x exp ( x 2 y 2) f (x) dx. (1.2) The following Laplace-type transforms which are the L 2n transform and the L 4n transform,

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1 Journal of Inequalities and Special Functions ISSN: , URL: Volume 8 Issue 517, Pages IDENTITIES ON THE E,1 INTEGRAL TRANSFORM NEŞE DERNEK*, GÜLSHAN MANSIMLI, EZGI ERDOĞAN Abstract. In the present paper, the authors prove a Parseval-Goldstein type theorem involving the E,1 generalized exponential integral transform. Then, the theorem is shown to yield a number of new identities involving several wellknown integral transforms. Using the theorem and its corollaries, a number of interesting infinite integrals are presented with illustrative examples. 1. Introduction, Definitions and Preliminaries The classical Laplace transform, L is defined as, L f x ; y L -integral trasform is introduced in [8] as, L f x ; y exp xy f x dx. 1.1 x exp x y f x dx. 1. The following Laplace-type transforms which are the L n transform and the L transform, L n f x ; y L f x ; y x n 1 exp x n y n f x dx, 1.3 x 1 exp x y f x dx, 1.4 are introduced by Dernek et al. [],[3]. The L transform, the L n transform and the Laplace transform are related by the formulas, L f x ; y 1 n L n f x 1/ ; y, 1.5 L f x ; y 1 L f x 1/ ; y Mathematics Subject Classification. Primary 44A1, 44A15; Secondary 33B15, 33B. Key words and phrases. ε,1 -transform, L -transform, P -transform, Laplace transform, Parseval-Goldstein type theorems. c 17 Ilirias Publications, Prishtinë, Kosovë. Submitted March 14, 17. Published October 5, 17. Communicated by Yilmaz Simsek. *Corresponding Author. 75

2 76 N. DERNEK, G. MANSIMLI, E.ERDOĞAN The P generalized Widder potential transform, the E,1 generalized exponential transform, the F s,n - generalized Fourier sine-transform and F c,n generalized Fourier cosine-transform are introduced in [3] as follows, P f x ; y E,1 f x ; y x 1 f x x dx, n N, y x 1 exp x y E 1 x y f x dx, 1.8 where E 1 x is the exponential integral function defined by and E 1 x E i x F s,n f x ; y F c,n f x ; y x exp u du u 1 exp xt dt, 1.9 t x n 1 sin x n y n f x dx, 1.1 x n 1 cos x n y n f x dx The P transform is related to the Widder potential transform and the Stieltjes transform [5] with the identity, P f x ; y 1 n P f x 1/n ; y n 1 S f x 1/ ; y, 1.1 where the Widder potential transform and the Stieltjes transform are defined by P f x ; y xf x x dx, y f x S f x ; y dx x + y The E,1 transform is related to the E,1 transform, which is defined in Brown et al. [1] by the following relation, f x 1/n ; y n E,1 f x ; y 1 n E,1 In this paper, we first show that the third iterate of the L transform 1.4 is a constant multiple of the E,1 transform 1.8. By using this iteration identity, we establish a Parseval-Goldstein type theorem. The identities proven in this article give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given as illustrations of the results presented here The Main Theorem. Lemma 1.1. The following identities, L 3 f x ; y 1 P L f x ; u ; y, 1.16 L 3 f x ; y 1 L P f x ; u ; y, 1.17 and L 3 f x ; y 1 16n E,1 f x ; y 1.18 hold true, provided that each member of the assertions exist.

3 IDENTITIES ON THE E,1 INTEGRAL TRANSFORM 77 Proof. The proof of the identities 1.16 and 1.17 easily follows from the relation, L f x ; y P f x ; y, 1.19 which is proven in [3] p.538, E. 33. Using the definition of the L transform 1.4 and the relation 1.19, we have L 3 f x ; y 1 u 1 exp u y [ x 1 ] f x x dx du u Changing the order of integration, which is permissible by absolute convergence of the integrals involving and using the definition of the P transform 1.7, we get L 3 f x ; y 1 [ x 1 u 1 exp u y ] f x u + x du dx 1 x 1 1 f x L u + x ; y dx. 1.1 Utilizing 1.6, the equation 1.1 can be written as L 3 f x ; y 1 x 1 1 f x L ; y dx. 1. u + x Making the change of variables u + x t and then ty ν on the right-hand side of 1., we arrive at the relation Example 1.. We show for Re ν >, Re y >, E,1 x ν 1 ; y E,1 x nν 1 ; y x ν 1 exp x y E 1 x y dx π cos ec [π ν 1] Γ ν y ν+1, 1.3 x nν+1 1 exp x y E 1 x y dx ν + 1 Γ π sec πν y nν Demonstration. We start with the proof of assertion 1.3 by setting f x x ν 1 into identity 1.18 of Lemma 1.1. E,1 x ν 1 ; y x ν 1 exp x y E 1 x y dx L 3 x ν 1 ; y L P x ν 1 ; u ; y, 1.5 where P x ν 1 x 1 x ν 1 ; u x + u dx. 1.6

4 78 N. DERNEK, G. MANSIMLI, E.ERDOĞAN Setting x t and then t w, we obtain u P x ν 1 ; u 1 t + u u ν 1 u ν 1 t ν 1 dt w ν w dw π π cos ecπν. 1.7 On the other hand, using the relation 1.6 and [5], p.137, E 1, we get π cos ec πν L u ν 1 ; y π cos ec [π ν 1] L u ν 1 ; y π cos ec [π ν 1] ν y ν, 1.8 where Re ν >. Substituting 1.8 into the relation 1.5, we arrive at 1.3. Assertion 1.4 immediately follows from substituting f x x nν 1 into the identity 1.18 of Lemma 1.1., then making the change of variables x t and t wu in obtained relation and using 1.6. Example 1.3. We show E,1 sin ax n ; y π Demonstration. We set x 1 expxy E 1 xy sin ax n dx [ ] πa a y a exp y6n 4y Erfc y n. 1.9 f x cos ax n x n 1.3 in the relation 1.18 of Lemma 1.1. We have P sin ax n ; u m 1 m a m+1 m + 1! 1 m a m+1 m + 1! P m 1 m π m x 1 xnm+1 dx x + u x nm+1 ; u 1 m a m+1 m + 1! S x m+ 1 ; u 1 m a m+1 m + 1! unm+1 π exp au n Using the relation 1.6 and the formula [5], p.146, E 31, we get π L e aun ; y π L exp au 1/ ; y [ π πa a y ] y a exp Erfc y6n 4 y n 1.3

5 IDENTITIES ON THE E,1 INTEGRAL TRANSFORM 79 where Re y >. Substituting 1.3, 1.31 and 1.3 into 1.18, we arrive at 1.9. Example 1.4. We show for Re y n >, cos ax n E n,1 ; y x 1 exp x y E 1 x y sin ax n dx x n π3/ a exp yn 4y n Demonstration. We set f x cos ax n x n 1.1. Using the formula [6], p.1, E 55, we have for a >, cos ax n P x n ; u 1 m a m x m! m x + u 1 1 m a m m! S x m 1 ; u m π 1 m a m m! m π au n m u n m! m a erfc y n in the relation 1.18 of Lemma 1 xnm 1 dx u [ m 1 cos ec π m + 1 ] π u n exp au n Using 1.6 and the known formula [5], p.147, E 33, we get π L u n exp au n ; y π L u 1/ exp au 1/ ; y π u 1/ e uy +au 1/ du Changing the variable of integration u 1/ t on the right-hand side of 1.35, we have from Lemma 1.1, cos ax n E,1 x n ; y π exp t y at dt π a n exp 4y exp [ y t + a ] y dt π3/ a a exp yn 4y Erfc y n, 1.36 where we set y n t + a y w on the right-hand side of the integral in Theorem 1.5. The following Parseval-Goldstein type identities; y 1 L f x ; y P g u ; y dy 1 x 1 f x E,1 g u ; x dx, 1.37

6 8 N. DERNEK, G. MANSIMLI, E.ERDOĞAN y 1 L f x ; y P g u ; y dy 1 u 1 g u E,1 f x ; u du 1.38 and x 1 f x E,1 g u ; x dx u 1 g u E,1 f x ; u du, 1.39 hold true, provided that the integrals involved converge absolutely. Proof. We only give the proof of 1.37, as the proof of 1.38 is similar. The identity 1.39 follows easily from the assertions 1.37 and By the definition 1.4 of the L transform, we have y 1 L f x ; y P g u ; y [ y 1 x 1 exp x y ] f x dx P g u ; y dy. 1.4 Changing the order of integration, which is permissible by absolute convergence of the integrals involved and using the relation 1.18, we find from 1.4 that y 1 L f x ; y P g u ; y dy 1 x 1 f x L P g u ; y ; x dx x 1 f x E,1 g u ; x dx Corollary 1.6. If the integrals involved converge absolutely, then we have for Re ν > 1, y nν+1 1 L fx; y dy 1 ν + 1 Γ fxdx x nν 1 + 1, 1.4 y n1 ν 1 P gu; y dy π sec πν u n1 ν 1 gudu Proof. We set g u u nν 1 in relation 1.37 of Theorem 1.5. Using the relations 1.7 and 1.8, we obtain the Parseval-Goldstein identity 1.4. Similarly, we put f x x nν 1 in relation 1.38 of Theorem 1.5 and use 1.7, then we have y 1 L x nν 1 ; y P g u ; y dy 1 where L x nν 1 ; y x 1 x nν 1 E,1 g u ; x dx, L x ν 1 ; y 1 Γ ν + 1 Setting the relations 1.4 and 1.18 in 1.38, we obtain y nν

7 IDENTITIES ON THE E,1 INTEGRAL TRANSFORM 81 Corollary 1.7. The following identities hold true for 1 > Re ν > 1; and y n1 ν 1 1 P g u ; y dy Γ ν+1 x n1+ν 1 E,1 g u ; x dx, u nν+1 1 E,1 f x ; u du π ν + 1 πν Γ sec 1 x provided that the integrals involved converge absolutely. Proof. Setting g u u nν 1 in 1.38, we get y 1 L f x ; y P u nν 1 ; y dy 1 f x 1.46 nν 1 1 dx, 1.47 u nν+1 1 E n,1 f x ; u du Using the relations 1.1 and the formula [6] p.16, E5, we have for 1 < Re ν < 1, P u nν 1 ; y 1 S u ν 1, y πν πy nν 1 sec Substituting 1.49 into 1.48, we arrive at Similarly, setting g u u nν 1 in relation 1.38, using 1.1 and the formula [6], p.16, E 5, we obtain The assertion 1.47 follows from the relations 1.46 and 1.4. Example 1.8. We show for 1/ < Re ν < 5/, E,1 x nν J ν ax n ; y x nν+ 1 e xy E 1 xy J ν ax n dx a ν 4y Γ ν, Γ ν + 1 ν+ n y ν+1 e a a 4y, 1.5 where J ν x is the Bessel function of first kind defined as 1 m x ν+m J ν x m!γ ν + m + 1 m Demostration. We set f x x nν J ν ax n in 1.18 of Lemma 1.1. Using the relation 1.1 and the formula [6], p.5, E 1, we have P x nν J ν ax n ; u 1 S x ν/ J ν ax 1/ ; u 1 n unν K ν au n, 1.5 where K ν x is Macdonald function. Then, using the relation 1.6 and the formula [5], p.18, E 36, we get E,1 x nν J ν ax n ; y nl u nν K ν au n ; y 1 n L u ν/ K ν au 1/ ; y Γ ν + 1 ν+ n a ν y ν+1 e a 4y Γ a ν, 4y, 1.53

8 8 N. DERNEK, G. MANSIMLI, E.ERDOĞAN where Γ a, b is incomplete gamma function [7]. Now, assertion 1.5 follows when we use relationship Corollary 1.9. The following identity, y nν+1 1 L f x ; y K ν z n y n dy u nν+1 1 J ν z n u n E,1 f x ; u du, 1.54 holds true, where Re z n >, 1 < Re ν < 3/, and the integrals involved converge absolutely. Proof. Setting g u u nν J ν z n u n in 1.38 of Theorem 1.5, then using 1.5, we obtain assertion We recall the definitions of the K ν,n transform, the H ν,n transform [5], K ν,n f x ; z H ν,n f x ; z x n 1 x n z n 1/ Kν x n z n f x dx, 1.55 x n 1 x n z n 1/ Jν x n z n f x dx, 1.56 respectively, where J ν is the Bessel function. K ν is the modified Bessel function of second kind. It is defined as K ν x π in which I ν x is the modified Bessel function defined as I ν x j I ν x I ν x, 1.57 sin πν 1 j 1 x ν+j j!γ ν + j + 1 Remark. We have the following identity involving the H ν,n transform, the K ν,n transform, L transform and the E,1 transform, K ν,n y nν+ 1 L f x ; y ; z H ν,n u nν+ 1 E,1 f x ; u ; z.. Examples Example.1. We show for < Re ν < 1, Arga < π, Re y >, y nν 1 exp a y Erfc a n y n dy a nν ν πν Γ sec..1 Demonstration. We set f x x + a 1/ in 1.4 of Corollary 1.6, we obtain for Arga < π, Re y >, x y nν+1 1 L + a 1/ ; y dy 1 ν + 1 x Γ + a 1/ dx. x nν 1+1.

9 IDENTITIES ON THE E,1 INTEGRAL TRANSFORM 83 Using the relation 1.6, we have x L + a 1/ ; y 1 x L + a 1/ ; y 1 x + a 1/ exp xy dx..3 Making change of variables x + a t, y t w and w z, respectively on the right-hand side of., we get x L + a 1/ ; y exp a y t 1/ exp y t dt a 1 exp a y y n w 1/ e w dw a y π y n exp a y Erfc a n y n..4 The integral on the right of. is evaluated by making the change of variable x t and using the relation [4], p.5, E.33, x nν 1 1 x + a 1/ dx 1 t nν 1 1 [t n + a n] 1/ dt 1 G n t nν+n 1 ; a 1 n ν 1 a nν B 1 ν; ν/,.5 where B a, b is the Beta function [7]. Now, making use of the well known formulas, B 1 ν; ν/ Γ 1 ν Γ ν Γ 1 ν,.6 ν ν + 1 Γ Γ ν 1 πγ ν,.7 π Γ ν Γ 1 ν sin πν,.8 ν Γ Γ 1 ν πν π sin,.9 the assertion.1 follows upon inserting.3 and into.. Remark. The identity.1 is obtained earlier in [1], p.1564, E.48, for n 1 as follows: y ν 1 exp a y ν πν Erfc ay dy a ν Γ sec..1 Example.. We show for Re λ < 1, Re ν < 1, y n1 ν 1 λ exp a y Γ λ, a y dy π πν Γ 1 λ sec Γ λ ν 1 a λ+ ν 1..11

10 84 N. DERNEK, G. MANSIMLI, E.ERDOĞAN Demonstration. We set g u u nλ exp a n u n in relation 1.43 of Corollary 1.6. Then, using the relation 1.1 and the formula [6], p.17, E 17, we have π y n1 ν 1 P u λ exp a u ; y dy u n1 ν 1 u λ exp a u sec P u λ exp a u ; y 1 S u λ exp a u ; y πν du,.1 Γ 1 λ y λ eau Γ λ, a y,.13 where Re a >, Re ν < 1, Γ a, b is incomplete gamma function [7]. Substituting the formula.13 into the identity.1 and using the relation 1.6, we obtain We know y n1 ν 1 Γ 1 ν y λ exp a u Γ λ, a y dy πν π sec u 1 u n1+ν+λ exp a u du πν π sec L u n1+ν+λ ; a π sec πν L u 1 1+ν+λ ; a..14 L u 1 1+ν+λ ; a a λ+ ν 1 Γ λ ν 1,.15 where Re λ + ν < 1, Re a >. Substituting.15 into.14, we arrive at the relation.11. Remark. If we set n 1, λ 1, ν µ in.11 and use 4 1 Γ, a y πerfc ay,.16 then we obtain the known relation [1], p.1564, E 4.1, y µ 1 exp a y 1 Erfc ay dy Γ µ/ q µ sec πµ..17 We conclude by remarking that many other infinite integrals could be evaluated by using the identities in this work. References [1] D. Brown, N. Dernek, O. Yürekli, Identities for the E,1 transform and their applications, Appl. Math. Comput. 187, 7, [] N. Dernek, F. Aylıkçı, Identities for the L n transform, the L n transform and the P n transform and their applications, J. Inequal. Spec. Funct. 5:4, 14, [3] N. Dernek, E. Ö. Ölçücü, F. Aylıkçı, New Identities and Parseval type relations for the generalized integral transforms L, P, F s,n and F c,n, Appl.Math.Comput. 69, 15,

11 IDENTITIES ON THE E,1 INTEGRAL TRANSFORM 85 [4] N. Dernek, F. Aylıkçı, G. Balaban, New identities for the generalized Glasser transform, the generalized Laplace transform and the E n,1 transform, International Eurasian Conference on Mathematical Sciences and Applications IECMSA, Book of Abstracts, 15, [5] A. Erdělyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms, Vol. I, Mac Graw- Hill Book Company Inc., New York, USA, [6] A. Erdělyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms, Vol. II, Mac Graw- Hill Book Company Inc., New York, USA, [7] W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical, Physics Springer- Verlag New York Inc., [8] O. Yürekli, Identities, inequalities,parseval type relations for integral transforms and fractional integrals., ProQuest LLC, Ann Arbor, MI, Thesis PhD. University of California, Santa Barbara. Neşe Dernek, Gülshan Mansimli, Ezgi Erdoğan Department of Mathematics, Faculty of Arts and Sciences, Marmara University, TR- 347, Kadıköy, Istanbul, TURKEY. address: ndernek@marmara.edu.tr address: gulsen.mensimli@yahoo.com address: ezgi.erdogan@marmara.edu.tr