The Integral Analog of the Leibniz Rule · the integral analog of the leibniz rule 907 (3.6) is the Parseval's formula [11, p. 50]. Thus, our integral analog of the Leibniz rule "extends

MATHEMATICS OF COMPUTATION, VOLUME 26, NUMBER 120, OCTOBER 1972

The Integral Analog of the Leibniz Rule

By Thomas J. Osier

Abstract. This paper demonstrates that the classical Leibniz rule for the derivative of

the product of two functions

DNuv =Y\ DN~kuDkvk-Q \kj

has the integral analog

D"uv = J ^ ^jDa-"uD"v da.

The derivatives occurring are "fractional derivatives." Various generalizations of the inte-

gral are given, and their relationship to Parseval's formula from the theory of Fourier

integrals is revealed. Finally, several definite integrals are evaluated using our results.

1. Introduction. The derivative of the function f(z) with respect to g(z) of

arbitrary (real or complex) order a is denoted by D"U)/(z), and called a "fractional

derivative". It is a generalization of the familiar derivative d"f(z)/(dg(z))a to values

of a which are not natural numbers. In previous papers (see [3]-[9]), the author

discussed extensions to fractional derivatives of certain rules and formulas for ordinary

derivatives familiar from the elementary calculus. This paper continues the previous

study and presents integral analogs of the familiar Leibniz rule for the derivative

of a product

DNu{z)v(z) = Y,(")DN-ku(z)Dkv(.z).k = Q \K I

We present below continually more complex extensions of the Leibniz rule.

Case I, The simplest integral analog of the Leibniz rule is

(1.1) d:u{z)v(z) = f {^jDrau(z)d:v(z)dw,

where (") = T(a + l)/r(a: — co + l)T(co + 1), and a is any real or complex number.

Notice that we integrate over the order of the derivatives.

Case 2. (1.1) can be generalized to the case where we differentiate with respect

to an arbitrary function g(z), and co is replaced by w + y, where 7 is arbitrary (real

or complex).

Received March 23, 1972.

A MS 1970 subject classifications. Primary 26A33, 44A45; Secondary 33A30, 42A68.Key words and phrases. Fractional derivative, Leibniz rule, Fourier transforms, Parseval rela-

tion, special functions.

Copyright © 1972, American Mathematical Society

903

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

904 THOMAS J. OSLER

(1.2) d:û(z)v(z) = J (w + J ö.^rVz) d°tJa(z) <*»■

Case 3. The product u(£)v(£) can be replaced by an arbitrary function of two

variables /(£, f):

(1.3) d:Mf(z, 2) = / a (w " 7) ß?ü,™ + 7/(£, 0 |{=,;r=2

where Z)°;^ is a generalization of the "partial derivative" operator da+t'/(dz)a(dw)t'.

Case 4. Our most general integral form of the Leibniz rule is

(1.4) d:mKz,z) = f /(«)<*«,

where

/(co) = ( " ) DZtän^wk, f)e.u)(f;z)9(ö"+T9(rr,""r"1] i«—ar-. *».\co -f- 7/

and where 0(j"; z) = (g(f) — g(z))c7(f). The three previous integrals are all special

cases of (1.4). The author's series form of the Leibniz rule [7]

(1.5) d:iz)f(z,z) = £ Kan)a

suggests the validity of (1.4) by letting a —* 0 (in which case a becomes dw, an becomes

co, and ^2 becomes J). If /(£, g~\0)) = 0, (1.4) assumes the simpler form

In previous papers [5], [7], [8], the author demonstrated that formulas familiar

from Fourier analysis are closely related to formulas involving fractional derivatives.

In this paper, we show that the familiar Parseval's integral formula [11, p. 50] is,

in a certain sense, a special case of our integral analog of the Leibniz rule (1.4).

A proof of (1.4) is presented, and the region in the z-plane in which the integral

converges is revealed.

Finally, several examples of (1.4) are given for specific functions /(£, f), g(z)

and q{l). A table of definite integrals (Table 5.2) emerges.

2. Fractional Derivatives. In a previous paper [3], the author explored the

definitions of the fractional derivatives of a function in detail. In this section, we

merely review the definitions so as to make the paper self-contained. We assume

that the reader is familiar with the motivation provided in [3].

The most common definition for the fractional derivative of order a found in

the literature is the "Riemann-Liouville integral"

d:m = r(-«r1 f /(0(z - ty-1 dt,Jo

where Re(a) < 0. The concept of a fractional derivative with respect to an arbitrary

function g(z), Z)"U)/(z), was apparently introduced for the first time in the author's

paper [3], while the idea appeared earlier for certain specific functions. The most


THE INTEGRAL ANALOG OF THE LEIBNIZ RULE 905

convenient form of the definition for our purposes is given through a generalization

of Cauchy's integral formula.

Definition 2.1. Let /(z) be analytic in the simply connected region R. Let g(z)

be regular and Univalent on R, and let g" '(0) be an interior or boundary point of R.

Assume also that Jc /(z)g'(z) dz = 0 for any simple closed contour C in R KJ {g_I(0)}

through g~ '(0). Then if a is not a negative integer, and z is in R, we define the frac-

tional derivative of order a of the function /(z) with respect to g(z) to be

T(a + 1) fu + ) /(f)gXf) <tf(2.1) D,U)j{z) = -—-j

'«•> (g(f) - g(z))° + I

For nonintegral a, the integrand has a branch line which begins at f = z and passes

through f = g~ '(0). The limits of integration imply that the contour of integration

starts at g~'(0), encloses z once in the positive sense, and returns to g~'(0) without

cutting the branch line or leaving R KJ |g-1(0)}.

If a is a negative integer —N, Y(a + 1) = co while the integral in (2.1) vanishes.

If we interpret (2.1) as the limit as a approaches — N, it then defines the derivative

of order — N, or perhaps we should say the "Mh iterated integral of /(z) with respect

to g(z)".

It is important to notice that with the restrictions on g(z) as given in Definition

2.1, the substitution w = g(z) maintains the equality Dlj{w) = D"(2)/(g(z)).

It is particularly interesting to set g(z) = z — a, for we find that

(2.2) ZCJ(z) = r(" + n f ' /(f)(f - *)—1 d[.

While ordinary derivatives with respect to z and z — a are equal, (2.2) shows that

this is not the case for fractional derivatives, since the value of the contour integral

depends on the point f = a at which the contour crosses the branch line.

We also require fractional partial derivatives.

Definition 2.2. Let /(z, w) be an analytic function of two variables for z and w

in the simply connected region R. Let g(z) be regular and Univalent on R, and let g~ \0)

be an interior or boundary point of R. Assume also that fc j(z, w)g'(w) dw = 0 and

/c Dß^w)f(z, vv)g'(z) dz = 0 for any simple closed contour C in R U lg_1(0)( through

g~ J(0). Then if a and ß are not negative integers, and z and w are in R we write

£>°('/).a(«)/(z. w) = jD"(,)[DJ(„)/(z, >v)]

(2'3) r(« + i)ros + i) fu+) gXO r'^ /(r, Qg'g) ̂A-MO) (g(f) - g(z))° + 1 J.- (g® - g(w))ß+1

Note. The expression (g(f) - g(z))a + 1 = exp[(a + 1) ln(g(f) - g(z))] appearing

in the above definitions is in general a multiple-valued function. To remove this

uncertainty, select a branch cut which starts at £ = z and passes through f = g~ '(0)

in the f-plane. Then let ln(g(f) — g(z)) assume values which are continuous and

single-valued on the cut f-plane and let ln(g(f) — g(z)) be real when g(f) — g(z) is

positive.

3. Connection with Fourier Analysis. In this section, we show that the special

case of our integral analog of the Leibniz rule (1.1) is formally a generalization of


906 THOMAS J. OSLER

the Parseval's formula [11, p. 50] familiar from the study of Fourier integrals.

Consider the definition of fractional derivatives (2.2), with a = 0. Take the contour

of integration to be the circle parametrized by the variable

(3.1) f = z + ze'', where — it < t < t.

We obtain at once

(3.2)

If we set

Tv+T) = YjJ{z + ze )e dt-

f[t] = /(z + ze") for —k < t < tt,

= 0 otherwise,

and denote the Fourier transform of /[/] by

(3.3) /*{«) = £ f_m dt,

we see that (3.2) becomes

(3.4) zaD:M/T(c* + 1) = /*(«)■

Thus, we see that the fractional derivative D"/(z) (modulo a multiplicative factor)

is in a certain sense a generalization of the Fourier transform. By varying z, the

circle (3.1) changes and, thus, the function f[t] changes. We say that we have "extended

the Fourier integral into the complex z-plane".

Now, consider (1.1) written in the form

z " d"u(z)v(z)

t(a + 1

Using (3.2), we get

(z) = r za-"Dr"u(z) z- d:v{z)

) J-c t(a - w + 1) T(co + 1)

(3.5)

— / u(z + ze'')e >lav(z + ze'1) dt'.it j_»

= J J u(z + zel)e "' ae "' dt ~~ J v(z + ze'')e "" dt^ rfco.

Using the notation

u[t] = u(z + ze'')e~'"* for — it < t < ir,

= 0 otherwise,

v[t] = v(z + ze'') for — it < t < it,

= 0 otherwise,

and the notation (3.3) for Fourier transforms, we convert (3.5) into

(3.6) u[t]v[t]dt = j u*(-u)ü*(o>) dco.


the integral analog of the leibniz rule 907

(3.6) is the Parseval's formula [11, p. 50]. Thus, our integral analog of the Leibniz

rule "extends the Parseval's formula into the complex z-plane".

We conclude this section by reviewing three connections observed in previous

papers between formulas from Fourier analysis, and new formulas involving frac-

tional derivatives.

L. The familiar Fourier series

m = £ Umt71= — 03

is (in a sense similar to that above) a special case of the generalized Taylor's series [5]

„rr„ T(an +7+1)

f ['" u[t]v[t] rf( = f f [uWKi dt ~ V" v[t]e-°nt dt,

2. The series form of the Parseval's relation

is a special case of the series form of the Leibniz rule [7]

D,"«(zMz) = £ at " ) Dr°n-yu(z)D7+Mz).„rr«, \an + 7/

3. The Fourier integral theorem

fit) = f ^ j'yiiyr4(u die'-1 d*

is a special case of the integral analog of Taylor's series [8]

«z) = j_ T{w+y+ f) (z - *.) do>.

We now leave our discussion of Fourier analysis, and present a rigorous deri-

vation of our integral analog of the Leibniz rule.

4. Rigorous Derivations. In the previous section, we saw that our integral

analog of the Leibniz rule is formally related to the integral form of Parseval's re-

lation from the theory of Fourier transforms. It would seem natural then that a

rigorous derivation would follow from known results on Fourier integrals. While

this is easily achieved for functions /(z, w) of a particular class, a derivation by this

method sufficient to cover all /(z, w) of interest has escaped the author. It appears

our integral form of the Leibniz rule is more difficult to prove than the series form

(1.5). In particular, it is necessary to restrict /(z, w) in the following Theorem 4.1

(see (i)) to a narrower class of functions than was necessary for the series form of

the Leibniz rule in which only the growth of / at the origin was restricted by |/(z, w)\ :£

M\z\p \z\Q, for P, Q, and P + Q in the interval (-1, ») [9].

Theorem 4.1. (i) Let /(£, f) = f{QF(t, f), where Re(F), Re(g), and Re(P + Q)

are in the interval (—1, °°), and let F(£, f) be analytic for £ and f in the simply con-

nected open subset of the complex plane R, which contains the origin.


908 thomas j. osler

(ii) Let 0(f; z) = (f — z)cj(f) be a given Univalent analytic function of£ on R such

that q(£) is never zero on R.

(iii) Assume that the curves C(z) = {f | |0(f; z)| = 10(0; z)|J are sz'w/?/e and closed

for each z such that C(z) C R-

(iv) Call S = \z I C(z) C77ze« /or z£ S, anc/ a// rea/ or complex a and y, the integral analog of the Leibniz

rule is

(4.1) ö27(z,z) = [ /(a>)J — CO

dw,

where

\co -+- 7/

a/u/ 0f(f; z) = rfÖ(f; z)/#.Remark.

( a )

\co + 7/T(a + 1)/T(a - co - 7 + l)r(co +7+1)

is, in general, undefined when a is a negative integer. In this case, dividing both

sides of (4.1) by T(a + 1) produces a valid expression.

Proof. The C(z) are the curves in the complex f-plane which pass through the

origin, over which the amplitude of 0(c:; z) is constant. For example, if 0(f; z) =

f — z, then C(z) is the circle centered at f = z passing through the origin. By re-

stricting z to S (described in (iv)), we insure that the curves C(z) are contained in

the region R on which /(£, f) is sufficiently regular for the manipulations which follow.

Using Definition 2.2 of fractional differentiation, we can express /(co) as

r( s = r(a + l) r('+) f('+) /«. f)gf(f;z)gg)"+1rg(r)"""1r"1 <*r

^ (2tt02 Jo Jo (I - z)" — T + 1(f - Z)" + T + 1

and since 0(f; z) = tfflß- - z) and /(£, f) - £PfQF(£, f), we get

f4 2) j( ) = r(a + *> ru+> fu+) tV^. r)<?(sr+10t(g; z)flf(f; z) # <ff(2irif Jo Jo 9({;z)"-"-T + 10(f;z)" + T + I0£(£;z)

Make the substitutions

(4.3) * - z = 0(f; z), s - z = 0(£; z)

and call b = z + 0(0; z). Since 0(£; z) is a Univalent function of £, we have

(44) tr^.y*' = (s _zf(t _zfns,0>0{(.?> z)

where 0£(£; z) is not zero and, thus, 5(s, r) is an analytic function. The curve C(z)

has now become the circle \t — z| = |0(O; z)| = |6 — z| in the /-plane, and \s — z\ =

|6 — z| in the s-plane. Thus, if z £ S, then SJ(s, r) is analytic for s and / inside and

on \t — z\ = \b — z\ and |j — z| = |o — z| and so

(4.5) JF(j, () = XI «»,„(* - z)m« - z)"


THE INTEGRAL ANALOG OF THE LEIBNIZ RULE 909

where the series converges absolutely and uniformly inside and on the circles just

mentioned. Thus, (4.2) becomes

/(«)T(a + 1) C(z + ) f(' + ) (s - bf(t - b)°S(s, t) dt ds

(2*if Jb Jb (J _ zy-«^+\t - zy^+1

Substituting the series for SF, (4.5), into this last integral, we can then integrate term-

wise since we have a uniformly convergent series times an integrable factor. We get

u v _T(<* + _

*£, T(a - u - 7 - m + l)T(co + y - n + 1)

T(a - co - 7 - m + 1) f<2 + ) fj - bf ds

1•I b2*i Jb (s - z)«-«-t--»+i

T(co + 7 - « + 1) fU + > 0 - bf dtf

T(B + 1) /-u + > - b) ds _ _ r(P + i)(z - fey(s - z)B+1 " D-'(z Ö) " T(P - B + 1)

2vi Jb (r-z)" + T""+1

It is well known that [6, p. 647]

and, thus, we get

(4.7) /(co) = £ <4»»G(co + m)//(co - n)m,n = 0

where

<4m„ = T(a + l)r(P + l)T(ß + l)amn(z - + +

G(co + m)

P-B

H(u - n) =

r(a — 7 — co - m + 1)T(P - a+ 7+ co+m+l)

_1_T(7 + co - n + 1)1X0 - 7 ~ « + « + D*

Assume for the moment that we can integrate (4.7) term by term (we will prove this

later) to get

(4.8) f /(co) du = £ ^„,„ f G(co + m)//(co - n) du.J-co m,n-0 •'-co

From [1, Vol. 2, p. 300, (21)], we get

J G(u + m)H(u — n) du

_ _T(P + Q + 1)_T(P + 1)T(Q + l)T(a - m - n + \)T(P + Q- a+ m + n+l)

and, thus, (4.8) becomes

f° , . v T(a + 1)T(P + Q + l)aw.(z - fey+<»-+"+-(4.9) /(co) c/co = ^ T(a_ m_n+imp + Q_a+m + n+ly


910 THOMAS J. OSLER

Using (4.6), we can write (4.9) as

J-„ m~0 2*i Jb (S - 2)"-—»+1

We can interchange summation and integration

P n w r(a + 1) fu + ) (s - b)p+Q A/ /(co) c/co = -—- /-— 2- amn(s - z) fife,

J-oo 2x1 Jj, (5 — z)" + 1 „,„_o

since we have an integrable factor times a uniformly convergent power series. Using

(4.5), we get

j _ co ^7t / j h

J_„ ^ M 2iri J0 0(£;z)a + 10e(£;

(s - z)

and using (4.3) and (4.4), we can rewrite this as

?t(K> z)

T(a + 1) f'2"' /(£, Q rf£^Jo r73^= D>^z)-2iri J0 (£ _ z)<"

Thus, the theorem is proved as soon as we verify the term by term integration in

(4.8).We know that (4.8) is correct provided

(4.10) £ \Amn\ [ \G(m + m)//(co - n)\ do*m.n = 0 j-co

converges [10, p. 45]. It is well known that T(a - co — l)_1r(co + b)'1 is an entire

function of co and that [2, Vol. 1, p. 33, (11)]

1

T(a - co + l)T(co + b)

T(co — a) sin t(co — a)= co ° b sin 7r(co — a)A(w)/ir

T(co + b)ir

where A{w) —> 1 as co —» + °°; and

T(l — co — b) sin ?r(fe + co) „_„ . , .= -=7-—tt- = (-co) sin ir(co + b)B(o>) it

T(a — co + 1)tt

where ß(co) —> 1 as co —> — 00 .

Thus,

(4.11) G(co + m) = ^1(co)co"p_1

and

(4.12) #(co - «) = BXcoJco"0-1,

where î(co) and 5,(co) are bounded as co —» ± «>. Let



(4.13) r = P + Q + 2.

Since by hypothesis, —1 < P + Q, we know that 1 < r. From (4.11) and (4.12),

we have

|<3(« + m)\ £ Z//(J3+1,(-co, oo) and |#(co - n)\ £ z//(Q + 1'(- °° , oo ).

Thus, we may apply Holder's inequality [10, p. 382] and obtain

f |G(co + m)H(po - n)\ dw

(4.14)

^ |j" |G(co + m)\r/{P + ,) Jco|<P + 1,Ay " |//(co - «)!"

since (P + 1)/V + (Q + 1)A = 1 by (4.13). But the R.H.S. of (4.14) is independentof m and n (since we can replace co by co — m in G and by « + « in //), and we denote

it by the constant M. Thus, combining (4.10) and (4.14), we get

QO »00 oo

£ |4J / \GH\ du ^ M £ M„|n,n = 0 J-od m,n = 0

and this last series converges. Thus, (4.8) is verified and the theorem is proved.

We complete our rigorous derivations by extending the Leibniz rule to the case

in which we differentiate with respect to an arbitrary function g(z). The proof of the

following Corollary 4.1 follows from the integral analog of the Leibniz rule (4.1)

in exactly the same way as the corresponding result [7, Corollary 4.2] follows from

the series form of the Leibniz rule. Thus, we omit the proof.

Corollary 4.1. Assume the hypothesis of Theorem 4.1 and the additional con-

ditions

(i) g(z) is regular and Univalent for z£g~ \R),

(ü) r) = m&, g(m(iii) e& z) = %(£); g(z)) = - g(z))<7*(9,(iv) <7(g(Ö) = <7*(Ö.

Then,

(4.15) ■] Li.' uq(v ,17(0

/.«. » d6*^ q*(0°+y

for z £ g '(S), and all a and y for which („ + 7) is defined.

If, in addition, we have

(v) g-\0)) = 0,then (4.15) can be simplified to

d:^z,z) = f ( l(4.16)


912 thomas j. osler

Having completed our rigorous analysis, we now turn to the evaluation of definite

integrals which are obtained from (4.15).

5. Examples. In this section, we give several examples of definite integrals

obtained from (1.4) by selecting specific functions for /(£, f), q({), g{z) and specifying

the parameters a and y. Table 5.1 lists our choices for /, q, g, a, y and Table 5.2

shows the resulting definite integrals. In computing the fractional derivatives in

(1.4), use was made of the extensive table of Riemann-Liouville integrals found in

[1, Vol. 2, pp. 181-200]. The notation used for the special functions is that of Erdelyi

et al. [1].

Attention is called to the various restrictions appearing in Table 5.2. These are

sometimes too strong. For example, it is well known that integral 8 requires only

3 < Re(A + B + C + D). The other two restrictions are not needed for the validity

of the integral. They emerge from item (i) of the hypothesis of Theorem 4.1 in which

we require -1 < Re(F) and -1 < Re(ß) so that Z>° ;£/(£, f) is defined. Since Table

5.2 is provided to illustrate our integral form of the Leibniz rule, all restrictions

emerging from the theorems of this paper are listed.

From the series form of the Leibniz rule (1.5), we see that in each integral in

Table 5.2 we can replace 'V by "an" and • • • du>" by "Er.-» ' •- fl" t0

obtain a valid expression. Series forms of integrals 1 through 13 were given in [7].

Table 5.1 Choices for functions and parameters in the integral analog of the Leibniz

rule (1.4) from which the definite integrals in Table 5.2 are derived.

No. /(£,f) 9(f) g(z) a y

1 f-\\-SrA 1 z B-C y2 eÂ'1 1 « A - B A - C

3 (cosf)/f 1 z2 -v-\ —\ — B

4 (coshf)/f 1 z2 -v - \ -\-B

5 (sinf)/f 1 z2 -v-\ -\-B

6 (sinhf)/f 1 z2 -v-\ -\-B

7 (1 - f2r 1 1 - z " + ß y

8 1 z A + c - 2 A - I

9 - Ö"'rB-,(l - fi~E 1 z b + B- d- DB-D

10 f-A-Y~'~X& + 1 ^ C-A-l C-l

11 t~Y + z a y12 $A~Y exp(fl) z a y

13 aAi-B-\FM, ■■■ ,ar; \ z a y

bu ■■■ , b,; f)

14 et+tf**-Y*<-» , z a + b - 2 a - I

15 mk + akYnr + ak)D 1 z 7



Table 5.2 Definite integrals obtained from the integral analog of the Leibniz rule-.

No. Definite integral

7t 2Ft(A, B; C; z)

T(C)T{B - C + 1)

sin((co + 7 + C - B)tt) 2F,{A, B; B - y - co; z)

(co + 7 + C - B)T(o> + 7 + DT(B - 7 - co)

Re(z) < I, 0 < Re(B).

7t ,F,(^; g; z) = r sin((co + £ - Ox) .F.H; C - co; z)

IX£)r(/l -5+1) i_„ (co + 5 - C)T(co + A - C + 1)T(C - co)

Re(A) > 0.

, cr/N rft - >) /zV-B r sin((co + v - Z?)x)Jfi.„(z) ( zY

3 ^) = ^T_l2j L (co + „ - P)r(co - B + *) W ^

through

6

where ff, = /„, /„, H„ and L„, respectively, for series 3, 4, 5 and 6.

c/co,

c/co,

irP"v(z) /"" sin((co + 7 — v — ;u)x)

r(f + it + 1) ~ JL. (co + 7 - v - p)

'T(co + 7 + i) \i + z/

-1 < Re(v), 0 < Re(z).

_T(A + B+ C+ D-3)_T(A + C - 1)T(A + D - l)T(ß + C - 1)T(5 + £>-!)

T(co + /i)T(co + B)T(C - co)r(D - co)

1 < Re(B + C), 1 < Re(A + D), 3 < Re(/4 + B + C + D).

T(b + B - 1) 2Fn(e + E, b + B - 1; ri + D - l;z)

T(rf + D - l)r(ft + B - d - + \)T(b)Y(B)

_2Fi(g, fe; ci + co; z) 2F,(£, B; D — co; z)_

r(co + P - D + l)T(co + fi?)r(6 d co + l)T(fl - co)

0 < Re{b), 0 < Re(ß), § > Re(z), 1 < Re(6 + B).


914 thomas j. osler

Table 5.2 (continued)


T(C+D- A-B-\) 3F5

10

E, (C+D- A-B-\)/2, (C+D- A-B)/2;

(D-B)/2AD-B+l)/2T(C - A)T(C - B)T(D - A)Y(D - B)

E, C - B, D - A; -z2

D + co, 1 — B — co

r(co + c)r\co + D>r(i - a - co)r(i - b - w)

0 < Re(D - a), 0 < Re(C - B), 1 < Re(C + D - a - B).

dw,

11 _T{A + B)_T(A + B - a)T(a + \)T(A)T(B + 1)

r-co — 7, { A}k; -zk/Pk'

fco + A + y — a}k .

co + 7, {B}k; -zk/P"

. {B - co - 7 + 1}*

rfco

r(co+7+l)r(co+^+7-a)r(a-7-co+l)r(S-7-co+l)

-1 < ReM), 0 < Re(B), k = 1, 2, 3, • • • .

12V(A + B)

V(A + B - a)T(a + l)T(^)r(ß + 1)

[B}„; -(co + y)zk

[B — a — y + IL= 1UU; (co + 7)zs

{co + ^ + 7 —

c/to

/_„ T(to+ /4+7-a)r(co+7+l)r(a-7-co+l)r(ß-7-co+l),*

1 < Re(A), 0 < Re(ß), /c = 1, 2, 3, • • • .

13

T(A + B) r+1Fs+1a + b, Ol, • • • , ßTi ?

^ + jB — a, 6ls • • • , b„

T(A B - a)T(a + l)T(A + l)T(B)

B, alt

B

■ ■ ■ , ar; z

co, fei, ■ • ■ , b.

r(co+7+l)T(co+ ^-a+7+l)r(a-7-co+l)r(ß-7-co)

1 < Re(A), 0 < Re(B).

dco,

* The notation {A}h stands for the set of k parameters A/k, (A + l)/k, (A + 2)/k,

,(A+ k - \)/k.


the integral analog of the leibniz rule

Table 5.2 (continued)

915


14T(a + b + c + d - 3) .Fâ + b + c + d - 3;c + d - 1; 2z)

r(c + d - 1)T(Ö + c - l)T(a + b - l)T(a + d - 1)

i F,(fr + c — 1; c + co; z) , Ft(a + — 1; t/ — co; z) dco

T(a + co)I\6 - co)T(c + co)T(c/ - co)

3 < Re(a + b + c + d), 1 < Re(fe + c), 1 < Re(a + d).

15

ä, {p + l}*; -z7«*~

{/? + <? — a + Mt_:T(p + q - a + l)T(a + l)T(p + \)Y(q + 1)

i+lFj— /4, {p + 1}*; —zk/a

\p — q + 7+ co + ljt.

t + i Fj-B, {<? + 1U; -z7«

. \q — 7 + to — ljt .

c/co

r(a-7-co+l)r(co+7+l)r(p-a+7+co+l)r(9-7-co+l)

1 < Re(p), -1 < Re(cj), -1 < Re(p + <?), A: = 1,2,3,

Department of Mathematics

Rensselaer Polytechnic Institute

Troy, New York 12181

1. A. Erdelyi, W. Magnus, F. Oberhettinger & F. G. Tricomi, Tables of IntegralTransforms. Vols. 1, 2, McGraw-Hill, New York, 1954, 1955. MR IS, 868; MR 16, 586.

2. Y. L. Luke, The Special Functions and Their Approximations. Vols. 1, 2, Math, inSei. and Engineering, vol. 53, Academic Press, New York, 1969. MR 39 #3039; MR 40#2909.

3. T. J. Osler, "Leibniz rule for fractional derivatives generalized and an applicationto infinite series," SIAM J. Appl. Math., v. 18, 1970, pp. 658-674. MR 41 #5562.

4. T. J. Osler, "The fractional derivative of a composite function," SIAM J. Math.Anal., v. 1, 1970, pp. 288-293. MR 41 #5563.

5. T. J. Osler, "Taylor's series generalized for fractional derivatives and applica-tions" SIAM J. Math. Anal., v. 2, 1971, pp. 37-48.

6. T. J. Osler, "Fractional derivatives and Leibniz rule," Amer. Math. Monthly, v.78, 1971, pp. 645-649.

7. T. J. Osler, "A further extension of the Leibniz rule to fractional derivatives andits relation to Parseval's formula," SIAM J. Math. Anal., v. 3, 1972, pp. 1-16.

8. T. J. Osler, "An integral analogue of Taylor's series and its use in computingFourier transforms," Math. Comp., v. 26, 1972, pp. 449-460.

9. T. J. Osler, "A correction to Leibniz rule for fractional derivatives." (To appear.)10. E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford Univ. Press, London,

1939.11. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed.,

Clarendon Press, Oxford, 1948.


Documents

The Integral Analog of the Leibniz Rule · the integral analog of the leibniz rule 907 (3.6) is the Parseval's formula [11, p. 50]. Thus, our integral analog of the Leibniz rule "extends