By an appeal to the Laplace transform and their inverses we have obtained five inversión formulae for some Bessel function transforms. The inversion formula for hyperqeometric transforms have been given whose kernel are the confluent hypergeometric function  1F1(γ - a;  γ; μx) and Gauss hypergeometric function  2F1( a, β, γ;  - 1/ μ x). The results given earlier by Gómez López [8] follow as special cases of our results given in section. 3

González de Galindo, S. E.

Kalla, S. L.

Spanish

Universidad Nacional De Colombia - Repositorio Institucional UN

Reui sta Colombiana de MatemtiticasVolu'men X (1976), pdgs, 83 - 92THE INVERSION ,FORMULAE' FOR SOME BESSEL AND HYPERGEOMETRICTRANSFORMSbyS.E. GONZALEZ DE GALINDO ane/S. L. KALLAABSTRACTBy an appeal to the Laplace transform andtheir inverses we have obtained five inversionformulae for some Bessel function transforms.The inversi on formul a for hyp er qe ome tr! c 'tran s-forms have been given ~hose kernel are the con-. fluent hypergeometri c function 1F 1(')' - a; ')'; fix}and Gau ss hypergeometri c functi on 2F 1( a, f3. 7/;- lIfl X). The results given earlier by GomezLopez [8] follow as special cases of our re-·sults given in section. 3.§ 1. Introduction. The Laplace transform of ¢(x) is defined by :(1)OJf { ¢ (x) } = J e -xt ¢(x) dx = f (t)o. -1If we denote with f the inverse Laplace transform, then we have-1f {f (t) } = ¢(x)with ¢(x) and fit) relatedasin(l);wedenotewith M{/(u)} orby F(s)the Mellin transform of /(u) .. genera!y u is real and positive, s is a complex va-83riable of the form s = CT + t r , where CT and Tare real, The Mellin transform of[Lu) [9, p. 46 J is defined by :(2)00 s-1M { [Lu) } = F (s) = J [Lu) u duos -1I(u) u E L2(O. CXJ). The inverse Mellin transform ofre Iate d as in (2), is written .as follows [9, p. 60J :F(s) with I(u) andwithF(s)(3) M-1{ F(s)} = [Lu) = _1_ J F(s) u-s ds. 2 TTi cwhere F(S)EL2(CT-iT,CT+iT);-CXJ<T<CXJ. and C is a suitable contour inthe complex.a-pl ane, If Hl s} and F(s) are, respectively, the Mellin transform ofh (u) and [iu}, then we have:(4)CXJJ h(u) I(u) du = -2 I. J H(s) FO-s) ds •o TT, Cknov v n as the Parseval theorem for MelIin transforms [9, p. 94 J • In this paper weuse !f and s:' operators to obtain five inversion formulae for some Bessel func-tion transforms and hypergcome trtc transforms whose kernels are IF 1 (y -o., y "f-L x) and 2F 1 (a. f3. y, _I x) • rc spect ive ly. In section 3 we give two resultsf-Lwhose particular cases where givcn earli er by Gomez Lopez. We lise the followingresults [4J . ,(5)TTif(s+v)f(s-i.)r . as r(s + i) 9l.,(s) > 19le( v ) 1,9l., (a) > 02v-Ir({-s)fO+tv +ts)(6) M [sen(ax)j (ax);s J = sv a fO+v-s)f(l-tv-ts)a> 0- 1 < :R,,-( v} < 9l.,(s) < t2v-1rrt-s)f(tv +ts)(7) M [cos(ax)'j (ax),'sl= -------=:----v as fet-tv -isWO +v -s )a > 0-9l., (v) < :R.,( s) < tr -ax(8) M LeI (ax); s Jv -84(1) .(9) AI [H (ax); slI-' .(1-i)2s-1r(ts+~I-') (/) (/) 3, .J\ -e. ( lJ) < .J\.e ( s} < 2"~lr (~I-' -J S + 1)We shall also use the following important theorem given by C. Fox [6,p.300 l.11: (i) 0. > 0 • (~ a + f3) > 0 , t> 0 I(ij) s = U + ifL'. o :and fL both real andF(S)E L(t-ioo, t +ioo)then(10) ~-1{2;r' J r (as+(3)F(s)t-aS-f3ds}= -21 . J F(s) xas+f3-1dst c 7Tl Cwhere, lor both integrals, the contour C maybe the line u = t , a line parallel tothe imaginary axis in the complex s-cplane, And(11) ~{_1_J H(s) xos+f3-1ds} =_1_ J H(s)t-a.s-f3ds.27Ti "Cr (as + 13) 27Ti C§ 2: Inversion Formulae [orBes sel function trans/arms. We shall obtain an in-version formula for the Following transform:(12)(D uxh(a) = Je KlJ (ax 1[Lx) dxawhere Kv(') is tile modified Bessel function of the second kind (2) •THEOREM 1 : IfI-' + t <0 , ~~ (s) > I~.e (v) I R (a) > 0 ,(i)(ii)(iii)(iv)(v)j(x) E L2(0, 00), thus h(a) is defined and also belongs to L2 (0, 00) ,s-v-tF.(l-S)EL2H-ioo, t +ioo),F(l-s) EL2 (t -i as, t + i 00) ,.. .y-'f(y) EL2(0. 00).where /(y) is 0/ bounded variation near the point y =x • then the inversion [ormu-la [or the trans/arm" (12)" is :(13) j(x) =7T-txvf-1{tl-'+t~ {x-v+tf:-1{(2afl-'h(a)}}}85Proof: We apply (4) to the right hand of (12) and by (5) we have:1 1. -s f(s+v)f(s-v)h(a) = -2 . J F(1-S)TT2(2 a) . ds ,TT, C f (s + t)(14)For large positive x and a> O. the kernel of our integral equation behaves asax ITT - te Kv(ax) "',VT (ax) .F or small positive x and a> 0, we have"axsince we have the condition v + t < 0 it follows that the expression e 'K)ax)belongs to L2(O,m) and since f(u) E L2 (0, m), the application of (4) to the righthand of (12) is justified. The contour C in thes-plane is the. straight I inc s = t tiTand T varies from -co to co , We shall now try to eliminate th"e other gamma func -tion [10, p. 27::JJ along the line s = t + ir , for large I T I we have:f(s+v)f(s-v) -1------F(1-s) = F(1-s) (1 + 0 (s »f (s + t)f(s-v) F(1-s) = s-v-tF(1-s) (1 +0(s-1»I" (s it)and the condition v < -t. Multiplying both sides of (l::J) by (2 at v and applying-1f .we have1.+ioo f )1 -v 1 1 2 (s -v s +v-1 }f- ((2a) h(a)}=7l{2-' J F(1-s) x. ds .TTl t-ioo f(s+t). +,. .Now multiplying this expression by x-v 2" and using the f operator, we get(15) f {x- v+tf-1 {(2a( vh( a)}}= {e -xtx -v+t {TTt ~ J F(1-s) r (s- v) Xs +v-1ds}dxo ~ TTiC r (s+t)The integral of the relation (15) is absolutely convergent and-v-1S 2 F(1-s) E Lit-im, t tim).86. ~frnce, integrating with n~sl'«~CI to x and eliminating r(s +j). we obtainMultiplying both sides by. ,v + j and lll'plying s:'. we gel:ReplaCing s by I-s, we have:(;)-1{. v+j(;) { -v+j -1{ -v ~ 1 -VI -s 1. -v .oJ.. I oJ.. x f.. (2a)h(a) }}}o'" --x Ft st x ds= 7T2x [t x}. 27Ti Cwhich is justified by virtue ofthe condition:1---2Y I(y) E L2(O.00). Hence-~ v(;)-l{ v+1 (;) - -v+j(;)-l{ -V}'}I(x) = 7T X oJ.. I 2 oJ.. {x oJ.. (2 a) h ta) J •Proceeding in the same way. we can easily establish the follOWing theorems.THEOREM 2: II(16)00'P (a) = J sen [ax}'] (ax) f(xJ dxo vand(i) v > 0 , a' > 0 ,Oi) f(x)EL2(O.00). thus ((l(a) is defined and also belongs to L2(O,oo) •(iii) s-jF(J-s) E L2( ~ -;00. j + (0) ,SV F(l-s) E L2 (j -;00, j + ;(0) •1. v+~. S2 • F(l-s)C.L2 (j -;00, j f ;(0) ,(io) F(1-s) E L2H-;00 ,~+ ;(0)(v) y-j fly) E L2(O, (0), .'where fly) is of bounded variation near tbe point y=x, then the inversion lormula87for the transform (16) is :-1 -3/2 -.Lv +.L -1 _.L v .(17) 2 {T [2{z' '2 {t '[2{a <:pea) }]x=zt}}]t=T-1}v-1 vvl tv t2 T Z f(z);THEOREM 3: l f :00(18) g(a) = Jocos (ax) J (ax) f(x) dxvand,(i) v > 0, a > 0 ,(ii) j(x) E L2(0, fJJ), thus g(a) is defined and also belongs to L2(0, (0) .(iii) s-t F(1 <s) E L2(t -ioo, t + i co ) rsVp(1-s)EL2(t-ioo :t+ioo),.!. V +.!.s' 'F(1-s) EL2(t-ioo" t +ioo)(iv) F(1-S)EL2(t-ioo, t+iOO)(o) y-t fry) E L2 (0,00) rwhere fry) is of bounded variation near the point y = x , then the inversion [ormu-la for the trans form (18) is :v -1 -Lv-I. 1.= 2 z· 'f(z')THEOREM 4 : If:(20)m (1)h(a) '" I /I (ax) f(x) dx. voand(i) (I>O"v';>O,88(ii) ((x) E L2(O, ro), thus h(a) is defined and also belongs to L2(O,ro),(iii).s-tp(l-S)E L2H-iro. j +iro) ,'-1 tI + 1. .S2 4p(l':"'S)EL20-iro, j+iro),(iv) P(l-s) E L2H -Lco , j + iai }',[u) y-j{(y) El..2(O,ro)',where {(y) is of bo',mded variation near the point y=x, then the inversion fOMnulafor the transform (20) is:,-1 -v..,1 jv . _.1+.1 V I(21) f {T [f{a [h(2a)] .1}] _I} =1::'-,x 2 • f(x2)ar a» t=T 2HU)(·) stands for the Hankel function of the second kind [2] .THEOREM 5: If(22)m() J -axga = e • 'v(ax) [t x} dxoandm «> o . );I>O~(ii} f(x)EL2(O,ro), thus gio.) is defined and also belongs to L2(O,ro),v + I(iii) s 2P(l-S)EL2(j-iro, j+iro),(iv) P(l-s) EL2(j-iro, j +iro),_.1(v) r ' fly) E £"2(0, ro) ,where {(y) is of bounded variation near the point y =x • then the inversion formulafor the trans form (22) is :(23)j-v -1 -v+j -1 j+v vf(x)=7T X f {T [f {t f{a g(!!) }}] r ..-- 2 X=T§ 3. Inversion formulae for bvpergeometric transform. In this section we esta-blish two theorems;which provide us the inversion formulae for two hypergeometrictransforms. Usinglhe following results [4] •(24) {{3 -u x -(s+{3) r(s.+a)·r(y)r(-s)M x e ]F1(y-a;y;jJ-x)}=u r (a) . r (y- s - a)(25) {p 1,2 [ ] /(1-a. 1)(1-[3. 1)} s i p r(s+p)r(a-s-p)r([3~s~p)M x H -- =u2,2 fiX (0.1) (1-y.1) r(y-s-p)~.,.{s + 1 +min (0,1 - y)} > 0g{.dmax(-a.-!3)+s+l} < 0and proceed ing in the same way as in the theorem 1, the following theorems can beproved easily,THEOREM 6 : IfcoJ f3-ux(26) r:p(u) = x e ]F] (')'-a, y; ux} f(x) dxaand(i) [3 > 0, ')'a>-2 a> O.(ii) f(x) E L2 (0, (0), thus r:p(u) is defined and also belongs to L2 (0, CO),(t't't') a- y+ {3 +1. (s 2F1-s)EL2(t-iCO.t+ico),as 'P(1-s) EL2(t-ico. t+ico).sa.- {3- t F(1-s) E L2 (t -ico . t + i co ) ,(iv) F(1-s) EL2(t-ico. t+ico),(v) y-tf(y)EL2(0.CO),where f(y) is of bounded variation near the point y=x, then the inversion formulafor the trans form (26) is :Particular case': If we set [3 = 0 . this theorem reduces to the one considered90recently by Gomez Lopez [8, (26)] .THEOREM 7: If(28)illcg(u) = I x' 2171 (0, (3, y .. - ~ x) /(x) dxuaand(i)(ii)P >0, p-a< 0, p-/~ < -1. a+;3-y > 0, a+{3-p> L a-l (3-2P>f(x) E 1~2(O,(JJ) thus g(u) is defined and also belongs to L2(O,W) ,a+8-ys' F(l-S)EL2(t-ilJJ,t+iW),a-p+,8·-tS F(l-s) E L2(t -Lco , t + ioo),a+(J-2p-1S F(1-S)EL2(t-ioo,t+iOO),8_p_l .s 'F(l-S)EL2(t-ioo,t+ioo).( iii)(iv) F(l-s) EL20-iOO, t+ioo),(v) y-t f(y) E L2(O,OO)where fry) is of bounded variation near the point y=x then the inversion formulafor the transform (28) isr (y ) ,8- p -2 -1----- x f(x) .r (a) r ((3)Particular case: If we put p=O th is theorem reduces to the one con s idcrr-d re-cently by Gomez Lopez [8,. (22) ] .References1. A. Erdelyi et AI., Higher Transcendental Functions, Vol.l, McGrow-Hill, NewYork, (1953),2. A. Erdeiy i et AI., Higher Transcendental Functions, Vol. II,McGrow-Hill, NewYork, (1953),3. A. Erdelyi, On Some Functional Transformations, Univ. Polit. Torino Rend.91Sem. Mat., 10( 1951), 2T7-234.4. A. Erdelyi et AI., Tables 0/ Integral Transforms, Val. I, McGraw-Hili, NewYork, (1954).Erdelyi et AI., Tables 0/ Integral Transforms; Vol.Il. McGraw-Hili. NewYork, (1954).-I ,Fox, Solving integral equation by f and f operators, Proc. Amer. Mat.Soc.29( 197)) 299-306.Fox. Appl ic ation s ofLaplace transforms and their inuers.es , Proc. Amer.Math. Soc. Vol. 3'5(i) (l!9'72l, 193-200.'8. A. M. M. de G. Lopez, The inversion formulae for some by perg eometric trans -5. A.6. C.7. C.forms, Tohoku Math. Journ., 26(1974), 315-325.9. E. C. T'itcbmarsb, lntroduc t ion to tbe theory of Fourier' integrals. ClarendonPress, Oxford. (1937).10. E. T. Whittaker, and G. N. Wa~.s.on, A course of modern analysis. Cambridge( 1963).Foc ult ad de Bioquimica, Quimiea y Farmaciarilldtac ultad de Ci.encias Ex act as y Te caologiaL'lIiversidad Nac ional de T'ucumdn"fuel/man, RepUblica Argentina, S.A ..(Re c ib ido en nouiembre de 1975).

The inversion, formulae for some bessel and hypergeometric transforms

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The inversion, formulae for some bessel and hypergeometric transforms

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