Menu Close

Let-go-toward-a-rational-order-of-derivation-Part-1-What-s-that-special-factor-Let-n-p-and-k-three-integer-different-of-zero-We-state-J-n-k-p-0-1-1-x-n-p-k-n-dx-and-C-n-

Tinku Tara
Pin




Question Number 65828 by ~ À ® @ 237 ~ last updated on 04/Aug/19
Let go toward a rational order of derivation Part 1 : What′s that special factor Let n , p and k three integer different of zero We state J_(n,k) (p)=∫_0 ^1 (1−x^n )^(p+(k/n)) dx and C_n (p)=Π_(k=0) ^(n−1) J_(n,k) (p) 1) a) Calculate C_1 (p) b) Prove that J_(n,k) (p)=(1/n)B((1/n),p+1+(k/n) ) and explicit C_n (p)in terms of n and p 2) Deduce that ∀ n>0 there exist a real a_n such as (na_n )^n C_n (p)= (1/(p+1)) 3) Study the convergence of the result suite (a_n )_n .Then show that lim_(n−>∞) na_n =1 Part 2: the rational order of derivation Let f ∈ C^1 (R,R) . We consider I_(1/n) (f) a function defined on R_+ by I_(1/n) (f)(x)= a_n ∫_0 ^x ((f(t))/((x−t)^(1−(1/n)) ))dt and D_(1/n) (f) = (I_(1/n) (f))^((1)) 1) a _ Prove that I_((1/n) ) (f)(x)= na_n x^(1/n) ∫_0 ^1 f(x(1−v^n ))dv then find D_(1/2) (t) b) Show that ∀ f∈C^1 (R,R) ∀ x∈R_(+ ) D_(1/n) (f)(x)= I_(1/n) (f)(x) + ((f(0))/((πx)^(1−(1/n)) )) 2)∀ p integer and k∈{0,...,n−1} explicit I_(1/n) (t^(p+(k/n)) ) in term of I_(n,k) (p) b) Prove that for polynomial function f the n− th composition I_(1/n) ._ ....I_(1/n) (f)(x)=∫_0 ^x f(t)dt , c) Deduce that ∀ f polynomial the function g =f −f(0) verify D_(1/n) ......D_(1/n) (g)(x) = g(x) 3) Widen that two formulas to all function that can be developp into integer serie 4) Try to find the relation between D_(1/n) .I_(1/n) (f) , I_(1/n) .D_(1/n) (f), and f 4) Show that ∀ x∈R_+ lim_(n−>∞) I_(1/n) (f)(x)= ∫_0 ^x f(t)dt pour g=f−f(0) lim_(n−>∞) D_(1/n) (g)(x)= g(x) conclusion the derivative of the function I_α (f) defined on R_+ by I_α (f)(x)= a_n ∫_0 ^x f(t)(x−t)^((1/n)−1) dt is called the derivative of order α
$$\:{Let}\:{go}\:{toward}\:{a}\:{rational}\:{order}\:{of}\:{derivation} \\ $$$$ \\ $$$${Part}\:\mathrm{1}\::\:\:{What}'{s}\:{that}\:{special}\:{factor}\:\: \\ $$$${Let}\:{n}\:,\:{p}\:{and}\:{k}\:{three}\:{integer}\:\:{different}\:{of}\:{zero} \\ $$$${We}\:\:{state}\:{J}_{{n},{k}} \left({p}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\left(\mathrm{1}−{x}^{{n}} \right)^{{p}+\frac{{k}}{{n}}} {dx}\:\:\:{and}\:\:{C}_{{n}} \left({p}\right)=\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\:{J}_{{n},{k}} \left({p}\right) \\ $$$$\left.\mathrm{1}\left.\right)\:{a}\right)\:{Calculate}\:{C}_{\mathrm{1}} \left({p}\right)\:\: \\ $$$$\left.\:\:\:\:{b}\right)\:{Prove}\:{that}\:\:{J}_{{n},{k}} \left({p}\right)=\frac{\mathrm{1}}{{n}}{B}\left(\frac{\mathrm{1}}{{n}},{p}+\mathrm{1}+\frac{{k}}{{n}}\:\right)\:\:\:{and}\:\:{explicit}\:\:{C}_{{n}} \left({p}\right){in}\:{terms}\:{of}\:\:{n}\:{and}\:{p}\: \\ $$$$\left.\mathrm{2}\right)\:{Deduce}\:{that}\:\forall\:{n}>\mathrm{0}\:\:\:\:{there}\:{exist}\:{a}\:{real}\:{a}_{{n}} \:{such}\:{as}\:\:\left({na}_{{n}} \right)^{{n}} {C}_{{n}} \left({p}\right)=\:\frac{\mathrm{1}}{{p}+\mathrm{1}}\: \\ $$$$\left.\mathrm{3}\right)\:{Study}\:{the}\:{convergence}\:{of}\:{the}\:{result}\:{suite}\:\left({a}_{{n}} \right)_{{n}} \:\:\:.{Then}\:{show}\:{that}\:{lim}_{{n}−>\infty} \:{na}_{{n}} \:=\mathrm{1} \\ $$$${Part}\:\mathrm{2}:\:\:{the}\:{rational}\:{order}\:{of}\:{derivation} \\ $$$${Let}\:\:{f}\:\in\:{C}^{\mathrm{1}} \left(\mathbb{R},\mathbb{R}\right)\:.\:{We}\:\:{consider}\:{I}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\:{a}\:{function}\:{defined}\:{on}\:\mathbb{R}_{+} \:{by}\:\:\: \\ $$$${I}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\left({x}\right)=\:{a}_{{n}} \int_{\mathrm{0}} ^{{x}} \:\:\frac{{f}\left({t}\right)}{\left({x}−{t}\right)^{\mathrm{1}−\frac{\mathrm{1}}{{n}}} }{dt}\:\:\:\:\:\:\:{and}\:\:{D}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\:=\:\left({I}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\right)^{\left(\mathrm{1}\right)} \:\: \\ $$$$\left.\mathrm{1}\right)\:{a}\:\_\:\:{Prove}\:{that}\:{I}_{\frac{\mathrm{1}}{{n}}\:} \left({f}\right)\left({x}\right)=\:{na}_{{n}} {x}^{\frac{\mathrm{1}}{{n}}} \:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{f}\left({x}\left(\mathrm{1}−{v}^{{n}} \right)\right){dv}\:\:\:{then}\:{find}\:{D}_{\frac{\mathrm{1}}{\mathrm{2}}} \left({t}\right) \\ $$$$\left.\:\:{b}\right)\:\:{Show}\:{that}\:\:\forall\:{f}\in{C}^{\mathrm{1}} \left(\mathbb{R},\mathbb{R}\right)\:\forall\:{x}\in\mathbb{R}_{+\:\:} \:{D}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\left({x}\right)=\:{I}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\left({x}\right)\:\:+\:\frac{{f}\left(\mathrm{0}\right)}{\left(\pi{x}\right)^{\mathrm{1}−\frac{\mathrm{1}}{{n}}} } \\ $$$$\left.\mathrm{2}\right)\forall\:{p}\:{integer}\:{and}\:\:{k}\in\left\{\mathrm{0},…,{n}−\mathrm{1}\right\}\:\:{explicit}\:\:{I}_{\frac{\mathrm{1}}{{n}}} \left({t}^{{p}+\frac{{k}}{{n}}} \right)\:{in}\:{term}\:{of}\:\:{I}_{{n},{k}} \left({p}\right) \\ $$$$\left.{b}\right)\:{Prove}\:{that}\:{for}\:{polynomial}\:{function}\:{f}\:\:\:{the}\:{n}−\:{th}\:{composition}\:\:{I}_{\frac{\mathrm{1}}{{n}}} ._{} ….{I}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\left({x}\right)=\int_{\mathrm{0}} ^{{x}} {f}\left({t}\right){dt}\:\:\:\:\:,\:\: \\ $$$$\left.\:{c}\right)\:{Deduce}\:{that}\:\forall\:\:{f}\:\:{polynomial}\:\:{the}\:{function}\:{g}\:={f}\:−{f}\left(\mathrm{0}\right)\:{verify} \\ $$$${D}_{\frac{\mathrm{1}}{{n}}} ……{D}_{\frac{\mathrm{1}}{{n}}} \left({g}\right)\left({x}\right)\:=\:{g}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{Widen}\:{that}\:{two}\:{formulas}\:{to}\:{all}\:\:{function}\:{that}\:{can}\:{be}\:{developp}\:{into}\:{integer}\:{serie} \\ $$$$\left.\mathrm{4}\right)\:{Try}\:{to}\:{find}\:{the}\:{relation}\:{between}\:\:{D}_{\frac{\mathrm{1}}{{n}}} .{I}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\:,\:\:{I}_{\frac{\mathrm{1}}{{n}}} .{D}_{\frac{\mathrm{1}}{{n}}} \left({f}\right),\:{and}\:\:{f}\:\: \\ $$$$\left.\mathrm{4}\right)\:{Show}\:\:{that}\:\forall\:{x}\in\mathbb{R}_{+} \:\:{lim}_{{n}−>\infty} \:\:{I}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} \:{f}\left({t}\right){dt}\:\:\:\:\:\:\: \\ $$$${pour}\:\:{g}={f}−{f}\left(\mathrm{0}\right)\:\:\:{lim}_{{n}−>\infty} \:\:{D}_{\frac{\mathrm{1}}{{n}}} \left({g}\right)\left({x}\right)=\:{g}\left({x}\right) \\ $$$${conclusion} \\ $$$$\:{the}\:{derivative}\:{of}\:{the}\:{function}\:{I}_{\alpha} \:\left({f}\right)\:\:{defined}\:{on}\:\mathbb{R}_{+} \:\:{by}\: \\ $$$${I}_{\alpha} \left({f}\right)\left({x}\right)=\:{a}_{{n}} \int_{\mathrm{0}} ^{{x}} \:{f}\left({t}\right)\left({x}−{t}\right)^{\frac{\mathrm{1}}{{n}}−\mathrm{1}} {dt}\:\:{is}\:\:{called}\:{the}\:\:{derivative}\:{of}\:{order}\:\alpha \\ $$$$ \\ $$
Commented by ~ À ® @ 237 ~ last updated on 04/Aug/19
I_α (f)(x)=a_n ∫_0 ^x f(t)(x−t)^(α−1) dt
$$\:\:\:{I}_{\alpha} \left({f}\right)\left({x}\right)={a}_{{n}} \int_{\mathrm{0}} ^{{x}} \:\:{f}\left({t}\right)\left({x}−{t}\right)^{\alpha−\mathrm{1}} {dt} \\ $$

Pin