u=0ν= ( −1) Γ ( a + ν)x− +xu du.− xu du( a ν ) ,and concluded [Ross] thatdνν( dx)x−aν= ( −1)Γ ( a + ν)xΓ()a−( a+ν)Many years later, the meaning of Fractional Derivatives is stillunclear.The common perception is that the Fractional Derivative Methodrepresents a new kind of Calculus, such as the Product Calculus,that is based on the Geometric Mean [Dan1],But so far, the Fractional Derivative Method has been developedonly as a refinement of the Arithmetic Means Calculus, and is nota new kind of Calculus, such as the Product Calculus is.Here, We interpret the Fractional Derivative in the context of theArithmetic Means Calculus in which it was presented.We notethat it can be similarly developed, and interpreted in the ProductCalculus.

We proceed with the simplest fraction q =1. The Fractionalintegral2and the derivative−12f−1D212d≡ f ,−( dx)12 121df( dx) 2≡ Df.