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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