A Class of Integral Transforms!

7/28/2019 A Class of Integral Transforms!

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A CLASS OF INTEGRAL TRANSFORMS!by JET WIMP

(Received 30th August 1963)1. Introduction

In this paper we discuss a new class of integral transforms and theirinversion formula. The kernel in the transform is a G-function (for a treatmentof this function, see ((1), 5.3) and integration is performed with respect to theargument of that function. In the inversion formula, the kernel is likewisea G-function, but there integration is performed with respect to a parameter.Known special cases of our results are the Kontorovitch-Lebedev transformpair ((2), v. 2; (3))

g(x) = -. x sinh (nx) f" rlKix(t)f(t)dt, (1.1)* Jo/ ( * ) = [" Ku(x)g{t)dt, (1.2)Jo

and the generalised Mehler transform pair (7)g(x) = - s inh( j tx)r ( i -*+ix)r ( i -*- ix) fP?x^{t)f(t)dt, ...(1.3)

J- * - i x ) f" P ? x ^

Ji/(*)= P Pi-^x)g{t)dt (1.4)Jo

These transforms are used in solving certain boundary value problems of thewave or heat conduction equation involving wedge or conically-shapedboundaries, and are extensively tabulated in (6).

Section 2 contains preliminary results and definitions, and Section 3 con-tains the derivation of our main result. Several examples are presented inSection 4.2. Preliminary definitions and results

The Mellin transform (2)9\s) I x J\X)UX jfl S\J\X)), (1)

t This paper covers research supported by the Aeronautical Research Laboratories, Office ofAerospace Research, U.S.A.F. The author is indebted to Yudell Luke and ProfessorArthur Erdelyi for their comments.E.M.S.


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A CLASS O F I N T E G R A L T R A N S F O R M S 35Then '-Til1v+s, 1 v s, a .(2.6)

Proof. We use the representation (2.3), with app rop riate change of variable,in (2.1). Condition (1) assures the separation of poles while condition (2)assures the absolute convergence of the doub le integral. Th us the order ofintegration may be interchanged and the result is (2.6).3. The main resultWe wish to solve the integral equation

g(x) = J (3-Dwhere the conditions on the parameters are those of Theorem 2. Replacex by iu . Multiply both sides of (3.1) by x~"du and integrate from ciooto c+ico. If we invoke (2.6) and let

4x = _ x= (2-x)-2jx} ( 3 2 )on the right-hand side of the equation, we have

.-v,i-v,2>3c-^(x)= PJo (3-3)Lwhere g(x) = ~ I x-ug(-iu)du (3.4)27tJc-iooUsing the notation of ((10), p. 315), we have

or(5(1 -i

_ s v ro41-s-v /


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36Also , from (2.5)

J. WIMP

n r{-bj+s) ftj = m +1 j = n+1

,...(3.7)

and so , by ((10), p. 316),

2t i J c_ I- o e S


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A CLASS OF INTEGRAL TRANSFORMSWe need the formulae

\ P,yJ Ma_Lt2l(fcl7l

37

.(4.1)

G\\(a>a\Let bt = ^ vK. Since

'p,qwe haveA(xeni | v it, v+ it)= G%1 ( xe"'

l-b

to (ffl). ...(4.2)' 2

(4.3)

( vi7, vqri'A -lofe'"' %K V \[xe T I = G}2 2 I,\ K + V \ J \ x 1 vq:t7, 1 v + it)~" x . .(4.4)

Using first the fact that

and then, ( * ) = -

e(m-*'M-Kt _m(x) = MK> _m(Jt f - M K , m ( x ) .

(4.5)

sin r(i+m-K)r(l-2m)J'(4.6)

(see [(8), (1.7)]), we arrive at the transform pairg(x) = r(i-ic-)r(i-K+ix) f00 , (4.7)/ W = ll2Xf * {X tsmh(2nt)WKiit(-\g(t)dt (4.8)i Jo \x/

In (4.7) replace t by \\t; in (4.8), x by 1/x, and then replace cf/2tv~*/(t"1) by/ ( / ) . The transform pair is

rix)r(i - K + ix) WK ix(t)f(i)dt, (4.9)Jotsmh(2nt)WKjt(x)g{t)dt (4.10)

Since

putting K = 0 in (4.9) and (4.10) we obtain the Kontorovich-Lebedev transforms(1.1), (1.2). For a theoretical study of how such transforms arise from secondorder differential equations, see (5), (9).


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A CLASS OF INTEGRAL TRANSFORMS 39details. After ap prop riate changes of variables, we get

g(x)= I"" S.JMfWt, (4-19)

/(*) =

Jo/ 1-^+iA / l- / j- tA

2 - 2 M - l Too \ 2 \ 2 I= Z-^~ t sinh nt -j { " LV 2 ; * v 2

...(4.20)Example 4. m = = 0, /> = gr = 1.Here a t = 1 v, 6 t = i - v ,no I l-v+ix, l-v-ix) = . / r y . " . [ J i j r* ) -^ ! -* ) ] , ...(4.2i)21 sinh (TIX)and

u v_r V~M )=p- c* -(z-*), ...(4.22)

where 8 = zrf/rfz [see [(1), (5.3(13)), (5.4(25), (26))]]. If c = 0 in (3.8) and wemake the necessary changes of variable, the transform pair is

sinh (nx) Jo (4.23)

fix) = ^ T e""* ~ W x ) W ( i i (4.24)2 J ^Example 5. m = 0, ^ = q = n = 1,a t = 1 - v , bx = | v.Since

| l - v + i x , l-v-ix)= r v /C f x (r*), (4.25)j2 | " + V> vli ""I - - V ^ M v - D K - 1 ^ ^ ) , 8 =^, ...(4.26)

andG (elementary manipulations with Bessel functions give

ff(x)=4= f"-K(O/(O*, (4.27)Jn Jo

= 4 = r < j - {/f.WMOdt (4.28)JnJ-oo


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40 J . W IM PIn closing, we note that the development in Sections 3 and 4 has been

formal only, and conditions for the validity of each transform pair must bedecided individu ally. W e defer these considerations to a future p ape r.REFERENCES

(1) A. ERDELYI, W. MAGNUS, F. OBERHETTINGER and F. G. TRICOMI, HigherTranscendental Functions, 3 vols. (M cGraw-Hill, 1953).(2) A. ERDELYI, W. MAGNUS, F. OBERHETTINGER and F. G. TRICOMI, Tables ofIntegral Transforms, 2 vols. (McGraw-Hill, 1954).(3) M. J. KONTOROVICH and N. N. LEBEDEV, On a method of solution of someproblems in diffraction theory, / . Phys. Moscow, 1 (1939), 229-241.(4) W. MAGNUS and F. OBERHETTINGER, Formeln und Satze fur die SpeziellenFunktionen der Mathematischen Physik, 2nd ed. (Springer-Verlag, Berlin, 1948),p. 189.(5) A. McD. MERCER, Integral transform pairs arising from second order differ-ential equations, Proc. Edin. Math. Soc. (2), 13 (1962), 63-68.(6) F. OBERHETTINGER and T. P. HIGGINS, Tables of Lebedev, Mehler andgeneralised Mehler transforms, Math. Note No. 246, Boeing Scientific ResearchLaboratories.(7) P. L. ROSENTHAL, On a Generalization of Mehler's Inversion Formu la and

Some of its Applications, dissertation (Oregon State University, 1961).(8) L. J. SLATER, Confluent Hypergeometric Functions (Cambridge UniversityPress, 1960).(9) E. C. TITCHMARSH, Eigenfunction Expansions Associated with Second-OrderDifferential Equations, 2nd ed. (Oxford, 1962).(10) E. C. TITCHMARSH, Introduction to the T heory of Fourier Integrals, 2nd ed.(Oxford, 1948).

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A Class of Integral Transforms!