- Email: [email protected]
Identities for the E2,1 -transform and their applications
Applied Mathematics and Computation 187 (2007) 1557–1566 www.elsevier.com/locate/amc
Identities for the E2;1-transform and their applications David Brown a, Nesße Dernek b, Osman Yu¨rekli a b
a,*
Department of Mathematics, Ithaca College, Ithaca, NY 14850, USA Department of Mathematics, University of Marmara, Istanbul, Turkey
Abstract In the present paper the authors introduce the E2;1 -transform with kernel the exponential integral function. It is shown that the third iterate of the L2 -transform is the exponential integral transform, and some identities involving the new transform, the L2 -transform and the Widder potential transform are given. Using the identities, Parseval–Goldstein type results involving these transforms are proved. Some illustrative examples are also given. Ó 2006 Elsevier Inc. All rights reserved. Keywords: E2;1 -transforms; L2 -transforms; Laplace transforms; Widder potential transform; Stieltjes transforms; Mellin transforms; K-transforms; Hankel transforms; Fourier sine transforms; Parseval–Goldstein type theorems
1. Introduction In this paper, we introduce the E2;1 -transform as Z 1 E2;1 ff ðxÞ; yg ¼ x expðx2 y 2 ÞE1 ðx2 y 2 Þf ðxÞ dx;
ð1:1Þ
0
where E1(x) is the exponential integral function defined as Z 1 xt Z 1 u e e du ¼ dt: E1 ðxÞ ¼ EiðxÞ ¼ u t x 1
ð1:2Þ
The L2 -transform L2 ff ðxÞ; yg ¼
Z
1
x expðx2 y 2 Þf ðxÞ dx;
0
*
Corresponding author. E-mail address: [email protected] (O. Yu¨rekli). URL: http://www.ithaca.edu/osman (O. Yu¨rekli).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.09.139
ð1:3Þ
1558
D. Brown et al. / Applied Mathematics and Computation 187 (2007) 1557–1566
is introduced in [12] and the Widder potential transform Z 1 xf ðxÞ dx; Pff ðxÞ; yg ¼ 2 þ y2 x 0
ð1:4Þ
is studied in [6]. It is well known that the second iterate of the L2 -transform is the Widder potential transform; that is, L22 ff ðxÞ; yg ¼ L2 fL2 ff ðxÞ; ug; yg ¼ Pff ðxÞ; yg;
ð1:5Þ
provided that integrals involved converge absolutely, (cf. [12, Eq. (2.1)]). These transforms were studied in [7,8,10,11]. The L2 -transform and the Laplace transform are related by the identity pffiffiffi 1 L2 ff ðxÞ; yg ¼ Lff ð xÞ; y 2 g; 2 in which the Laplace transform is defined as Z 1 Lff ðxÞ; yg ¼ expðxyÞf ðxÞ dx:
ð1:6Þ
ð1:7Þ
0
The Widder potential transform and the Stieltjes transform are related by the identity pffiffiffi 1 Pff ðxÞ; yg ¼ Sff ð xÞ; y 2 g; 2 in which the Stieltjes transform is defined as Z 1 f ðxÞ dx: Sff ðxÞ; yg ¼ x þy 0
ð1:8Þ
ð1:9Þ
In this paper, it is shown that the third iterate of the L2 -transform is a constant multiple of the E2;1 -transform as defined in (1.1). Using the identities, Parseval–Goldstein type theorems relating these integral transforms are proved. As an application of the identities and theorems, some illustrative examples are given. 2. The main theorem In the following lemma, we show that the third iterate of the L2 -transform is essentially the E2;1 -transform. Lemma 1. The identities 1 1 L32 ff ðxÞ; yg ¼ PfL2 ff ðxÞ; ug; yg ¼ L2 fPff ðxÞ; ug; yg; 2 2
ð2:1Þ
1 L32 ff ðxÞ; yg ¼ E2;1 ff ðxÞ; yg; 4
ð2:2Þ
and
hold true, provided that the integrals involved converge absolutely. Proof. The proof of identity (2.1) easily follows from (1.5). In order to prove (2.2), we use (2.1) 1 L32 ff ðxÞ; yg ¼ L2 fPff ðxÞ; ug; yg: 2 Using the definitions of the L2 -transform (1.3) and the Widder potential transform (1.4) we have Z 1 Z 1 1 u2 y 2 xf ðxÞ 3 L2 ff ðxÞ; yg ¼ ue dx du: 2 0 x 2 þ u2 0
ð2:3Þ
ð2:4Þ
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Changing the order of integration, which is permissible by absolute convergence of the integrals involved, and using definition (1.4) once again, it follows from (2.4) that Z 1 Z Z 1 1 u expðu2 y 2 Þ 1 1 3 L2 ff ðxÞ; yg ¼ xf ðxÞ du dx ¼ xf ðxÞPfexpðu2 y 2 Þ; xg dx: ð2:5Þ 2 0 x 2 þ u2 2 0 0 Utilizing (1.8), the last equation (2.5) can be written as Z 1 1 3 L2 ff ðxÞ; yg ¼ xf ðxÞSfexpðuy 2 Þ; x2 g dx: 4 0
ð2:6Þ
Making use of the formula [2, Entry (11), p. 217] and (1.2) 2
Sfeuy ; x2 g ¼ expðx2 y 2 ÞEiðx2 y 2 Þ ¼ expðx2 y 2 ÞE1 ðx2 y 2 Þ;
ð2:7Þ
where RðyÞ > 0. Now assertion (2.2) follows upon inserting (2.7) into (2.6) and then using definition (1.1) of the E2;1 -transform. h Some illustrations of the identities in Lemma 1 are contained the following examples. Example 2. We show that mp m 1 p m1 E2;1 fx ; yg ¼ sec C þ y m1 ; 2 2 2 2
ð2:8Þ
where 1 < RðmÞ < 1. Proof. If we set f ðxÞ ¼ xm1
ð2:9Þ
in Lemma 1, and then substitute (2.9) and the formula mp p Pfxm1 ; yg ¼ sec y m1 ; 2 2 where 1 < RðmÞ < 1, (cf. [9, (A1), p. 248]), into identity (2.1) we obtain mp m 1 p 3 m1 L2 fx ; yg ¼ sec C þ y m1 : 8 2 2 2
ð2:10Þ
ð2:11Þ
Now assertion (2.8) follows when we use (2.11) in (2.2). h The following example contains results involving the error functions. The error function and the complementary error function are defined as Z x 2 ErfðxÞ ¼ pffiffiffi expðt2 Þ dt; ð2:12Þ p 0 and 2 ErfcðxÞ ¼ 1 ErfðxÞ ¼ pffiffiffi p
Z
1
expðt2 Þ dt;
ð2:13Þ
x
respectively. Example 3. We show that 2 p p1=2 a a a exp Erfc E2;1 fsinðaxÞ; yg ¼ 2 1 2y 2y 2y 4y 2
ð2:14Þ
and E2;1
cosðaxÞ ;y x
2 p3=2 a a ¼ exp Erfc 2 2y 2y 4y
ð2:15Þ
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Proof. We set f ðxÞ ¼ sinðaxÞ
ð2:16Þ
in Lemma 1. Using the formula p PfsinðaxÞ; ug ¼ expðauÞ; ð2:17Þ 2 (cf. [9, (A3), p. 248]), then applying the L2 -transform to both sides of (2.17), we obtain p L2 fPfsinðaxÞ; ug; yg ¼ L2 fexpðauÞ; yg: ð2:18Þ 2 Using identity (1.6) on the right-hand side of (2.18), we get p L2 fPfsinðaxÞ; ug; yg ¼ Lfexpðau1=2 Þ; y 2 g: ð2:19Þ 4 Substituting (2.16) and (2.18) to identity (2.1) of Lemma 1 then using the formula [1, Entry (31), p. 146] we deduce 2 p p1=2 a a a 3 exp : ð2:20Þ L2 fsinðaxÞ; yg ¼ 2 1 Erfc 8y 2y 3 2y 4y 2 Now assertion (2.14) follows when we use relationship (2.2) of Lemma 1 in (2.20). The proof of assertion (2.15) is similar. We set f ðxÞ ¼
cosðaxÞ x
ð2:21Þ
in Lemma 1 and make use of the formulas [2, Entry (55), p. 221] and [1, Entry (33), p. 147]. h In the next example we evaluate the E2;1 -transform of the Bessel function of the first kind Jm(x) defined as m 1 x mþ2m X ð1Þ : ð2:22Þ J m ðxÞ ¼ m!Cðm þ m þ 1Þ 2 m¼0 Example 4. We show that E2;1 fxm J m ðaxÞ; yg ¼
2 am Cðm þ 1Þ a a2 exp C m; ; 4y 2 4y 2 2mþ1 y 2mþ2
ð2:23Þ
where RðmÞ > 1. Proof. We set f ðxÞ ¼ xm J m ðaxÞ
ð2:24Þ
in Lemma 1. Using the formula Pfxm J m ðaxÞ; ug ¼ um K m ðauÞ;
ð2:25Þ
(cf. [9, (A8), p. 249]), then applying the L2 -transform to both sides of (2.25), we obtain L2 fPfxm J m ðaxÞ; ug; yg ¼ L2 fum K m ðauÞ; yg;
ð2:26Þ
where Km(x) is the Macdonald function. It is defined as K m ðxÞ ¼
p I m ðxÞ I m ðxÞ ; 2 sinðmpÞ
in which Im(x) is the modified Bessel function defined as 1 x mþ2m X ð1Þm : I m ðxÞ ¼ m!Cðm þ m þ 1Þ 2 m¼0
ð2:27Þ
ð2:28Þ
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Using identity (1.6) in the right-hand side of (2.26), we get 1 L2 fPfxm J m ðaxÞ; ug; yg ¼ Lfum=2 K m ðax1=2 Þ; y 2 g; ð2:29Þ 2 Substituting (2.24) and (2.26) to identity (2.1) of Lemma 1 then using the formula [1, Entry (36), p. 199], we deduce 2 am Cðm þ 1Þ a a2 L32 fxm J m ðaxÞ; yg ¼ mþ3 exp C m; ; ð2:30Þ 4y 2 4y 2 2 y 2mþ2 where RðmÞ > 1. Now assertion (2.23) follows when we use relationship (2.2) of Lemma 1 in (2.30).
h
We now state our main result. Theorem 5. The Parseval–Goldstein type relations Z Z 1 1 1 yL2 ff ðxÞ; ygPfgðuÞ; yg dy ¼ xf ðxÞE2;1 fgðuÞ; xg dx 2 0 0 and
Z 0
1
1 yL2 ff ðxÞ; ygPfgðuÞ; yg dy ¼ 2
Z
ð2:31Þ
1
ugðuÞE2;1 ff ðxÞ; ug du
ð2:32Þ
0
hold true, provided that the integrals involved converge absolutely. Proof. We only give the proof of (2.31), as the proof of (2.32) is similar. From definition (1.3) of the L2 transform, we obtain Z 1 Z 1 Z 1 yL2 ff ðxÞ; ygPfgðuÞ; yg dy ¼ yPfgðuÞ; yg x expðx2 y 2 Þf ðxÞ dx dy: ð2:33Þ 0
0
0
Changing the order of integration, which is permissible by absolute convergence of the integrals involved, we find from (2.33) that Z 1 Z 1 Z 1 yL2 ff ðxÞ; ygPfgðuÞ; yg dy ¼ xf ðxÞ y expðx2 y 2 ÞPfgðuÞ; yg dy dx: ð2:34Þ 0
0
Now assertion (2.31) follows from (2.1) and (2.2).
0
h
3. Useful corollaries Interesting consequences of the main theorem will be given in this section. Useful Parseval-type identities for the L2 -transform and the Widder potential transform are contained in Corollary 6. If the integrals involved converge absolutely, then we have Z 1 Z 1 1 mþ1 f ðxÞ y m L2 ff ðxÞ; yg dy ¼ C dx; 2 2 xm 0 0 where RðmÞ > 1, Z 1 mp Z 1 gðuÞ PfgðuÞ; y g p sec dy ¼ du; ym 2 2 um 0 0 where 1 < RðmÞ < 1, and Z 1 Z 1 Pff ðxÞ; yg 1 m dy ¼ C y m L2 ff ðxÞ; yg dy; ym 2 2 0 0 where RðmÞ < 1.
ð3:1Þ
ð3:2Þ
ð3:3Þ
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Proof. We set gðuÞ ¼ um1
ð3:4Þ
in relation (2.31) of Theorem 5, then use (3.4), (2.8), and (2.10) in the Parseval–Goldstein identity (2.31). Similarly, we put f ðxÞ ¼ xm1
ð3:5Þ
in relation (2.32) of Theorem 5, then use (3.2), (2.8), (1.6), and the identity pm 1 m 1 m C þ C ¼ p sec 2 2 2 2 2 for the gamma function in the Parseval–Goldstein identity (2.32).
ð3:6Þ
h
Remark 7. The Mellin transform is defined as Z 1 Mff ðxÞ; yg ¼ xy1 f ðxÞ dx:
ð3:7Þ
0
If we use the definition above, then the relationships in Corollary 6 can be expressed as 1 mþ1 MfL2 ff ðxÞ; yg; m þ 1g ¼ C Mff ðxÞ; m þ 1g; 2 2 where RðmÞ > 1, MfPfgðuÞ; yg; m þ 1g ¼
mp p sec MfgðuÞ; m þ 1g; 2 2
ð3:8Þ
ð3:9Þ
where 1 < RðmÞ < 1, and
1 m MfPff ðxÞ; yg; m þ 1g ¼ C MfL2 ff ðxÞ; yg; m þ 1g; 2 2
ð3:10Þ
where RðmÞ > 1. More useful Parseval-type identities for the L2 -transform, the Widder potential transform and the E2;1 -transform are contained in Corollary 8. If the integrals involved converge absolutely, then we have Z 1 Z 1 m 1 PfgðuÞ; yg xm E2;1 fgðuÞ; xg dx ¼ C þ dy; 2 2 0 ym 0 where RðmÞ > 1, and Z 1 mp Z 1 um E2;1 ff ðxÞ; ug du ¼ p sec y m L2 ff ðxÞ; yg dy; 2 0 0 where 1 < RðmÞ < 1, Z 1 mp m 1 Z 1 f ðxÞ p m u E2;1 ff ðxÞ; ug du ¼ sec dx; C þ 2 2 2 2 0 xm 0
ð3:11Þ
ð3:12Þ
ð3:13Þ
where 1 < RðmÞ < 1. Proof. Setting f ðxÞ ¼ xm1
ð3:14Þ
in relation (2.31) and gðuÞ ¼ um1
ð3:15Þ
D. Brown et al. / Applied Mathematics and Computation 187 (2007) 1557–1566
1563
in relation (2.32), then using relationship (1.6), we obtain assertions (3.11) and (3.12). Assertion (3.13) is obtained when we combine (3.1) and (3.12). h Remark 9. Using the definition of the Mellin transform given in (3.7), the relationships in Corollary 8 can be expressed as mþ1 MfE2;1 fgðuÞ; xg; m þ 1g ¼ C MfPfgðuÞ; yg; m þ 1g; ð3:16Þ 2 where RðmÞ > 1, MfE2;1 ff ðxÞ; ug; m þ 1g ¼ p sec
mp 2
MfL2 ff ðxÞ; yg; m þ 1g;
ð3:17Þ
where 1 < RðmÞ < 1, and mp 1 m MfE2;1 ff ðxÞ; ug; m þ 1g ¼ p sec C þ Mff ðxÞ; m þ 1g; 2 2 2
ð3:18Þ
where 1 < RðmÞ < 1. In the following corollary, a Parseval–Goldstein type identity is given. The identity relates the L2 -transform, E2;1 -transform, the Bessel function of the first kind Jm(x) and the modified Bessel function of the second kind Km(x) as defined in (2.22) and (2.27), respectively. Corollary 10. We have Z Z 1 1 1 mþ1 y mþ1 K m ðzyÞL2 ff ðxÞ; yg dy ¼ u J m ðzuÞE2;1 ff ðxÞ; ug du 2 0 0
ð3:19Þ
provided that the integrals involved converge absolutely, RðmÞ > 1=2, and Re(z) > 0. Proof. We set gðuÞ ¼ um J m ðzuÞ
ð3:20Þ
in Theorem 5, then use (3.20) and (2.25) in the Parseval–Goldstein type relationship (2.32), we obtain assertion (3.19). h Remark 11. If we recall the definitions of the Hankel-transform Z 1 pffiffiffiffiffi Hm ff ðxÞ; yg ¼ xy J m ðxyÞf ðxÞ dx;
ð3:21Þ
0
where Jm(x) is the Bessel function of the first kind as defined in (2.22) and the K-transform Z 1 pffiffiffiffiffi Km ff ðxÞ; yg ¼ xy K m ðxyÞf ðxÞ dx;
ð3:22Þ
0
where Km(x) is the modified Bessel function of the second kind as defined in (2.27), we obtain the following identity involving the Hankel transform, the K-transform, the L2 -transform, and the E2;1 -transform n 1 o 1 n 1 o Km y mþ2 L2 ff ðxÞ; yg; z ¼ Hm umþ2 E2;1 ff ðxÞ; ug; z : ð3:23Þ 2 Remark 12. If we recall the definition of the Fourier sine transform Fs ff ðxÞ; yg ¼
Z 0
1
sinðxyÞf ðxÞ dx;
ð3:24Þ
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and the special cases of the Bessel function of the first kind (2.22) and the K-transform rffiffiffiffiffi rffiffiffiffiffi 2 2 expðxÞ and K 1=2 ðxÞ ¼ sinðxÞ; J 1=2 ðxÞ ¼ px px respectively, then identity (3.18) takes the form n 1 o 1 n 1 o L y mþ2 L2 ff ðxÞ; yg; z ¼ Fs umþ2 E2;1 ff ðxÞ; ug; z : 2
ð3:25Þ
ð3:26Þ
4. Illustrative examples An illustration of the Parseval–Goldstein type relation (3.1) of Corollary 6 is given in the following example. Example 13. We have Z 1 pl al l C y l1 expða2 y 2 ÞErfcðayÞ dy ¼ sec ; 2 2 2 0
ð4:1Þ
where 0 < RðlÞ < 1. Proof. We set 1 f ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ a2
ð4:2Þ
in relation (3.2) of Corollary 6, then using relation (1.6) and the formula [1, Entry 18, p. 135], we have o pffiffiffi p 1 n 2 1=2 2 expða2 y 2 ÞErfcðayÞ: ;y ¼ ð4:3Þ L2 ff ðxÞ; yg ¼ L ðx þ a Þ 2y 2 Substituting (4.2) and (4.3) into (3.2) of Corollary 6, we find Z 1 Z 1 1 l 1 xl pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx: þ y l1 expða2 y 2 ÞErfcðayÞ dy ¼ pffiffiffi C 2 2 0 p x 2 þ a2 0 The integral on the right-hand side has the value (cf. [3, Eq. (2), p. 171], and [5, Eq. (29), p. 163]) Z 1 xl al 1l l pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ ; ; B 2 2 2 x 2 þ a2 0
ð4:4Þ
ð4:5Þ
where B(x, y) is the beta function and it is defined as Bðx; yÞ ¼
CðxÞCðyÞ Cðx þ yÞ
If we use definition (4.6) of the beta function, then (4.4) simplifies to Z 1 al 1 l l 1 l þ C C y l1 expða2 y 2 ÞErfcðayÞ dy ¼ C : 2 2 2 2 2 2p 0
ð4:6Þ
ð4:7Þ
Using formula (3.6) for the C-function on the right-hand side of (4.7), we obtain assertion (4.1). h Remark 14. The formula 2 Z 1 l pl x x xl1 exp sec Erfc pffiffiffi dx ¼ 2ðl=2Þ1 C 2 2 2 2 0 pffiffiffi from [1, Entry (9), p. 325] is obtained when we set a ¼ 1= 2 in result (4.1) of Example 13.
ð4:8Þ
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Example 15. We have Z 1 pm p mþ2k1 1 2k m 1 2km 2 2 2 2 y expða y ÞCðk; a y Þ dy ¼ a ½Cð1 kÞ C sec ; 2 2 2 0
ð4:9Þ
where 1 < RðmÞ < 1, RðkÞ < 1, and Rð2k þ mÞ < 1. Proof. We set gðuÞ ¼ u2k expða2 u2 Þ
ð4:10Þ
in relation (3.2) of Corollary 6, then using relation (1.4) and the formula [2, Entry (17), p. 217], we have Cð1 kÞ 2k 1 y expða2 y 2 ÞCðk; a2 y 2 Þ: PfgðuÞ; yg ¼ S uk expða2 uÞ; y 2 ¼ 2 2
ð4:11Þ
Substituting (4.10) and (4.11) into (3.2) of Corollary 6, we find Z 1 mp Z 1 p 2km 2 2 2 2 sec y expða y ÞCðk; a y Þ dy ¼ u2km expða2 u2 Þ du: Cð1 kÞ 2 0 0
ð4:12Þ
Using definition (1.3) of the L2 -transform and (1.6), the integral on the right-hand side has the value 1 1 1 2k m 2kþm1 2km1 ð2km1Þ=2 2 L2 fu ; yg ¼ Lfu ;y g ¼ C : ð4:13Þ y 2 2 2 Assertion (4.9) follows upon substituting (4.13) into (4.12). h Remark 16. Setting k = 1/2, m = l in (4.9) and using pffiffiffi 1 C ; a2 y 2 ¼ p ErfcðayÞ 2
ð4:14Þ
we obtain result (4.1) of Example 13. Example 17. We have Z 1 am 1 m m1 2 2 p ffiffiffi y expða y ÞErfcðayÞ dy ¼ m C CðmÞ; 2 2 2 p 0
ð4:15Þ
where 0 < RðmÞ < 1. Proof. We set f ðxÞ ¼
cosðaxÞ x
ð4:16Þ
in relation (3.13) of Corollary 8, and upon using identity (2.15) and the definition of the Mellin transform (3.7), we have
2 Z 1 lp l 1 cosðaxÞ p3=2 a a p sec þ ; l : ð4:17Þ exp yl dy ¼ C M Erfc 2y 2 2 2 2 x 2y 4y 2 0 Simplifying and using the formula [1, Entry (21), p. 319] yields 2 Z 1 a a al l 1 p ffiffiffi þ C y l1 exp dy ¼ CðlÞ; Erfc 2y 2 2 4y 2 p 0 where 1 < RðlÞ < 0. Making the substitutions m = l and x = 1/2y, we obtain assertion (4.15).
ð4:18Þ h
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Remark 18. Setting l = 2z and combining the results of Examples 13 and 17, we have a derivation of the wellknown identity pffiffiffi 22z1 CðzÞCðz þ 1=2Þ ¼ pCð2zÞ ð4:19Þ (cf. [4, p. 3]). We conclude that many other infinite integrals can be evaluated in this manner by applying the lemma, the theorem and its corollaries considered here. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
A. Erdelyi et al., Tables of Integral Transforms, vol. 1, McGraw-Hill Book Company, 1954. A. Erdelyi et al., Tables of Integral Transforms, vol. 2, McGraw-Hill Book Company, 1954. M.L. Glasser, Some Bessel function integrals, Kyungpook Math. J. 13 (1973) 171–174. N. Lebedev, Special Functions and their Applications, Prentice-Hall, 1965. H.M. Srivastava, O. Yu¨rekli, A theorem on Stieltjes-type integral transform and its applications, Complex Var. Theory Appl. 28 (1995) 159–168. D.V. Widder, A transform related to the Poisson integral for a half-plane, Duke Math. J. 33 (1966) 355–362. S. Wilson, O. Yu¨rekli, A new method of solving Bessel’s differential equation using the L2 -transform, Appl. Math. Comput. 130 (2002) 587–591. S. Wilson, O. Yu¨rekli, A new method of solving Hermite’s differential equation using the L2 -transform, Appl. Math. Comput. 145 (2003) 495–500. O. Yu¨rekli, A Parseval-type theorem applied to certain integral transforms, IMA J. Appl. Math. 42 (1989) 241–249. O. Yu¨rekli, New identities Involving the Laplace and the L2 -transforms, Appl. Math. Comput. 99 (1999) 141–151. O. Yu¨rekli, Theorems on L2 -transforms and its applications, Complex Var. Theory Appl. 38 (1999) 95–107. O. Yu¨rekli, I. Sadek, A Parseval–Goldstein type theorem on the Widder potential transform and its applications, Int. J. Math. Math. Sci. 14 (1991) 517–524.
Identities for the E2;1-transform and their applications David Brown a, Nesße Dernek b, Osman Yu¨rekli a b
a,*
Department of Mathematics, Ithaca College, Ithaca, NY 14850, USA Department of Mathematics, University of Marmara, Istanbul, Turkey
Abstract In the present paper the authors introduce the E2;1 -transform with kernel the exponential integral function. It is shown that the third iterate of the L2 -transform is the exponential integral transform, and some identities involving the new transform, the L2 -transform and the Widder potential transform are given. Using the identities, Parseval–Goldstein type results involving these transforms are proved. Some illustrative examples are also given. Ó 2006 Elsevier Inc. All rights reserved. Keywords: E2;1 -transforms; L2 -transforms; Laplace transforms; Widder potential transform; Stieltjes transforms; Mellin transforms; K-transforms; Hankel transforms; Fourier sine transforms; Parseval–Goldstein type theorems
1. Introduction In this paper, we introduce the E2;1 -transform as Z 1 E2;1 ff ðxÞ; yg ¼ x expðx2 y 2 ÞE1 ðx2 y 2 Þf ðxÞ dx;
ð1:1Þ
0
where E1(x) is the exponential integral function defined as Z 1 xt Z 1 u e e du ¼ dt: E1 ðxÞ ¼ EiðxÞ ¼ u t x 1
ð1:2Þ
The L2 -transform L2 ff ðxÞ; yg ¼
Z
1
x expðx2 y 2 Þf ðxÞ dx;
0
*
Corresponding author. E-mail address: [email protected] (O. Yu¨rekli). URL: http://www.ithaca.edu/osman (O. Yu¨rekli).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.09.139
ð1:3Þ
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is introduced in [12] and the Widder potential transform Z 1 xf ðxÞ dx; Pff ðxÞ; yg ¼ 2 þ y2 x 0
ð1:4Þ
is studied in [6]. It is well known that the second iterate of the L2 -transform is the Widder potential transform; that is, L22 ff ðxÞ; yg ¼ L2 fL2 ff ðxÞ; ug; yg ¼ Pff ðxÞ; yg;
ð1:5Þ
provided that integrals involved converge absolutely, (cf. [12, Eq. (2.1)]). These transforms were studied in [7,8,10,11]. The L2 -transform and the Laplace transform are related by the identity pffiffiffi 1 L2 ff ðxÞ; yg ¼ Lff ð xÞ; y 2 g; 2 in which the Laplace transform is defined as Z 1 Lff ðxÞ; yg ¼ expðxyÞf ðxÞ dx:
ð1:6Þ
ð1:7Þ
0
The Widder potential transform and the Stieltjes transform are related by the identity pffiffiffi 1 Pff ðxÞ; yg ¼ Sff ð xÞ; y 2 g; 2 in which the Stieltjes transform is defined as Z 1 f ðxÞ dx: Sff ðxÞ; yg ¼ x þy 0
ð1:8Þ
ð1:9Þ
In this paper, it is shown that the third iterate of the L2 -transform is a constant multiple of the E2;1 -transform as defined in (1.1). Using the identities, Parseval–Goldstein type theorems relating these integral transforms are proved. As an application of the identities and theorems, some illustrative examples are given. 2. The main theorem In the following lemma, we show that the third iterate of the L2 -transform is essentially the E2;1 -transform. Lemma 1. The identities 1 1 L32 ff ðxÞ; yg ¼ PfL2 ff ðxÞ; ug; yg ¼ L2 fPff ðxÞ; ug; yg; 2 2
ð2:1Þ
1 L32 ff ðxÞ; yg ¼ E2;1 ff ðxÞ; yg; 4
ð2:2Þ
and
hold true, provided that the integrals involved converge absolutely. Proof. The proof of identity (2.1) easily follows from (1.5). In order to prove (2.2), we use (2.1) 1 L32 ff ðxÞ; yg ¼ L2 fPff ðxÞ; ug; yg: 2 Using the definitions of the L2 -transform (1.3) and the Widder potential transform (1.4) we have Z 1 Z 1 1 u2 y 2 xf ðxÞ 3 L2 ff ðxÞ; yg ¼ ue dx du: 2 0 x 2 þ u2 0
ð2:3Þ
ð2:4Þ
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1559
Changing the order of integration, which is permissible by absolute convergence of the integrals involved, and using definition (1.4) once again, it follows from (2.4) that Z 1 Z Z 1 1 u expðu2 y 2 Þ 1 1 3 L2 ff ðxÞ; yg ¼ xf ðxÞ du dx ¼ xf ðxÞPfexpðu2 y 2 Þ; xg dx: ð2:5Þ 2 0 x 2 þ u2 2 0 0 Utilizing (1.8), the last equation (2.5) can be written as Z 1 1 3 L2 ff ðxÞ; yg ¼ xf ðxÞSfexpðuy 2 Þ; x2 g dx: 4 0
ð2:6Þ
Making use of the formula [2, Entry (11), p. 217] and (1.2) 2
Sfeuy ; x2 g ¼ expðx2 y 2 ÞEiðx2 y 2 Þ ¼ expðx2 y 2 ÞE1 ðx2 y 2 Þ;
ð2:7Þ
where RðyÞ > 0. Now assertion (2.2) follows upon inserting (2.7) into (2.6) and then using definition (1.1) of the E2;1 -transform. h Some illustrations of the identities in Lemma 1 are contained the following examples. Example 2. We show that mp m 1 p m1 E2;1 fx ; yg ¼ sec C þ y m1 ; 2 2 2 2
ð2:8Þ
where 1 < RðmÞ < 1. Proof. If we set f ðxÞ ¼ xm1
ð2:9Þ
in Lemma 1, and then substitute (2.9) and the formula mp p Pfxm1 ; yg ¼ sec y m1 ; 2 2 where 1 < RðmÞ < 1, (cf. [9, (A1), p. 248]), into identity (2.1) we obtain mp m 1 p 3 m1 L2 fx ; yg ¼ sec C þ y m1 : 8 2 2 2
ð2:10Þ
ð2:11Þ
Now assertion (2.8) follows when we use (2.11) in (2.2). h The following example contains results involving the error functions. The error function and the complementary error function are defined as Z x 2 ErfðxÞ ¼ pffiffiffi expðt2 Þ dt; ð2:12Þ p 0 and 2 ErfcðxÞ ¼ 1 ErfðxÞ ¼ pffiffiffi p
Z
1
expðt2 Þ dt;
ð2:13Þ
x
respectively. Example 3. We show that 2 p p1=2 a a a exp Erfc E2;1 fsinðaxÞ; yg ¼ 2 1 2y 2y 2y 4y 2
ð2:14Þ
and E2;1
cosðaxÞ ;y x
2 p3=2 a a ¼ exp Erfc 2 2y 2y 4y
ð2:15Þ
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Proof. We set f ðxÞ ¼ sinðaxÞ
ð2:16Þ
in Lemma 1. Using the formula p PfsinðaxÞ; ug ¼ expðauÞ; ð2:17Þ 2 (cf. [9, (A3), p. 248]), then applying the L2 -transform to both sides of (2.17), we obtain p L2 fPfsinðaxÞ; ug; yg ¼ L2 fexpðauÞ; yg: ð2:18Þ 2 Using identity (1.6) on the right-hand side of (2.18), we get p L2 fPfsinðaxÞ; ug; yg ¼ Lfexpðau1=2 Þ; y 2 g: ð2:19Þ 4 Substituting (2.16) and (2.18) to identity (2.1) of Lemma 1 then using the formula [1, Entry (31), p. 146] we deduce 2 p p1=2 a a a 3 exp : ð2:20Þ L2 fsinðaxÞ; yg ¼ 2 1 Erfc 8y 2y 3 2y 4y 2 Now assertion (2.14) follows when we use relationship (2.2) of Lemma 1 in (2.20). The proof of assertion (2.15) is similar. We set f ðxÞ ¼
cosðaxÞ x
ð2:21Þ
in Lemma 1 and make use of the formulas [2, Entry (55), p. 221] and [1, Entry (33), p. 147]. h In the next example we evaluate the E2;1 -transform of the Bessel function of the first kind Jm(x) defined as m 1 x mþ2m X ð1Þ : ð2:22Þ J m ðxÞ ¼ m!Cðm þ m þ 1Þ 2 m¼0 Example 4. We show that E2;1 fxm J m ðaxÞ; yg ¼
2 am Cðm þ 1Þ a a2 exp C m; ; 4y 2 4y 2 2mþ1 y 2mþ2
ð2:23Þ
where RðmÞ > 1. Proof. We set f ðxÞ ¼ xm J m ðaxÞ
ð2:24Þ
in Lemma 1. Using the formula Pfxm J m ðaxÞ; ug ¼ um K m ðauÞ;
ð2:25Þ
(cf. [9, (A8), p. 249]), then applying the L2 -transform to both sides of (2.25), we obtain L2 fPfxm J m ðaxÞ; ug; yg ¼ L2 fum K m ðauÞ; yg;
ð2:26Þ
where Km(x) is the Macdonald function. It is defined as K m ðxÞ ¼
p I m ðxÞ I m ðxÞ ; 2 sinðmpÞ
in which Im(x) is the modified Bessel function defined as 1 x mþ2m X ð1Þm : I m ðxÞ ¼ m!Cðm þ m þ 1Þ 2 m¼0
ð2:27Þ
ð2:28Þ
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1561
Using identity (1.6) in the right-hand side of (2.26), we get 1 L2 fPfxm J m ðaxÞ; ug; yg ¼ Lfum=2 K m ðax1=2 Þ; y 2 g; ð2:29Þ 2 Substituting (2.24) and (2.26) to identity (2.1) of Lemma 1 then using the formula [1, Entry (36), p. 199], we deduce 2 am Cðm þ 1Þ a a2 L32 fxm J m ðaxÞ; yg ¼ mþ3 exp C m; ; ð2:30Þ 4y 2 4y 2 2 y 2mþ2 where RðmÞ > 1. Now assertion (2.23) follows when we use relationship (2.2) of Lemma 1 in (2.30).
h
We now state our main result. Theorem 5. The Parseval–Goldstein type relations Z Z 1 1 1 yL2 ff ðxÞ; ygPfgðuÞ; yg dy ¼ xf ðxÞE2;1 fgðuÞ; xg dx 2 0 0 and
Z 0
1
1 yL2 ff ðxÞ; ygPfgðuÞ; yg dy ¼ 2
Z
ð2:31Þ
1
ugðuÞE2;1 ff ðxÞ; ug du
ð2:32Þ
0
hold true, provided that the integrals involved converge absolutely. Proof. We only give the proof of (2.31), as the proof of (2.32) is similar. From definition (1.3) of the L2 transform, we obtain Z 1 Z 1 Z 1 yL2 ff ðxÞ; ygPfgðuÞ; yg dy ¼ yPfgðuÞ; yg x expðx2 y 2 Þf ðxÞ dx dy: ð2:33Þ 0
0
0
Changing the order of integration, which is permissible by absolute convergence of the integrals involved, we find from (2.33) that Z 1 Z 1 Z 1 yL2 ff ðxÞ; ygPfgðuÞ; yg dy ¼ xf ðxÞ y expðx2 y 2 ÞPfgðuÞ; yg dy dx: ð2:34Þ 0
0
Now assertion (2.31) follows from (2.1) and (2.2).
0
h
3. Useful corollaries Interesting consequences of the main theorem will be given in this section. Useful Parseval-type identities for the L2 -transform and the Widder potential transform are contained in Corollary 6. If the integrals involved converge absolutely, then we have Z 1 Z 1 1 mþ1 f ðxÞ y m L2 ff ðxÞ; yg dy ¼ C dx; 2 2 xm 0 0 where RðmÞ > 1, Z 1 mp Z 1 gðuÞ PfgðuÞ; y g p sec dy ¼ du; ym 2 2 um 0 0 where 1 < RðmÞ < 1, and Z 1 Z 1 Pff ðxÞ; yg 1 m dy ¼ C y m L2 ff ðxÞ; yg dy; ym 2 2 0 0 where RðmÞ < 1.
ð3:1Þ
ð3:2Þ
ð3:3Þ
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Proof. We set gðuÞ ¼ um1
ð3:4Þ
in relation (2.31) of Theorem 5, then use (3.4), (2.8), and (2.10) in the Parseval–Goldstein identity (2.31). Similarly, we put f ðxÞ ¼ xm1
ð3:5Þ
in relation (2.32) of Theorem 5, then use (3.2), (2.8), (1.6), and the identity pm 1 m 1 m C þ C ¼ p sec 2 2 2 2 2 for the gamma function in the Parseval–Goldstein identity (2.32).
ð3:6Þ
h
Remark 7. The Mellin transform is defined as Z 1 Mff ðxÞ; yg ¼ xy1 f ðxÞ dx:
ð3:7Þ
0
If we use the definition above, then the relationships in Corollary 6 can be expressed as 1 mþ1 MfL2 ff ðxÞ; yg; m þ 1g ¼ C Mff ðxÞ; m þ 1g; 2 2 where RðmÞ > 1, MfPfgðuÞ; yg; m þ 1g ¼
mp p sec MfgðuÞ; m þ 1g; 2 2
ð3:8Þ
ð3:9Þ
where 1 < RðmÞ < 1, and
1 m MfPff ðxÞ; yg; m þ 1g ¼ C MfL2 ff ðxÞ; yg; m þ 1g; 2 2
ð3:10Þ
where RðmÞ > 1. More useful Parseval-type identities for the L2 -transform, the Widder potential transform and the E2;1 -transform are contained in Corollary 8. If the integrals involved converge absolutely, then we have Z 1 Z 1 m 1 PfgðuÞ; yg xm E2;1 fgðuÞ; xg dx ¼ C þ dy; 2 2 0 ym 0 where RðmÞ > 1, and Z 1 mp Z 1 um E2;1 ff ðxÞ; ug du ¼ p sec y m L2 ff ðxÞ; yg dy; 2 0 0 where 1 < RðmÞ < 1, Z 1 mp m 1 Z 1 f ðxÞ p m u E2;1 ff ðxÞ; ug du ¼ sec dx; C þ 2 2 2 2 0 xm 0
ð3:11Þ
ð3:12Þ
ð3:13Þ
where 1 < RðmÞ < 1. Proof. Setting f ðxÞ ¼ xm1
ð3:14Þ
in relation (2.31) and gðuÞ ¼ um1
ð3:15Þ
D. Brown et al. / Applied Mathematics and Computation 187 (2007) 1557–1566
1563
in relation (2.32), then using relationship (1.6), we obtain assertions (3.11) and (3.12). Assertion (3.13) is obtained when we combine (3.1) and (3.12). h Remark 9. Using the definition of the Mellin transform given in (3.7), the relationships in Corollary 8 can be expressed as mþ1 MfE2;1 fgðuÞ; xg; m þ 1g ¼ C MfPfgðuÞ; yg; m þ 1g; ð3:16Þ 2 where RðmÞ > 1, MfE2;1 ff ðxÞ; ug; m þ 1g ¼ p sec
mp 2
MfL2 ff ðxÞ; yg; m þ 1g;
ð3:17Þ
where 1 < RðmÞ < 1, and mp 1 m MfE2;1 ff ðxÞ; ug; m þ 1g ¼ p sec C þ Mff ðxÞ; m þ 1g; 2 2 2
ð3:18Þ
where 1 < RðmÞ < 1. In the following corollary, a Parseval–Goldstein type identity is given. The identity relates the L2 -transform, E2;1 -transform, the Bessel function of the first kind Jm(x) and the modified Bessel function of the second kind Km(x) as defined in (2.22) and (2.27), respectively. Corollary 10. We have Z Z 1 1 1 mþ1 y mþ1 K m ðzyÞL2 ff ðxÞ; yg dy ¼ u J m ðzuÞE2;1 ff ðxÞ; ug du 2 0 0
ð3:19Þ
provided that the integrals involved converge absolutely, RðmÞ > 1=2, and Re(z) > 0. Proof. We set gðuÞ ¼ um J m ðzuÞ
ð3:20Þ
in Theorem 5, then use (3.20) and (2.25) in the Parseval–Goldstein type relationship (2.32), we obtain assertion (3.19). h Remark 11. If we recall the definitions of the Hankel-transform Z 1 pffiffiffiffiffi Hm ff ðxÞ; yg ¼ xy J m ðxyÞf ðxÞ dx;
ð3:21Þ
0
where Jm(x) is the Bessel function of the first kind as defined in (2.22) and the K-transform Z 1 pffiffiffiffiffi Km ff ðxÞ; yg ¼ xy K m ðxyÞf ðxÞ dx;
ð3:22Þ
0
where Km(x) is the modified Bessel function of the second kind as defined in (2.27), we obtain the following identity involving the Hankel transform, the K-transform, the L2 -transform, and the E2;1 -transform n 1 o 1 n 1 o Km y mþ2 L2 ff ðxÞ; yg; z ¼ Hm umþ2 E2;1 ff ðxÞ; ug; z : ð3:23Þ 2 Remark 12. If we recall the definition of the Fourier sine transform Fs ff ðxÞ; yg ¼
Z 0
1
sinðxyÞf ðxÞ dx;
ð3:24Þ
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and the special cases of the Bessel function of the first kind (2.22) and the K-transform rffiffiffiffiffi rffiffiffiffiffi 2 2 expðxÞ and K 1=2 ðxÞ ¼ sinðxÞ; J 1=2 ðxÞ ¼ px px respectively, then identity (3.18) takes the form n 1 o 1 n 1 o L y mþ2 L2 ff ðxÞ; yg; z ¼ Fs umþ2 E2;1 ff ðxÞ; ug; z : 2
ð3:25Þ
ð3:26Þ
4. Illustrative examples An illustration of the Parseval–Goldstein type relation (3.1) of Corollary 6 is given in the following example. Example 13. We have Z 1 pl al l C y l1 expða2 y 2 ÞErfcðayÞ dy ¼ sec ; 2 2 2 0
ð4:1Þ
where 0 < RðlÞ < 1. Proof. We set 1 f ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ a2
ð4:2Þ
in relation (3.2) of Corollary 6, then using relation (1.6) and the formula [1, Entry 18, p. 135], we have o pffiffiffi p 1 n 2 1=2 2 expða2 y 2 ÞErfcðayÞ: ;y ¼ ð4:3Þ L2 ff ðxÞ; yg ¼ L ðx þ a Þ 2y 2 Substituting (4.2) and (4.3) into (3.2) of Corollary 6, we find Z 1 Z 1 1 l 1 xl pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx: þ y l1 expða2 y 2 ÞErfcðayÞ dy ¼ pffiffiffi C 2 2 0 p x 2 þ a2 0 The integral on the right-hand side has the value (cf. [3, Eq. (2), p. 171], and [5, Eq. (29), p. 163]) Z 1 xl al 1l l pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ ; ; B 2 2 2 x 2 þ a2 0
ð4:4Þ
ð4:5Þ
where B(x, y) is the beta function and it is defined as Bðx; yÞ ¼
CðxÞCðyÞ Cðx þ yÞ
If we use definition (4.6) of the beta function, then (4.4) simplifies to Z 1 al 1 l l 1 l þ C C y l1 expða2 y 2 ÞErfcðayÞ dy ¼ C : 2 2 2 2 2 2p 0
ð4:6Þ
ð4:7Þ
Using formula (3.6) for the C-function on the right-hand side of (4.7), we obtain assertion (4.1). h Remark 14. The formula 2 Z 1 l pl x x xl1 exp sec Erfc pffiffiffi dx ¼ 2ðl=2Þ1 C 2 2 2 2 0 pffiffiffi from [1, Entry (9), p. 325] is obtained when we set a ¼ 1= 2 in result (4.1) of Example 13.
ð4:8Þ
D. Brown et al. / Applied Mathematics and Computation 187 (2007) 1557–1566
1565
Example 15. We have Z 1 pm p mþ2k1 1 2k m 1 2km 2 2 2 2 y expða y ÞCðk; a y Þ dy ¼ a ½Cð1 kÞ C sec ; 2 2 2 0
ð4:9Þ
where 1 < RðmÞ < 1, RðkÞ < 1, and Rð2k þ mÞ < 1. Proof. We set gðuÞ ¼ u2k expða2 u2 Þ
ð4:10Þ
in relation (3.2) of Corollary 6, then using relation (1.4) and the formula [2, Entry (17), p. 217], we have Cð1 kÞ 2k 1 y expða2 y 2 ÞCðk; a2 y 2 Þ: PfgðuÞ; yg ¼ S uk expða2 uÞ; y 2 ¼ 2 2
ð4:11Þ
Substituting (4.10) and (4.11) into (3.2) of Corollary 6, we find Z 1 mp Z 1 p 2km 2 2 2 2 sec y expða y ÞCðk; a y Þ dy ¼ u2km expða2 u2 Þ du: Cð1 kÞ 2 0 0
ð4:12Þ
Using definition (1.3) of the L2 -transform and (1.6), the integral on the right-hand side has the value 1 1 1 2k m 2kþm1 2km1 ð2km1Þ=2 2 L2 fu ; yg ¼ Lfu ;y g ¼ C : ð4:13Þ y 2 2 2 Assertion (4.9) follows upon substituting (4.13) into (4.12). h Remark 16. Setting k = 1/2, m = l in (4.9) and using pffiffiffi 1 C ; a2 y 2 ¼ p ErfcðayÞ 2
ð4:14Þ
we obtain result (4.1) of Example 13. Example 17. We have Z 1 am 1 m m1 2 2 p ffiffiffi y expða y ÞErfcðayÞ dy ¼ m C CðmÞ; 2 2 2 p 0
ð4:15Þ
where 0 < RðmÞ < 1. Proof. We set f ðxÞ ¼
cosðaxÞ x
ð4:16Þ
in relation (3.13) of Corollary 8, and upon using identity (2.15) and the definition of the Mellin transform (3.7), we have
2 Z 1 lp l 1 cosðaxÞ p3=2 a a p sec þ ; l : ð4:17Þ exp yl dy ¼ C M Erfc 2y 2 2 2 2 x 2y 4y 2 0 Simplifying and using the formula [1, Entry (21), p. 319] yields 2 Z 1 a a al l 1 p ffiffiffi þ C y l1 exp dy ¼ CðlÞ; Erfc 2y 2 2 4y 2 p 0 where 1 < RðlÞ < 0. Making the substitutions m = l and x = 1/2y, we obtain assertion (4.15).
ð4:18Þ h
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Remark 18. Setting l = 2z and combining the results of Examples 13 and 17, we have a derivation of the wellknown identity pffiffiffi 22z1 CðzÞCðz þ 1=2Þ ¼ pCð2zÞ ð4:19Þ (cf. [4, p. 3]). We conclude that many other infinite integrals can be evaluated in this manner by applying the lemma, the theorem and its corollaries considered here. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
A. Erdelyi et al., Tables of Integral Transforms, vol. 1, McGraw-Hill Book Company, 1954. A. Erdelyi et al., Tables of Integral Transforms, vol. 2, McGraw-Hill Book Company, 1954. M.L. Glasser, Some Bessel function integrals, Kyungpook Math. J. 13 (1973) 171–174. N. Lebedev, Special Functions and their Applications, Prentice-Hall, 1965. H.M. Srivastava, O. Yu¨rekli, A theorem on Stieltjes-type integral transform and its applications, Complex Var. Theory Appl. 28 (1995) 159–168. D.V. Widder, A transform related to the Poisson integral for a half-plane, Duke Math. J. 33 (1966) 355–362. S. Wilson, O. Yu¨rekli, A new method of solving Bessel’s differential equation using the L2 -transform, Appl. Math. Comput. 130 (2002) 587–591. S. Wilson, O. Yu¨rekli, A new method of solving Hermite’s differential equation using the L2 -transform, Appl. Math. Comput. 145 (2003) 495–500. O. Yu¨rekli, A Parseval-type theorem applied to certain integral transforms, IMA J. Appl. Math. 42 (1989) 241–249. O. Yu¨rekli, New identities Involving the Laplace and the L2 -transforms, Appl. Math. Comput. 99 (1999) 141–151. O. Yu¨rekli, Theorems on L2 -transforms and its applications, Complex Var. Theory Appl. 38 (1999) 95–107. O. Yu¨rekli, I. Sadek, A Parseval–Goldstein type theorem on the Widder potential transform and its applications, Int. J. Math. Math. Sci. 14 (1991) 517–524.