Fractional Calculus - Gauge-institute.org
Fractional Calculus - Gauge-institute.org
Fractional Calculus - Gauge-institute.org
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<strong>Gauge</strong> Institute Journal, Volume 5, No 1, February 2009H. Vic Dannon6.3 The Fundamental Theorem of the <strong>Fractional</strong> Product<strong>Calculus</strong> 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 <strong>Fractional</strong> Productderivative of order 1 2 .6.4 The Geometric Mean of the Convolution TransformThe <strong>Fractional</strong> Product Derivative of order 1 of f ( x ) at x is the2Geometric Mean of12( −G) ( x )over [ xx , + dx].25
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