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


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

$I_{α} (f) (x) = a_{n} \int_{0}^{x} f (t) {(x - t)}^{α - 1} d t$
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