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Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannon6.The Fractional Product CalculusWe developed the Product Calculus, or Geometric Mean Calculusin [Dan1]. The Product Derivative and the Product Integral ofthat Calculus are related by the Fundamental Theorem of theProduct Calculus [Dan1]The Fractional Derivative, and the Fractional Integral can bedefined in the context of the Product Calculus to yield a FractionalProduct Calculus.We outline here the development of the Fractional ProductCalculus of order 1 , with its Fundamental Theorem. We keep the2notations of [Dan1]6.1 Definition of Fractional Product Integral of Order 1 211 122 D−f x −2(0) −ln ( )( )( D ) f( x) ≡e ≡ G ( x)6.2 Definition of Fractional Product Derivative of Order 1 21212(0) ln ( )D f x( D ) f( x) ≡ e24
Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannon6.3 The Fundamental Theorem of the Fractional ProductCalculus of Order 1 2Proof:−{ }1 12 2(0) (0) (0)( D ) f( x) = D ( D ) f( x)⎧−1⎫(0)Dlog ⎪( D ) 2 f( x)⎪1⎨⎬(0) (0) − 2⎪⎩⎪⎭D ( D ) f( x)= e= e= e= eD⎧⎪⎨⎪⎩12 log ( )log D −⎪ef x−1DD 2 log f ( x )1D 2 log f ( x )(0)12= ( D ) f( x)⎫⎪⎬⎪⎭This leads to the interpretation of the Fractional Productderivative of order 1 2 .6.4 The Geometric Mean of the Convolution TransformThe Fractional Product Derivative of order 1 of f ( x ) at x is the2Geometric Mean of12( −G) ( x )over [ xx , + dx].25
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