basics of 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

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

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