Basics of Fractional Calculus
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2712 이이이 FRACTIONAL CALCULUS
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2712 Fractional CalculusPart I. PreliminariesDefinition : mapping between two vector spaces ,
Linear Operator(Homogenity of degree 1)(Additivity)
Properties
Functional Square RootHalf Iterate (Functional Square Root) of a function
Examples
Notation :
Part II. Fundamentals & NatureFractional DerivativeFunctional square root of differentiation operator?
Let linear operator
Fractional DerivativeGeneralization
If
7Gamma FunctionDefinition
Examples
,
Fractional DerivativeExample
,
Fractional IntegralLet operator
satisfyThus,
10Repeated Integration FormulaCauchy formula for repeated integration
Proof) By mathematical inductionSee the boardFractional IntegralProperty
Proof)idea:
(1)(2)MerciLa Fin