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Question Number 65828 by ~ À ® @ 237 ~ last updated on 04/Aug/19

$$Letgotowardarationalorderofderivation$$$$$$$$Part1:{What}^{\prime }sthatspecialfactor$$$$Letn,pandkthreeintegerdifferentofzero$$$$Westate{J}_{n,k}\left(p\right)={\int }_0^1{(1-x^n)}^{p+\frac{k}{n}}dxand{C}_n\left(p\right)=\underset{k=0}{\prod ^{n-1}}{J}_{n,k}\left(p\right)$$$$1)a)Calculate{C}_1\left(p\right)$$$$b)Provethat{J}_{n,k}\left(p\right)=\frac{1}{n}B(\frac{1}{n},p+1+\frac{k}{n})andexplicit{C}_n\left(p\right)intermsofnandp$$$$2)Deducethat\mathrm{\forall }n>0thereexistareal{a}_{n}suchas{\left({na}_n\right)}^n{C}_n\left(p\right)=\frac{1}{p+1}$$$$3)Studytheconvergenceoftheresultsuite{\left(a_n\right)}_n.Thenshowthat{lim}_{n->\mathrm{\infty }}{na}_n=1$$$$Part2:therationalorderofderivation$$$$Letf\in C^1(\mathbb{R},\mathbb{R}).Weconsider{I}_{\frac{1}{n}}\left(f\right)afunctiondefinedon{\mathbb{R}}_{+}by$$$$I_{\frac{1}{n}}\left(f\right)\left(x\right)=a_n{\int }_0^x\frac{f\left(t\right)}{{(x-t)}^{1-\frac{1}{n}}}dtand{D}_{\frac{1}{n}}\left(f\right)={\left(I_{\frac{1}{n}}\left(f\right)\right)}^{\left(1\right)}$$$$1)a\mathrm{\_}Provethat{I}_{\frac{1}{n}}\left(f\right)\left(x\right)={na}_n{x}^{\frac{1}{n}}{\int }_0^{1}f\left(x(1-v^n)\right)dvthenfind{D}_{\frac{1}{2}}\left(t\right)$$$$b)Showthat\mathrm{\forall }f\in C^1(\mathbb{R},\mathbb{R})\mathrm{\forall }x\in {\mathbb{R}}_{+}{D}_{\frac{1}{n}}\left(f\right)\left(x\right)=I_{\frac{1}{n}}\left(f\right)\left(x\right)+\frac{f\left(0\right)}{{\left(\pi x\right)}^{1-\frac{1}{n}}}$$$$2)\mathrm{\forall }pintegerandk\in \{0,\dots ,n-1\}explicit{I}_{\frac{1}{n}}\left(t^{p+\frac{k}{n}}\right)intermof{I}_{n,k}\left(p\right)$$$$b)Provethatforpolynomialfunctionfthen-thcomposition{I}_{\frac{1}{n}}{.}_{}\dots .I_{\frac{1}{n}}\left(f\right)\left(x\right)={\int }_0^{x}f\left(t\right)dt,$$$$c)Deducethat\mathrm{\forall }fpolynomialthefunctiong=f-f\left(0\right)verify$$$$D_{\frac{1}{n}}\dots \dots D_{\frac{1}{n}}\left(g\right)\left(x\right)=g\left(x\right)$$$$3)Widenthattwoformulastoallfunctionthatcanbedeveloppintointegerserie$$$$4)Trytofindtherelationbetween{D}_{\frac{1}{n}}.I_{\frac{1}{n}}\left(f\right),I_{\frac{1}{n}}.D_{\frac{1}{n}}\left(f\right),andf$$$$4)Showthat\mathrm{\forall }x\in {\mathbb{R}}_{+}{lim}_{n->\mathrm{\infty }}{I}_{\frac{1}{n}}\left(f\right)\left(x\right)={\int }_0^{x}f\left(t\right)dt$$$$pourg=f-f\left(0\right){lim}_{n->\mathrm{\infty }}{D}_{\frac{1}{n}}\left(g\right)\left(x\right)=g\left(x\right)$$$$conclusion$$$$thederivativeofthefunction{I}_{\alpha }\left(f\right)definedon{\mathbb{R}}_{+}by$$$$I_{\alpha }\left(f\right)\left(x\right)=a_n{\int }_0^{x}f\left(t\right){(x-t)}^{\frac{1}{n}-1}dtiscalledthederivativeoforder\alpha$$$$$$

Commented by ~ À ® @ 237 ~ last updated on 04/Aug/19

$$I_{\alpha }\left(f\right)\left(x\right)=a_n{\int }_0^{x}f\left(t\right){(x-t)}^{\alpha -1}dt$$

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