AN EXAMPLE FOR THE GENERALIZATION OF THE INTEGRATION OF SPECIAL FUNCTIONS BY USING THE LAPLACE TRANSFORM

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1 Journal of Inequalitie Special Function ISSN: 7-433, URL: Volume 6 Iue 5, Page 5-3. AN EXAMPLE FOR THE GENERALIZATION OF THE INTEGRATION OF SPECIAL FUNCTIONS BY USING THE LAPLACE TRANSFORM J. L. G. SANTANDER Abtract. By uing the convolution theorem of the Laplace tranform Faltung theorem, we can generalize the calculation of integral involving pecial function. In thi paper, the following integral involving the modified Beel function i calculated, τ α+n I α a τ t τ β+m I β a t τ dτ. A a conitency tet of the reult obtained, etting the parameter n, m to particular value, ome integral reported in the literature are recovered.. Introduction Thi paper ue the convolution theorem of the Laplace tranform in order to generalize ome integral of pecial function given in the literature. Depite the fact we are going to conider an integral of a pecial type, involving only the modified Beel function of the firt kind, the method decribed here can be extended to many other type of integr. In thi way, in the literature we can find the following integral [, Eqn..5.9 ] τ α I α aτ t τ β+±/ I β a t τ dτ. Γ α + Γ β + ± πa Γ α + β + 3± I α+β+ at, tα+β+±/ Re α >, Re β > 3 ± 4, t >, where I ν z denote the modified Beel function of the firt kind [, Chap. 5] Γ z the gamma function [, Chap. 43]. By uing the beta function [3, Eqn. 5..] B a, b t a t b Γ a Γ b dt Γ a + b,. Mathematic Subject Claification. 33C, 44A. Key word phrae. Modified Beel function; Integral of pecial function; Convolution theorem of Laplace tranform. c 5 Iliria Publication, Prihtinë, Koovë. Submitted March 3, 5. Publihed May 7, 5. J. L. G. Santer wa partially upported by Generalitat Valenciana Spain under grant GV/5/7. 5

2 6 J. L. G. SANTANDER it i worth noting that we can rewrite. a τ α I α aτ t τ β+±/ I β a t τ dτ tα+β+±/ πa B α +, β + ± Re α >, Re β > 3 ± 4 I α+β+ at,, t R, where we can extend the parameter t to non-poitive value. paper i jut to generalize. to the following integral The cope of thi τ α+n I α aτ t τ β+m I β a t τ dτ, n, m,,,....3 by uing the convolution theorem of the Laplace tranform [4, Eqn. 7..5] L {L [f t] L [g t]} f τ g t τ dτ,.4 where the Laplace tranform the invere Laplace tranform are defined a [4, Sect. 7.] F L [f t] f t L [F ] πi e t f t dt,.5 γ+i γ i e t F d.6 being γ a contant that exceed the real part of all the ingularitie of F. Thi paper i organized a follow. Section perform the calculation of the integral.3 by mean of the convolution theorem.4. The reult i expreed a a finite um of term. A a conitency tet, Section 3 particularize the integral calculated in the previou Section for the cae n m n, m, o we recover the integral reported in the literature... Calculation of the integral Theorem.. For n, m non-negative integer, Re α > n+ m+, Re β > t R, τ α+n I α aτ t τ β+m I β a t τ dτ. b µ Γ α + ñ + u + Γ β + m + v + ñ + u! m + u! t µ+r uñ + v m + r b k c k ñ, u, m, v, α, β µ + r Γ µ + r + k F µ + r + k, + µ + r + k k b,

3 AN EXAMPLE FOR GENERALIZATION OF INTEGRATION OF SPECIAL FUNCTIONS 7 where being c k ñ, u, m, v, α, β n ñ,. {, n odd u n ñ, n even,.3 m m,.4 {, m odd v m m, m even,.5 µ α + β +,.6 r ñ + m + u + v,.7 b at,.8 b l ñ, u, α Proof. According to.4, we have minñ,k lmaxk m, b l ñ, u, α b k l m, v, β,.9 u ñ l + l! ñ + u l! Γ + α + l.. τ α+n I α aτ t τ β+m I β a t τ dτ L { L [ t α+n I α at ] L [ t β+m I β at ]},. thu, according to.5, let u calculate firt L [ t α+n I α at ]. a e t t α+n I α at dt α Γ α + n + α+n+ Γ + α F Re α > n +, α + n+, α + n + α + where we have applied the following reult [, Eqn..5.3 ] a, x λ e px I ν cx dx.3 p λ ν c ν Γ ν + λ λ+ν Γ ν + F, λ+ν+ c ν + p, Re λ + ν >, Re p > Re c, being F a, b; c; z the Gau hypergeometric function [, Chap. 6]. According to..3, let u et for convenience n ñ + u,

4 8 J. L. G. SANTANDER thu L [ t α+n I α at ].4 a α Γ α + ñ + u + α + ñ + u + α+ñ+u+ Γ + α F, α + ñ + a α +. In order to reduce the above hypergeometric function to an elementary function, let u apply [, Eqn. 6:4:] where F a, c + n c n,,,... x x a+n x k n k n c ak x k, k c k Γ x + k,.5 Γ x denote the Pochhammer ymbol, which ha the following property [, Eqn. 8:5:] x k k x k + k. Therefore, we have α+4ñ+u+ a α+ñ+u+/ a ñ + u + α + ñ + +u + F, α + ñ + α + ñ ñ k k k k + α k a k..6 Subtituting back.6 in. taking into account the definition of the Pochhammer ymbol.5 the following formula [, Eqn. 49:4:3] Γ + n n! π,.7 4 n n! we can rewrite. a Notice that L [ t α+n I α at ].8 a α u Γ α + ñ + u + ñ + u!ñ! a α+ñ+u+/ ñ + u! ñ ñ + u k! a k ñ k k! ñ k! ñ + u k! Γ + α + k, k Re α > n +. x! x + u!, u x +, u ux +,

5 AN EXAMPLE FOR GENERALIZATION OF INTEGRATION OF SPECIAL FUNCTIONS 9 o we can implify.8, arriving at L [ t α+n I α at ].9 a α u Γ α + ñ + u + ñ + u! R α a α+ñ+u+/ ñ,u, uñ + Re α > n +, where we have defined the following polynomial R α ñ,u ñ k [u ñ k + ] a k ñ k k! ñ + u k! Γ + α + k. Therefore, ubtituting in. the reult given in.9, we get τ α+n I α aτ t τ β+m I β a t τ dτ a α+β Γ α + ñ + u + Γ β + m + v + ñ + u! m + u! uñ + v m + [ u+v L R ñ,u α R β m,v ],. a α+β+ñ+ m+u+v+ Re α > n +, Re β > m +, where, imilar to..3, we have defined.4.5. According to the reult given in the Appendix A.3, we have Rñ,u α R β m,v ñ+ m k a k ñ+ m k c k ñ, u, m, v, α, β,. where we have defined the coefficient c k a the finite um given in.9-.. Subtituting now. in. we arrive at τ α+n I α aτ t τ β+m I β a t τ dτ a α+β Γ α + ñ + u + Γ β + m + v + ñ + u! m + u! uñ + v m + [ ] r a k ck ñ, u, m, v, α, β L r k a µ+r,. k Re α > n +, Re β > m +,

6 J. L. G. SANTANDER where we have et µ r a.6.7 repectively. In order to calculate the invere Laplace tranform given in., we apply [5, Eqn...58] L [ λ a ν ] t λ ν ν Γ λ ν F λ λ ν, ν Re λ + ν <, Re > Re a, a t 4 where p F q i the generalized hypergeometric function [3, Eqn. 6..] a,..., a p pf q b,..., b q z a k a p k z k b k b q k k!..3 k Therefore, we finally obtain., a we wanted to prove. It i worth noting that the numerical integration of. i lower than the computation of the above formula. For intance, taking rom parameter uch a α. + i, β.3 i, n, m 5, a.4 +.i t.9 the numerical integration i 9 time lower., 3. Particular cae A a conitency tet, we are going to particularize. in order to recover the reult reported in the literature tated in Example. Taking the ign in., we have τ α I α aτ t τ β I β a t τ dτ 3. Γ α + Γ β + I πa Γ α + β + α+β+ at, tα+β+/ Re α >, Re β >, which correpond to the cae n m i.e. ñ m u v r in., o that, taking into account.6,.8.3, we have at τ α I α aτ t τ β I β a t τ dτ α+β Γ α + Γ β + t α+β+ c,,,, α, β Γ α + β + F Re α >, Re β >. 3 + α + β a t, 4 Applying now the formula [6, Sect. 9.4] F γ z z γ/ Γ γ I γ z, 3.

7 AN EXAMPLE FOR GENERALIZATION OF INTEGRATION OF SPECIAL FUNCTIONS knowing that, according to.9, we have then c,,,, α, β Γ + α Γ + β, τ α I α aτ t τ β I β a t τ dτ 3.3 Γ α + Γ β + Γ α + β + 3 a Γ α + β + Γ + α Γ + β tα+β+/ I +α+β at, Re α >, Re β >. Taking into account now the duplication formula of the gamma function [6, Eqn...3] Γ z Γ z + π z Γ z, 3.4 we have Γ α + Γ α + Γ α +, 3.5 π α Γ α + β + 3 Γ α + β + π α β Γ α + β +, o that 3.3 i reduced to 3., a we wanted to check. 3.. Example. Taking the + ign in., we have τ α I α aτ t τ β I β a t τ dτ 3.6 Γ α + Γ β + 3 I πa Γ α + β + α+β+ at, tα+β+3/ Re α >, Re β > 3, which correpond to the cae n, m i.e. ñ m u, v, r in., o that, taking into account.6,.8.3, we have! τ α I α aτ t τ β+ I β a t τ dτ 3.7 at α+β Γ α + Γ β + t α+β+ c,,,, α, β Γ α + β + 3 F Re α >, Re β > α + β a t, 4 Applying now 3. knowing that, according to.9, we have c,,,, α, β! Γ + α Γ + β,

8 J. L. G. SANTANDER then τ α I α aτ t τ β I β a t τ dτ Γ α + Γ β + Γ α + β + 3 a Γ α + β + 3 Γ + α Γ + β tα+β+3/ I +α+β at, Re α >, Re β > 3. Taking into account now the duplication formula of the gamma function 3.4, we have 3.5 alo Γ β + Γ β + Γ β + 3, π β Γ α + β + 3 π α β Γ α + β + 3 Γ α + β +, o that 3.7 i reduced to 3.6 a we wanted to check. Acknowledgment. The author would like to thank the anonymou referee for hi/her comment that helped u improve thi article. Reference [] A.P. Prudnikov, Y.A. Brychov, O.I. Marichev, Integral Serie. Vol. : Special Function, New York, Gordon Breach Science Publiher, 986. [] K. Oldham, J. Myl, J. Spanier, An Atla of function. Second Edition, New York, Springer, 8. [3] F.W.J. Olver, D.W. Lozier, R. F. Boivert, C. W. Clark editor, NIST Hbook of Mathematical Function, New York, Cambridge Univerity Pre,. [4] I. S. Gradthteyn, I. M. Ryzhik, Table of integral, erie product. Seventh edition, New York, Academic Pre Inc., 7. [5] A. P. Prudnikov, Y.A. Brychov, O. I. Marichev, Integral Serie. Vol. 5: Invere Laplace Tranform, New York, Gordon Breach Science Publiher, 986. [6] N.N. Lebedev, Special Function their Application, New Jerey, Prentice-Hall Inc., 965. Appendix A. Product of two finite um Let u conider the product of two finite um n m P a i a b + + a n b m. i j b j A. We can view the umm of the above product a the element of following rectangular matrix a a a n b a b a b a n b b a b a b a n b.... b m a b m a b m a n b m A.

9 AN EXAMPLE FOR GENERALIZATION OF INTEGRATION OF SPECIAL FUNCTIONS 3 We can group thee umm by antidiagonal a follow P a b }{{} i+j + a b + a b }{{} i+j + + a n b }{{ m, } i+jn+m thu we can rewrite A. a a double finite um, taking a firt index k i + j P n+m k l max ll min a l b k l. Notice that in the matrix A. we have {, k m l min k m, k > m max k m,, l max Therefore, we can write finally n m a i i j b j { k, k n n, k > n min k, n. n+m maxk,n k lmink m, a l b k l. J. L. G. Santer Univeridad Católica de Valencia an Vicente mártir, 46, Valencia, Spain. addre: martinez.gonzalez@ucv.e A.3