On functional relations of differintegrals involving trigonometric functions

              Rev. Acad. Canar. Cienc., XVI (Núms. 1-2), 15-27 (2004) (publicado en j ulio de 2005)
ON FUNCTIONAL RELATIONS OF DIFFERINTEGRALS
INVOLVING TRIGONOMETRIC FUNCTIONS
Marleny Fuenmayor, Susana Salinas de Romero & K. Nishimoto
Centro de Investigación de Matemática Aplicada (C.l.M.A.)
Facultad de Ingeniería. Universidad del Zulia
Apartado 10482. Maracaibo-Yenezuela
Abstract
Based on Nishimoto's fractional calculus [7, Vol. 1], the main object in
the present paper is to calculate the differintegral to arbitrary order /3 the
sorne relations functionals involving trigonometric functions of argument z.
In addition sorne illustrative examples are shown.
O. Introduction. (Definition of fractional calculus).
The fractional calculus deals with derivatives and integrals of arbitrary
order. The origin of fractional calculus is based in extension of the derivatives
concept of integer order n, supposing n real or complex, called fractional di­fferintegrals.
Recently, by applying the Nishimoto's definition [3,4,13,14,17,19]
of fractional differintegral of order v E IR, many authors have obtained di­fferintegrals
of different functions.
Riemann-Liouville [1-2] defines a fractional integration of /3 order as:
Re(/3) ~ O
a ::; X < b and r is the Gamma function.
Nishimoto [7, Vol. 1] defines the fractional differintegration of the func­tion
of a variable as follows:
15
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I. Definition.
If J(x) is an analytic function with no internal branching points and on
e (e = {e_' e+}) is an integral curve along the section joining two points z
and -oo + i Im( z) and C+ is an integral curve alorrg the section joining two
points z and ( +oo + i Im(z))
f = f ( ) = r(v + 1) J J(~)d(O ( E lR. d z-)
V e V z 27ri e(~ - z)v+l' V ' V 'F (1)
Í-n = lím cfv(n E z+)
y->-n
where ~-# z, -7r ::::; arg(~ - z) ::::; 7r for C_ and O ::::; arg(~ - z) < 27r for C+,
then fv( v > O) is the derivative of fractional order v and fv( v < O) is the
integral of fractional order lvl , if fv exists.
II. Lemma.
Sorne properties of the Fractional Calculus [3, Vol. 1, p. 21,22,28,30, 50]
and [4, p. 41-46]
1) (sin az)v = av sin ( az + ~v)
2) (cosaz)v = av cos ( az + ~v)
3) (zª )v = e-i7rv r(v - a) zª-v
r(-a) 1
r(v - a) 1
r(-a) < 00
4) a) (z)~
b) (z)~
= 0 f orn E zt ; V ~ Z
O for n E zt ; k E z+ and k > n
00 r(v+ l)
5) (uv)v = L k! f(l +V_ k) Uv-k Vk
k=O
zn
6) (1)-n = 1
n.
16
(2)
(3)
(4)
(5)
(6)
(7)
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l.- Differintegrals of functions relations.
Theorems 1:
We have
') ( (sin z)a + (sin z)_ª ) . ( 7r (.1) i = sm z + - /J
2 cos %0: f3 2
where o: # 2n + 1 and n = 0, ±1, ±2, ....
" ) ((sin z)a - (sin z)_ª ) ( 7r (.1) 11 = cos z + - /J
2 cos %0: f3 2
where o:# 2n and n = O, ±1, ±2, ....
Proof of the theorem li):
Using the lemma 1, we have
(sin z)a =sin (z +~a)
and
(sin z)-a =sin ( z - ~o:)
We know that
sin( A± B) = sin A cos B ±sin B cos A
Using (12), we have
(sin z )-a = sin ( z - ~o:) = sin z cos ~o: - sin ~o: cos z
Adding (13) and (14), we have
Then,
7r
(sin z)a + (sin z)-a = 2sin z cos 2ª
(sin z)a + (sin z)-a .
------ = sm z
2 cos ~a
17
(8)
(9)
(10)
(11)
(12)
(.13)
(14)
(15)
(16)
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Afterward, applying lemma 1, we obtain (8).
Proof of the theorem lii:
We have this theorem from theorem li (see the proof of previous. Sub­tracting
(14) from (13) and applying lemma 2).
Theorem 2:
We have
i) A(z,a,,B) = ({(sin z)a + (sin z)-a}2) = -2.B-1 cos (2z +!'._,e) (17)
4 cos2 IIa 2
2 .B
where a -=f 2n + 1 and n =O, ±1, ±2, .... ,,B ~ Z0 .
.. ) B( ,B) _ ({(sinz)a - (sin z)-a}2 ) _ 2.B- l (2 7f ,B) 11 z, a, - . 2 rr - cos z + -
4sm 2a .B 2
where a -=f 2n , n = 0,±1, ±2, .... and ,B ~ Z0.
iii) A(z, a, ,B) + B(z, a, ,B) =O
Proof of the theorem 2i:
From (15), we have
Then,
7f
{(sin z)a + (sin z)-a}2 = 4sin2 zcos2 2"ª
( {(sin z)a + (sin z)-a}2) (. 2 )
4 cos2 IIa = sm z .B
2 (3
Using the trigonometric identity
. 2 1 - cos 2z
sm z = 2
we have
(sin' z)p = G) P - (co;2z) 1
Applying lemmas (2) and (4a) , we obtain (17).
18
(18)
(19)
(20)
(21)
(22)
(23)
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Proof of the theorem 2ii:
We have this theorem from theorem 2i (see the proof of previous). Sub­tracting
(14) from (13) and using the trigonometric identity
(
2 1 + cos2z )
cos z = 2 (24)
we obtain (19) adding (17) and (18).
Theorem 3:
If (3 ~ Z0, we have
i) C(z,a,(3) = ((sin z)a(sin z)-ci)13 = -2!3-1 cos (2z + ~f3) (25)
ii) D(z, a, (3) = ( ( cos z )ci( cos z )- a) 13 = 2/3-l cos ( 2z + ~(3) (26)
iii) C(z, a, (3)+ D(z, a,(3) =O (27)
Proof of the theorem 3i:
Multiplying (13) and (14), we have
7r 7r
(sin z)ci (sin z)-ci = cos2 2asin2 z - sin2 2acos2 z (28)
Then
7r 7r
((sin z)a(sin z)- a)13 = cos2 2"ª (sin2 z)13 - sin2 2a(cos2 z )13
Using (22); (23) and applying lemmas 2 and 4a, we obtain (25).
Proof of the theorem 3ii:
Applying lemma 2, we have
( cos z) ci = cos ( z + ~a)
(cos z)-a = cos ( z - ~a)
19
(29)
(30)
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Using the trigonometric identity
cos(A ± B) = cosAcosB ::¡::sin A sin E (31)
Applying (31) in (29) and (30) , we have
7r . . 7r
( cos z)a = cos z cos 2a - sm z sm 2a (32)
7r . . 7r
(cos z)-a = cos z cos 2a+sm z sm 2a (33)
Multiplying (32) and (33) , we havc
7r 7r
(cos z)a(cos z)-a = cos2 z cos2 2a - sin2 z sin2 2a (34)
Afterward, using (22) and (24) , calculating the differintegral from (34)
applying lemmas 2 and 4a, we obtain (26) .
Proof of the theorem 3iii: Adding (25) and (26) , we obtain (27).
Theorem 4: We have
') ((cos z)a + (cos z)_ª ) ( 7r /3) 1 = cos z + -
2cos !!:a 2 2 {3
(35)
where a f. 2n + 1, and n =O, ±1, ±2, ....
u") ((cos z)-a - (cos z)ª ) = sm. ( z + -7r /3)
2 sin ~a /3 2
(36)
where a f. 2n and n =O, ±1 , ±2, ....
"')E( /3) = ({(cos z)a + (cos z)-a}2
) = 213-1 ( 2 + 2/3) u1 z, a , 2 7r cos z 2 4cos 2a
{3
(37)
where a f. 2n + 1, n =O, ±1, ±2, .... , and f3 ~ Z0.
iv) F(z, a,/3) = ({(cos z)-a - (cos z)a}2
) = -2,6-l cos (2z + 213) (38)
4cos2 !!:a 2
2 {3
20
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where a -=f 2n, n = O, ±1, ±2, .... , and (3 ~ Z0.
v) E( z,a,(3)+F(z, a, (3) = O
Proof of the theorem 4i:
Adding (32) and (33), we have
7T'
(cosz)a + (cosz)-a = 2cos 2acos z
Then
( (cosz)a + (cos z)_ª ) ( ) = cos z r.i 2 cos :!!:a /J
2 /3
Applying lcmma 2, we obtain (35).
Proof of the theorem 4ii:
(39)
(40)
(41)
We ha ve this theorem from theorem 4i ( see the proof of previous. Sub­tracting
( 32) from ( 33) and using lemma 1.)
Using (40), (24) and lemmas 2 and 4a, we obtain (37).
The proof of ( 38) is similar from theorem ( 4iii) using ( 22) and lemmas 2
and 4a.
Theorem 5: We have
i) ( ( cos z)a( cos z)a - (sin z)a (sin z)-a )13 = 213 cos ( 2z + ~(3) ( 42)
ii) ((sin z)ª( cos z)-a + ( cos z)a (sin z)-a )13 = 213 sin ( 2z + ~(3) ( 43)
m. a) ((sinz)a(cosz)-a + (cosz)a(sin z)-aLn = (sin2 z )-(n- 1) (44)
b) ((sinz)a(cosz)-a + (cos za (sin z)-aLn =
where n E z+.
21
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Proof of the theorem 5i:
Subtracting (28) from (34), we have
((cosz)a(cosz)-a - (sinz)a(sinz) - a)p = (cos2 z)p - (sin2 z)p
Afterward, using (22) , (24) and lemmas 2 and 4a obtain (42).
Proof of the theorem 5ii:
Multiplying (13) for (33) and (32) for (14), we have
(sinz)a(cosz)-a + (cosz)a(sinz)-a = 2cos z sin z (46)
Using the trigonometric identity
sin2z = 2sin zcosz (47)
and lemma 1, we obtain (43).
Proof of the theorem 5iiia and 5iiib:
Calculating the differintegral of integer order -nin (46), we have
((sinz)a(cosz)-a + (cosz)a(sinz)-a)_n = (2sin z cos z)-n
We know that
(sin z )i = 2 sin z cos z
Then
(sin zh-n = (sin2 z)-(n-1) (48)
Afterward, using (22) and lemmas 2 and 6 in (48) we obtain (45).
Theorem 6: We have
i) ( ((sin z)z)ª + ((sinz)z)_ª ) =sin (z + ?!_/3)
2 { z cos ~a+ a sin ~a} 2
¡3
(49)
ii) ( ( ( COS z) z) a + ( ( COS Z) Z )_ª) = COS (z + ?!._ /3)
2 { z cos ~a + a sin ~a} 2
¡3
(50)
1 •
1 •
1 .
) ((sin z)z)ª + ((sinz)z) _ª
~~~~~~~~ =tan z
((cos z)z)a + ((cosz)z)-a
(51)
22
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Proof of the theorem 6i:
Using lemma 5 with u= sin z and v = z (52)
Afterward, developing the serie and using lemmas 1 and 4b, we have
((sin z)z)a = zsin (z + ~o: ) +o:sin (z +~(o: -1))
(53)
Then
((sin z)z)ª = z sin (z +~o:) - o:cos (z +~o:) (54)
Using lemmas 1 and 5, we have
((sin z)z)_ª = z sin (z - ~o:) + o:cos ( z - ~a) (55)
Afterward, adding (54) and (55), we have
( (sin z )z )ª + ( (sin z )z La = 2 sin z ( z cos ~a + a sin ~a) (56)
Calculating the differintegral to order /3 in (56) and using lemma 1, we
have obtain (49).
Proof of the theorem 6ii:
Using lemma 2 and 5, we have
((cosz)z)ª=zcos(z+~a) +asin(z+ ~a) (57)
and
((cosz)z)_ª = z cos (z - ~a) - a sin (z - ~a) (58)
Adding (57) and (58), we have
((cos z)z)ª + ((cos z)z)_ª = 2 { z cos ~a + a sin ~a} cos z (59)
Afterward, calculating the differintegral of (59), using lemma 2, we obtain
(50).
Obtain (51) using (56) and (59).
23
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Theorem 7: We have
i) ({((sin z)z)a}2 + {((cos z)z)a}2) 13 = O
where /3 ~ íZ.
ii) ( { ( (sin Z) Z) a} 2 + { ( ( cos z) z) a} 2 )_n =
- (- )nr(-2 - n) 2+n 2Zn
- 1 r(-2) z +a n!
where n E íZci .
Proof of the theorem 7i:
Adding (54) and (57), we have
{((sin z)z)a}2 + {((cos z)z)a}2 = z2 + ci
Then
(60)
(61)
(62)
({((sin z)z)a}2 + {((cos z)z)a}2) 13 = (z2)13 + (0:2)13 (63)
Afterward, applying lemma (4a) in (63), we obtain (60).
Proof of the theorem 7ii:
Calculating the differintegral of integer order -n in (62), we have
Afterward, applying lemmas (3) and (6) in (64) we obtain (61).
EJERCICIOS
) (
(sin z)v'3 + (sinz)_v'3) . .
1 v'3 =sm(z+37r)=-sm z
2 cos 271' 6
2) ( (sin z)5 - (sin z)_5 ) ( 71') v12 ( . )
. ir = cos z - - = - sm z + cos
2sm52 _112 4 2
24
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3) (
{(sin z)- 1¡3 - (sinz)i¡3}2) 1¡3 ( 7f) = - r cos 2z + -3
4cos2 ~
. 6 2/3
= -r513(vÍ3cos2z - sin2z)
4) ((cos z)J2(cos z)_J2)_1/3 = r 413cos (2z- ~)
= r 1!2 (cos2z+vÍ3sin2z)
5) ((sin z) 1;2(cos z)- i;2 + (cosz)i¡2(sinz) _1;2) _7 = (sin2 z)-6
6) (((sin z)z)a+(sin z)z)-a ) = sin (z+~37f)
2 { cos ~a+ a sin ~a} 413
1 z6 r 7 cos2z + --
2 6!
= ~ ( J3 cos z - sin z)
7) ( { ( (sin z) z) V2} 2 + { ( ( cos z) z) V2} 2) _8 =
r(-10) io zs
= r(-2) z + 28!
ACKNOWLEDGEMENT
The authors are thankful to CONDES, Universidad del Zulia, for provi­ding
a research grant.
25
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BIBLIOGRAFÍA
1) A.C. MCBRIDE: Fractional Calculus and Integral Transforms of gene­ralized
functions, Research Notes, Vol. 31 (1979), Pitman.
2) IBRAHIM A. G. AND AHMED M. A EL-SAYED: Definite integral of
fractional order for set valued functions. J. Frac. Caleulus (to appear) Vol.
11, May, 1997.
3) MATERA J. AND NISHIMOTO K.: N-Fractional Caleulus of sorne ele­mentary
functions II. J. Frac. Cale. Vol. 14, Nov. 1998, 23-31.
4) MATERA J., PRIETO A. , SALINAS S. AND NISHIMOTO K.: N­Fractional
Caleulus of sorne elementary functions II. Jour. Frac. Cale. Vol.
12, 1997.
5) MILLER K. S. AND ROSS B.: An Introduction to the Fractional Caleulus
and Fractional Differential Equations. John vViley & Sons. Inc. New York,
1993.
6) NISHIMOTO K.: An Essence of Nishimoto 's Fractional Caleulus ( Caleu­lus
of the 21St. Century); integrals and differentiations of Arbitrary Order,
1991, Descartes Press, Koriyama, Japan.
7) NISHIMOTO K.: Fractional Caleulus, Vol. 1, 1984; Vol. 2, 1987; Vol. 3,
1989; Vol. 5, 1996, Descartes Press, Koriyama, Japan.
8) NISHIMOTO, K.: Unification of the integrals and derivatives (A serendi­pity
in Fractional Caleulus). J. Frac. Cale. Vol. 6, 1994, 1-14.
9) OLDHAM K.B. AND SPANIER J.: The Fractional Caleulus, Academic
Press, 1974.
10) OLDHAM K. B. AND SPANIER J.: The Fractional Caleulus. Theory
and Applications of Differentiation and integration to arbitrary order. New
York, 1974.
11) PODLUBNY IGOR AND EL-SAYED, AHMED M.A.: On two Defini­tions
of Fractional Caleulus. Slovak Academy of Sciences Institute of Expe­rimental
Physics U.E.F. 03-96 ISBN 80-7099-252-2, 1996.
12) ROSS B. (ED): Fractional Caleulus and its Applications. Lecture Notes,
Vol. 457, 1975. Springer, Verlag.
13) SALINAS S., KALLA S. AND NISHIMOTO K.: N-Fractional Caleulus
of sorne Functions. Jour. Frac. Cale. Vol. 9, 1996, 33-39.
14) SALINAS S., MATERA J. AND NISHIMOTO K.: N-Fractional Caleulus
of sorne elementary functions III. J. Frac. Cale. Vol. 13, May, 1998, 57-62.
15) SAMKO S.G. , KILBAS A.A. AND MARICHEV 0.I.: Fractional Inte­grals
and Derivatives and sorne of their Applications, Gordon and Breach,
26
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
1993.
16) SPIEGER MURAMY R. Y ABELLANAS LORENZO: Fórmulas y tablas
de matemática aplicada. McGraw-Hill, 1992.
17) TU SHIH-TONG AND NISHIMOTO K.: On the Fractional Calculus
((x - a).B(x - b)v)°'. J. Fractional Calculus, Vol. 4, November 1993, 15-16.
18) TU, SHIH-TONG AND CHYAN DING-KUO: A certain family of infinite
series, differintegrable functions and Psi functions. J. Fractional Calculus,
Vol. 7; May, 1975, 41-16.
19) WU, TSU-CHEN AND NISHIMOTO K.: N-Fractional Calculus of sorne
trigonometric functions, J. Fractional Calculus, Vol. 14, November 1998,
83-94.
Marleny Fuenmayor and
Susana Salinas
Centro de Investigación de
Matemática Aplicada (C.I.M.A.)
Facultad de Ingeniería. L.U.Z.
Apartado 10482.
Maracaibo, Venezuela
27
K. Nishimoto
Institute of Applied
Mathematics
Descartes Press Co.
2-13-10, Kagüike, Koriyama
963-8833, Japan.
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