ENGINEERING RESEARCH INSTITUTE'IUNIVERSITY F MICHIGAN ANN ARBOR PRELIMINARY REPORT HERMITE AND ILAGUERRE INTEGRAL TRANSFORMS By JOSEPH C. McCULLY R.. Vo CHURCHILL Supervisor Project 2137 ORDNANCE CORPS, U..I S. ARMY CONTRACT DA-20-018-ORD-12916 Agust, 1953

TABLE OF CONTENTS Page INTRODUCTION 1 I. HERMITE TRANSFORMS 4 1. The Transform 4 2.* The Basic Operational Property 3* Effect of- Iermite Transform on Derivatives 6 4. The Transform of xr 6 5. Tables 7 II. FURTE PROPERTIES AND EXAMPLES OF HEBMITE TRANSFORMP 1. EXample 11 2. Inverse Operator 11 35. Order and Continuity Properties of F, F', and Ft 12 4. Example 14 5. Some Simple Transforms 15 6. A Possible Use of Some Previous Results 15 III. APPLICATIONS 17 1. Problem 17 2. Remarks 19 IV. LAGURRE'TRANSFORMS 20 1. The Transform 20 2. The Basic Operational Property 21 3. Laguerre Transforms and Derivatives 21 V. THE INVERSE:OPERATOR AND THE LAER.RE TRANSFORM 23 1. The Inverse Operator 23 2. Properties of F (x) 24 3. Summary 24 REFERENCES 25 ii

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN PRELIMINARY REPORT HERMITE AND LAGUERRE INTEGRAL TRANSFORMS INTRODUCTION When a function K(a,x) is a known function of two variables a and x, and the integral If (a) = f r(x) K(a,x) dx b is convergent, then If(a) defines a function of the variable a. This function is called the integral transform of the function f(x) with kernel K(a,x). When the limits b, c are both finite we speak of If(a) as being a finite transform of f(x). We are going to consider two possible choices of the kernel K(a,x). They are as follows: 2 K1 (nx) = eX Hn (x), n 0, 1, 2,..., and K2 (n,x) = e'x Ln (x), n = 0, 1, 2,... where Hn (x) is the Hermite polynomial and Ln (x) is the sinmple Laguerre polynomial. The literature contains more than one definition for the Hermite polynomials. We will give two of'the most common here. In [1] the Hermite polynomial of degree n is defined by (1) (x) = ( nx d -x (1) Hn (x)'= (-1)n'x2 dx n _ In [2] the Hermite polynomial.of degree n is defined by (2) Hn (x) = ex2/2 (_l)n dn e-x2/2 In this paper we will consider (1) as defining the Hermite polynomial. Formulas such as (1) and (2)'are often referred to as Rodriguez formulas.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Following the definition (1) the first few polynomials are: Ho (X) - 1, H1 (x) = 2x, (3) H2 (x) = 4x 2, H3 (x) 8 8 - 12x, LH4 (x) = 16x4- 48x2 + 12 In what follows we shall be interested in some particular properties of the He-rmite- polynomials. Among these the Hernite differential equation -and two differential recurrence relations., namely (4) Hn" (x) - 2xHn' (x) + 2n Hn (x) -- O, (5) Hn' (x) = 2n Hn-i (x) (6) 2xHn (x) - Hn' (x) = Hn+1 (x),: n = O l:, 2,. The differential recurrence relations (5) and (6) can be used to arrive at the differentia -eqation (4). To see how one might arrive at relations (5) and (6) let us consider another definition of the Hermite polynomial of degree n. Consider the expansion in powers of t of the function exp (2xt - t2)o Since exp (2xt - t2) = exp (2xt) exp (-t2), the coefficients of'the powers of t in the expansion will be polynomials in.x We will define HE (x) by (7) exp (2xt - t2) Hn (x). n=Q The polynomials Hn (x) will be Hermite polynomials.'From (7) we can readily obtain the result (8) H (x) = k' (2k)n"2k k. (n-2k) k=O where [n/2] is the greatest integer in (n/2). The relation (7) is valid for -all x and t in the finite t plane. By differentiati"ng both sides of (7)'with respect to x one can.ar rive at (5) and the relations 2'

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN (9) Ho (x) = 0, (5) HnI' (x) = 2n Hn.i (x), n = 1,2,3,.... If, on the other hand, one differentiates both sides of (7) with respect to t, uses known expansions, and compares coefficients, one arrives at relation (6). The Hermite polynomials are orthogonal over the interval (-oooo). The weight function is e-X2 and,0 m f n (10) e X Hn (x) Hm (x) dx = - 2n n, m = n We define Ln (x) as n (11) Ln (x) n' (-x)k Ln (n' k): (k:)2 We shall be interested in the Laguerre differential equation (12) xLn" (x) + (1 - x) Ln' (x) + nLn (x) = O. Other properties which will be of interest are as follows: (13) xLn' (x) = nLn (x) - nLn-1 (x), (14) Ln' (x) = LA-1 (x)- Ln-1 (x), n-1 (15) Ln' (x) Lk (x) k=O (16 ) dLn (xn (xn e-x) n: dxn o 00 o if m n. (~17) o~ir e-X Ln (x) Lm (x) dx = 1 if m =n. From (17) we see that the Laguerre polynomials are orthogonal ove.r the interval (O,oo). The weight function is e-X 35

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN IT HER.IE:TTRANSFORM4S l. The Transform We will define the Hermite transformation of the function F(x) to be (1.1) Tn (F (x)@ = () F (x) eHn (X -W00 We would, of course, expect some restrictions on the function F (x) since, by definition, the integral'must necessarily be convergent. More will be said later -as to the character of F (x). For an integral transformation, it would be desirable to have for an inversion process a TFauberian theorem}, however, in our -case we will settle.for an Abelian theorem. Since we are dealing with a set of orthogonal polynomials we are led to consider the expansion of an "".arbitrary" function in an infinite series of these polynomials as a possible means of obtaninig an inversion formul, a. If we assume 00 (1.2) F (x)'= an (x).,.o x o oo, n=0 we can find the coefficients tan. If we assume the interchange of integration and sumation and take advantage of orthogonality property (10) of Hermite polynomials we will have Tn (F} (1.3) an =, 2n n='We therefore have an inversion formulae (1.4) F (x) n Tn {F} n (x) ~n0O The fund..ental problem here is seen to be the determination of the conditions which will assure the convergaence of the series 4

-- ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN oo Hn (x) n=O to F (x): In answer to this: question, much has been written. Probably the most complete Work being done in Chapter 9 of Szego [31] (More work is to be done later in order to obtain a' workable set of conditions on F (x).,) 2. The Basic,ional Prorty The:.:ermite polynomial is known to satisfy the differential equation (4). This leads us to try our transformation on the expression (1*5) L [v] = V" (x) - 2xVt (x) We will aSui:eihnwhat follows that V (x), v' (x) are continuous and that V" (x) is bounded and integrable- on each finite interval. We also assume V (x) and V' (x) are such that IV (x) < Meax2 and IV' (x)l < Me aX for large values of x and where a < 1/2. Successive integration by parts of the integral 00 (1.6) ki ~ L [V] e-X2 Hn (x) dx -00 along with the information from (4) that (1.7) [e-x2 Ht (x)]' - -e-X2 2niln (x) will lead to th'e "fundamental property of the Hermite transform 0(1.) Tn [V 0 (1.8) 2nTn n = 1, 2, 3, *, For a modification.-of forla (1,8) suppose V' (x) has an ordinary discontinuity at x = xo, A' process similar to the one: followed in arriving at formula (1.8) gives (1.9) Tn [L [V = e-xO2 [V1 (Xo + 0) - Vl (xo - 0)], n = 0, e'XO" n (xo) [V' (xo + 0) - V' (xo - 0)] - 2n Tn [V], n = 1, 2, 3 _ _ _ _ _ _ _ _ 5o

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 3, Effect of Hermite Transform on Derivatives We assume in th'e following that F (x) has continuous derivatives F(m-1) (x) and a sectionally continuous derivative F(m) (x). We will assume F (x2 and its continuous F(m-1) (x) derivatives all are such that IF (x)l < MeGx for large values of x and a < 1/2. One integration by parts of the integral 00 (1.10) )ex F' (x) En (x) dx -00 and the use of property (6) leads to (1.11) Tn {F' (xX) Tn+i e (x), n = O 1, 2, Now, applying (1. 11) to F, we have Tn {F" (x) = Tn+i F (x = Tn+2 (F (x) Continued inductively, this process gives the property (1.12) Tn {F(m) (x)} = Tn+m (F (xX, n = 0, 1, 2,... 4, The Transform of xr It is known that when r is a; fixed integer the following expression is true, r 1 [/1 r' Hr-2k (x). (1.13) xr L k (r - 2k). k=O where [r/2] indicates the greatest integer in (r/2) Now let us consider TE (xr) for possible choices of r and n (r fixed while n = 0, 1, 2,.-.)* Case 1 n > r =o __ Hn (x) [r/21 r' Hr.2k (x) (4r - (1.14) Tn f__rX n x) 2r kS (r 2 o;o k=O Since n > r we see that in the finite sum ho- Hs (x) can acciur with an index equal to n, Hence by property (10) we have the following formula: 6

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN (1.15) Tn (xrI) = 0 n > rr Case 2. n-. r. From (1.14) we see t'hat Tn n) = J4 n' n rCase. n < r. ~. Z~.~4.~~ We see from equation (1.14) that in order to obtain a contribution, n = r- 2k or k= 2'When n = r - 2k k -= 0 1, 2,..., [r/2], n - 0O 1, 2,.., n <-r -e;quation (1.14) gives (1.16) Tn xr3 = for n = 1, 2. [(r -n)/2]. n < r n =r -'2k If however, n =:r- 2k - 1 there will be no contribution and we will have (1.17) Tn r = 0 n ='0, 1 2,.. n < -r, n = r - 2k i - where k - 0, 1, 2,-.. [r/2] 1 5.'Tables The remaining pages of Chapter I contain tables of some simple trans-forms:and of simple operations which have already been considered or will be considered in Chapter II. 7,

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Table of Tratnsforms 00 f (n) - e2' (x) F (x) x f (n) F (x) 0 n O 0 (i{ Ns (1) 1 n = 0 (2) Kf (n) (2) KF (x) 0 n. r O l:r n = r < (3)r [(rr'n)/21 n' r-2k n — r. n = r-2k-i where k =O l, 2,..o. Ir/21 n = 2p'0p = (4') (4) sin la. (-1l)Pne-(k2/4) i, n = 2p+ 1 ('1-Pkn+f'e- (k2/4)<,& np (5) O n -= 2p +:1 (5).cos kx where p = 0, 1,2....o.

ENGINEERING RESEARCH INSTITUTE. UNIVERSITY OF MICHIGAN Table of Trafo, (contf) f (n) F.(x) (6) qneq2/4N, n = 0, 1,.2,.. (6) eqx (7:) -2ne1/4, n.= 0,, 2,. (7) eX (1 - 2x)...~~1/ (8) (.l)n+"2ne/4,I n = 0O, 1, 2,,. (8) e'X (1 + 2x)'(-1 "":~'.'.2n.e"'(k2/4) 2nk"+ -e~F~ n.= 2p' ~(9) "~'.,..n'= 2p.+. (9) 2kx sin kx k2 cos kx 0 n ix 2p +1 whtere p = 0, 1, 2,.0 n -= 2p (lO (_.~p",.nk~.-.(k2/4)~F..= 2'2.Snk 2 (10). (P(-IF+1 2nke(k /4)1 n 4I + 1 (10) -(2 sinkx+ kx cos kx) where p- = 0, 1, 2,. T~be of. is F (x) ~ (n).00 (1) F (x) (1) e n (x) F (x) dx Fn02 (2) F' (x) (2) f (n'+-1) n'.= O, 1, 2,.. (3) F (x) (3) f (n+.m), n = 0, 1,.2,.,, ~dxm

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Table of Operations (cont ) f (x) f (n) (4) (t)dt 4 e x F (t)dt dt d, n = O O I _~-~00 0 f (n), n= 1,2,3, (5) F (x)' + K a Constant (5) f (0) + l/1 n = 0 f (n') n = 0,2,3, (6) F (x) + kx, Ka constant (6) Lf (n) +K'h, n = [f (n) nm (7) F (x)+ m (x)'(7) f (n)+ 2nn', n ='m -2n f (n) n = 12,3, (8) F" (x) - 2x Ft (x) (8 lO: Mn= 0 *This is (1.8). See condition8 on F, d/dx(F),' d2/" (F)there. 10

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN II. FT PRQP IES MANM APES 1. lEze:`ig,>Fl| V (i) T We will assuume V (t) continuous and IV (t) < Meat2 for lArge t and a < 1/2. x'(2,1) F (x) V(t)dtThen F (x') is contimnous "and Sinee V (t) has the: desired orer property it is easi y aeen,,by the us..of asimple properties of the Rimann inte that I IF (x)I <M2' r large xa:d <1/2, Also F' (x) =V (x), ThenT {F'(x)]'= Tn {V (x)}, but si'ne- F' (x) is dontin s..and F (x) po-sesses the -cor ret order property we.an.,use property (1.12) on transforms of derivatives and write Tn SF () t F(x n 1,2 3t,. (2.2) {f tX.d 00 X T F (t) dt =' /..eX 0 "V'(t) dt dx n O. t,0 2,-. I.nverse..peratorz We. -cosider now the opertor L [V] gi.n' by (15.5) -wre. V (x.)' V' (x) axe ontinuous a.d IV (x)I <IM. [V' (x)l <'NebX.' f: gs of x and -a, b< 1/2. -Also V" (x) is ed atd int aon.ch finte interval. We are now interested in considering Tn L'" ['VJ]. Proeeding forx.ally we have L'1 [V] r= F(x).o V = L [F:]'(2.3) F" (x) - ax F' (x) = V (x)' 11

ENGINEERING, RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN We now apply the Hermite Tranform to equation (23). If F, F d'"t possess: the ness y properties we.apply (18) and write -n f (n) - v (n) qr (2.,4) f (n) =, 1 2 3 2n For the ea n = 0 we have 0.. v (30) In,.other words, the zero transform of V (x)''st be zero:,. We see then that if we are to be able to use (1.8). V (x).mst be such that 00 (2,5) /' e.x2 V (x) dx 0= o, We will eneounter equlation (2.5) agin when we investigate the orer property of F' (x). It will be seen that unless.eq.ation (2.5) is tr e no fi'.nction F(x) will exist for eation (2.3) suh tt F' (x) will hKve the neessaryor O: property As x +,,. Or d' Cont.y orties of, F F" We will -obtain F (x) from the. differential equation (2.6) [e-x2 F' (x)]' V (x) ex2 It now follows that X (2.,7) e:x2F' (x) _' V (t) e-t2dt+ C1. 0 where C1 is a -.constant of integration W. e seetht if F' (x) is to.have the c:orrect or-er property as X + o,'00 (2.8) F' (x) V (t) -t dt x From'equation (2.8) we- arrive at 12

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN x.co (2.9) F F(x) =.: eY2./ v (t) e-t dt dx + C2,,0 Y where C2 is a: onstant of integration. It is:esily seen from eation (2,8) that F' (x) hAs te -cTect order.property as x -+.+o. When x- + - we have (2,10) eX2 F (x) + K.~ 0 -and. hence F' (x) does not possess the eird order property when x + ue (2:.5) is true.. That F (x) has the -eret order proprty ollows iediately from -the brder property of F' (x).'Continity properties.of. F, and follow from properties of V (t) and of the Riemann integrl. - e lnoe we ivt th following re.lt. "T.'.em1', If V (x), V' (x) " re continuous and. V"' (x) isbounde-d f.d integrbe orn eaich finite inte.rvel, and.if C0 Iv (x) < M1 2, M1,. contants 2-fn (2*11) 00 x oo00 To: X{L 1 [V]} 2' ee2dtdy Q y00 lr~ce V'..:..ot where L1[ = -] " eY V (t) e=t dt,y3 P Y'2n'

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 4 Examle: v (x) = xr in Equation (2. ) We see that only when'r is an-odd integer will the differential equation have solutions vfth the desired rmoperties necessary for- the use of formula (1.8). To.see that this is true it is only ne'cessary to consider the' integral -. n'eVen (2,12), et2n dt = n -, n even, n o.dd -, _Yo Hence, using transform (3) in the:Table of Transforms and formula (2.,11) from the previous section, we:Can write some new transforms. For.equua. tion (2.3)';with V (x) = xr we find. (2.13) F () = Y e tr dy + C, 0 y where C2 is a:constant:of integration.'From transform (3) of the Table of Transforms we: find T ( = 0 n>r Tn F (rci 2r nl= r Tn {F cxi)' = 21l 12r f n< (2*14) nnr< i Tn F (x -: n < r n [(r -n)/2]' =r. 2k, T(F n J<-r -1. n-= r 2k -1 where n -= 1, 2, 35,-.,..., r.is an odd integer, and; k _- O 1, 2, *,., [r/2]', When.n = O, (2.15) TOy (x) = f eey t dy dx 2 e 00 wr~here 0C2 /''X dx = *01 Coo 14

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN From (5) of the Table of Operations, i 9 we see that we hae here| an infinity of funcetions whih will hve the / tranform at n - l 2 35, Henee whn n,- 0 we: ta.e C2 ai zor 5, S TrSifole T.et F (x) 1, < < i.<+O. Sie H(x) 1 we uset t orthOgolity property of i.mite pLyoials to rive at ~j O, n.~ O-., Since. Tn is a linear trMsf_"''ormtion:' we..an' wr.ite. the transform of-.a.constat K. (.o:I,. n i o:X| (2:.17) Tn {n} "K~, n O.:0. -,,. A Posib:l. U:...f S ei.' ReltS usitng'fo.ta" (1 ). w (2-.18):n $- X} = T*1 Isin xx (20.19) -Tn. sin x} 3 Tn+l tM ~.and (2921) Tn (in = -.T js, in.x, n, 2..... We. also,,obin, sing property (112), 15

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN (2.22) Tn(} -, n 0 O, f 2,.e., which tells us that Tn ex) i onsstant. Similarly, (2.23) -Tn = ) Tn+l -,and (2.i24) rTln 1' e-XJ + ~aTn r= Tn+l {r ex Since 00 (2.295) S 2 e-x:sin kxAdx 0,'.00 n-if n0..., x (2,27) -.e'X cos..kx 2x a = O we rn wite th e:actul transfotrm s for. sin kx afnd cOs kx: Tn {sin k = O for n = O, p, 1,,.... (2.28).Tn {sin kx3 = (-1)P k-n e'(k2/4) Jh, n = 2p + 1 Tn (Cos kx (e )P kn+ e(k2/4)p T n 2p, Tn os n + = 0, n. 2p =4 1 Since- 00 q...:-x2+ qx dx (q2/4) -00 we can obtain the transforms -of eqx and'-eX (2.30) TF,n e9. oe(/),,1 2,,..e.. 16

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 1 Prodex In -this:setion- we shll diseuss.-a type ofry "bvalUe problem for -whi.h it might b- advantageus. to us. the I'rmite transfo y.a. 0 o- <:x < +00 Fig. 1.| C ionsr the stdyett two-4imensio-l case of the he t e-quation (3.1) JxK ( n. + (K... P (x. where the theral.-..ondctiity K is prortio. l to.e:X2 ad P (x) is a icontin:us sor ce of heat within the solid. -W' h l. -conslidr -the cae in whi'ch P (x) =.:e-X2 G (x), here IG (x) <: 2 for arge x an.a < l/2. In light:of' th praedin g eqution, (3ol) beomes (3i:2) 2x t =t _ (x1), 3X2 dx dY2 |where U (x,y)l < ~ebx2 for large x, b <1/2. We will assume bo- - eCnon-d tions (3.3). "U (x,O) = O;, U (x,a) = O The transfarmed problem eomes 17

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN (3,4) - u (n,y). - 2n u (n,y)- = g(n) (.3'5) u (n,O)': O, u (n,:a-) ":= 0 UTsing:onditio (3.,5) we find;2 2 i (n J) Therefore, (3*7) u(ny). = 2...'(.,n = 1,2, 3, * W1Vnf n 0, 1'u (..y)..'g Cn:' + C.. Y + C' ~ Again -isi:i onition( (5.5) wn find 00 (3~9) U (x,y.).I u (0,y) +~ t 2nLLu (n2y) H. (x) UT(xyu) i- [y (Y - a)x)dx (3.10) 00: C'. +uM (oay)- =.g (o)3,.n2.H' e'..n'2e,.0'"-:2 n:... Herioe ~ ~ ~ ~ ~ ~~(,~ ~ (x''.~l.~~~~~~~~(......u (X-~~~~Y) ~L8 (

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Whre (n E- + s. inh a':" 2.. i..r~.. The' pl.e t.aor-:ti-on pro ess is not.uited to the abave problem beu the partial iffentl eqti in.- sthe. t sfor "f a seeaonl dRitivae:ia- this t form-oles the. init ue v of" both the f.ction a. its dertivtive. For Se ati.on'of "iables we woul. nee.d. P (x).W e no te: e'xpression [g(n)/:2n1 c:rring. If w:ewo- de.and:00 we might then be able to. ue (2-11) i:n o uirnversion pro.esS:'This problem-l.o: points out the need of ai:ng'.a cvolution property [5]. re is- to be done with this. As the work in obtaining (5.10)'as fo averifition.of o th solution w:ould n.w be. ne.es sy It - an esily be:se-n thbt the boumey'e nditions are atisfied. Also if e. all difff.reLntiation of' infhite seri, u (xjy) will atisfy,the diffeential eqton. We would hoever,y like to h soe citionS on u (n.i'y) hich uld ens.: re the.existenbe of U (x,y). At present wCan offer ees ry -con -dition on u (nsy): nEoo. 0.. nA This follows from (31cn/") Jin!e(X /2) C': -constait, for -. vlues of x::.and n [6]. More is to be. done tod finding ~ondition:on u (n,y). 19

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN IV,' IUE TRANSFiRNS'. The Transform We will define the laguerre Transform -of the function F (x) to be 00 (4.1) Tn (F (x)3 = F (x) eX (x) dx. AAgain as in the case of the Hermite transf orm we r:e faced with the necessity of having- some kind of an inversion proess.- We will'proceed:-as ~before and consider the possibility of the expansion of an'"abitrary" fuction in an infinite series, of Laguerre polynomials. If we ass,.ume.Q00 (4.2) F "(x) = Z'an Ln (x)' o x < L,, n-=O where 00 (4-3) an = F (x) e Ln (x) dx, 0 our..inversion formula for the transform is'00 (4,4) F (x) =' Tn F (x~ Ln (x') n-0 This is arrived at by a process anrlogous to the one used for'Hermite polyno-. mials. We seem again to have the followin fundamental problem: Wiat -conditions..assure the -convergene of the series 00 Z.n Ln (x) n-O to the funtion F (x)? We again refer to [3]. 20

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 2. The Basic Operational Property Consider the self -adjoint form (4.5) [IV] VI [xe'X V' (x)]' eX The Laguerre polynomial is known to satisfy the differential equation (12)which can also be written (4.6) [xe'x L':(x)]' + ne'X L (x) = 0 n Hence it is natural to try our transform on L [V]. We will assume in what follows that V (x), V' (x) are continuous and that V" (x) is bounded and integrable on each finite interval. We assume V (x) and V' (x) are such that JV (x)l < Meax and IV' (x)l <MeaX for x large and a <1/2. Successive integration by parts of the integral 00 (4.7) (xe-x V')' Ln (x) dx 0'as well as the use of equation (4.6) leads to 0 Y [', n = O, (4.8) Tn L [V] = | n Tn V ("x, n = 1, 2, 3, A modification of (4.8) is obtained if V' (x) is permitted to have an ordinary discontinuity at x = xo: Xo eXo [dL V (xo + O) d V (xo -O)], n = 0, Xo eXo..Ln(xo) [d V (Xo + ~) d V (Xo -0O)] -nTn{T(xW x dx dx n = is 2,.... 3. Laguerre Transforms and Derivatives We will assume F (x) continuous and F' (x) bounded and integrable. Also we assume assume IF (x) <Meax for large x and a < 1/2. One integration by parts of the integral l _ _ __ _ _ __ _ __ _ _ _ _ ~ 21

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 00 (4.10) F' (x) e~'IXL (x) dx and.the use of propety (15) leads. to- the foUlowing fomtu-: (.) T JF' ( (0) T ) 1220

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN V. TIE''SE AND THE -. Ta. 1. The nvere Opetr We consid.r here the operator (5.1) L [V] x V" (x) + (1Z-x)V'I (x), where V,.V'.ae continuous V". (x) bounded and integrable on -each finite interval, and Iv < M eaX, (5-2) Ivl < M2 eb for'M and. M2 constant a, b < 1/2, and x + + In section 2 of Chapter IV it.has been- sho- that (48) TnL [V = — n Tnf n 1 2,,.., (4.8) TO {L [VB) O, n. 0:,. We -are now intere-ted in e-onsiderieng (5'3) Tn[ L- [V] Proceeding..forally, we obtain the differential equation (S,4) xF F" (x) + (1'- x) Ft (x) = V (x), where F (x)'= L1 [V] We apply the Laguierre Tran.sff'om ":nd obtain (5.5) Tn L [FI~} Tn { f If F (x) and F' (x)' re. ontinuous, F" (x) bounded -and integrable on each fin-. ~ite interval and F'and F1 satisfy property (5.2), we can use property (4.8) and'write. (5.6) -n f (n) v (n), n 1, 2, 3,... For the case n O we obtain v (O) = 0, which indicates that the zero transform of'V (x)' r:st be zero in order tht such a fnction F (x) exist.' 23

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 2P. Pro perties of (X).. We bcan obtain F (x) from the differential equation (5.7) (xex F')' ~ Ve-x: It follows from:equation (5.7) that 00 (5.8) xewx F' - (t)et dt dhC1 -X where C1 is a eonstnt of integrtion. We see that C1 munst be zero if F' is to satisfy a property simiar to property (5.2). When x + 0 we see t-hat 00 (5-9) v (t)et dt 0. Etiori (5.9) is the condition on the zero transform of V (x) which we noted ini:section 1 of this chapter. That F (x) and F' (x) hae the cect rder ~property (5.2) is a... onsequen-ce o f the order property of V (x) and simple properties of the Rienn -inter,. The contia ity properties follow: as a onse: quenee of elementry properties of they' P f inte.gal.: If V (x), V' (x)are continuous, V" (x) bounde:d and integrable on ec~h finite. intervl, if O4 and V (x) Vt (x) satisfy property (5'2), then Tn -l[v - -, n L= f 2, 3,...., n (5-10) x.00 T'LVI [Vi _ e- e' v(t)..et dt d dx, 0 0 y where x L- LV];_ _ 1 e r V (t)e"'t dt day:24

RERERENCES 1. Titchmarsh, E. C. Theory of Fourier Integrals Oxford, 1937, p. 76. 2. Jahnke, E., and Emde, F., Tables of HigerFunctions, Leipzig, 1952, p. 26 53 Szego, G.: Orthogonal Polynomials, Amer. Math. Soc. Collogium Publication, 1939. 4. de Haan, B., NQuvelles Tables d' Integrals, New York, 1939. 5. Churchill, RR. V., Modern Qperational Mathematics in Engineering, New York, 1944. 6. Hille, E., "A Class of Reciprocal Functions", Annals of Math. 27 (1925-26), pp. 427-464. 25

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