the integral analog of the leibniz rule - american mathematical
TRANSCRIPT
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:^u(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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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)"
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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 ^i(co) and 5,(co) are bounded as co —» ± «>. Let
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
the integral analog of the leibniz rule 911
(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)
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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 e^A'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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
the integral analog of the leibniz rule 913
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).
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
914 thomas j. osler
Table 5.2 (continued)
No. Definite integral
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.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
the integral analog of the leibniz rule
Table 5.2 (continued)
915
No. Definite integral
14T(a + b + c + d - 3) .F^a + 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.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use