On Ruby s solid angle formula and some of its generalizations

Transcription

1 On Ruby s soli angle formula an some of its generalizations Samuel Friot arxiv:4.3985v [nucl-ex] 5 Oct 4 Abstract Institut e Physique Nucléaire Orsay Université Paris-Su, INP3-NRS, F-945 Orsay eex, France Using the Mellin-Barnes representation, we show that Ruby s soli angle formula an some of its generalizations may be expresse in a compact way in terms of the Appell F 4 an Lauricella F functions. Keywors: Ruby s formula, Soli angle, Detectors, Mellin-Barnes representation, Lauricella functions. Introuction Ruby s formula, giving the soli angle subtene at a isk source by a coaxial parallel-isk etector [Ruby-Rechen 968], is the following: G = R D R S k e k k J kr S J kr D, wherer S anr D arerespectively theraiusofthesourceanoftheetector, is the istance between the source an the etector an J x is the Bessel function of first kin an orer. Until [onway 6], where an expression in terms of complete an incomplete elliptic integrals has been given, it seems that Ruby s formula ha not been expresse in a close form. A ouble series representation ha been previously mentione in [Ruby 994] but it was conclue in [Pommé 4] that the convergence region of this ouble series is restricte in a way Part of this work has been one at Institut e Physique Nucléaire e Lyon, Université Lyon, INP3-NRS, F-696 Villeurbanne eex, France. Preprint submitte to Nucl. Instr. an Meth. in Phys. Res. A October 3, 8

2 that when the etector is too close to the source, one ha to compute the integral by means of numerical methos. In the next section, we will see that with the help of the Mellin-Barnes MB representation metho see e.g. [Paris-Kaminski ] for an introuction, Ruby s formula may be expresse in a compact way in terms of the Appell function F 4. Generalizations of Ruby s formula, which have been treate mainly in [onway 6] but not always obtaine in close form, will also be consiere within the same approach, in a subsequent section where their expressions in terms of the Lauricella function F will be given.. Ruby s formula The Bessel function J z has the following MB representation, vali for z >, see [Paris-Kaminski ]: J z = iπ c+i c i z s s s s, where, for absolute convergence of the integral, the constant c, which is the real part of s since the chosen integration path is a vertical line in the s- complex plane has to belong to the interval ], [ [Paris-Kaminski ]. Inserting twice this integral representation in eq. we get G = R D 4 c+i c +i s t iπ c i c i RS s t RD s t s t k k s t e k, 3 where c = Rs ], [ an c = Rt ], [. Performing the k-integral leas to the following -fol MB representation: G = RD 4 iπ c+i c +i s t RS RD st s t s t, s t c i c i with the constraint Rs+t <, which is fulfille.

3 To compute this integral one can irectly apply the general metho escribe in [Friot-Greynat ]. If one follows this approach, one will fin three ouble series representations one of them being the one mentione in [Ruby 994] an stuie in [Pommé 4], converging in three ifferent regions of values of the parameters R S, R D an. It is however even simpler to notice that, by using the uplication formula for the Euler gamma function s = π s s one has G = RD c+i s π iπ c i st s t s t s+, 5 c +i c i 3 s t s t RS RD t, 6 which, since 3 = π, is nothing but the MB representation of the Appell F 4 function [Appell-Kampé e Fériet 96] RD G = F 4, 3,,; RS RD, 7 from where, by efinition, one gets the ouble series representation RD m+n 3 m n m+n m+n RS RD G =, 8 m!n! m= n= m n where a m = a+m is the Pochhammer symbol. a The series 8 is the same as the one stuie in [Pommé 4] with convergence region given by R S + R D <. This confirms the analysis performe in [Pommé 4]. The convergence regions of the ouble series representations of the Appell functions are well-known: it is straightforwar to get them by Horn s metho, see for instance [Friot-Greynat ] or [Appell-Kampé e Fériet 96]. 3

4 The well-known analytic continuation formula [Appell-Kampé e Fériet 96] F 4 a,b,c,;x,y = b a ab y a F 4 a,a+,c,a+ b; x y, y + a b ba y b F 4 b+,b,c,b+ a; x y, 9 y an the symmetric relation obtaine by exchanging x with y an c with allow toobtain ouble series representations vali inthe regions R S + < R D an R D + < R S. Series representations vali in other ranges of values of the parameters than those given above may be foun in [Exton 995], where a full analytic continuation stuy has been performe. 3. Generalization The same technique may be use to compute the more general integral I l,,...,m N = N k k l e k J mj kr j, where l an the are such that the integral converges, an R j > for all j {,...,N}. In this case, we use the following MB representation for the Bessel functions vali for z > : J m z = iπ ] [ where c = Rs., Rm c+i c i z m s s s +m s, Notice that this integral is efine only when m > but we will see that one can relax this constraint at the en of the calculations by appeal to analytic continuation. Inserting eq. in eq. we get I l,,...,m N = [ N iπ cj +i c j i s j Rj mj s j s j ] + s j k e k k l+ N s j. 4

5 The k-integral gives l N s j +l + N s j with the constraint R +l+ N s j >. Let us suppose that this constraint is fulfille in all particular cases consiere in [onway 6], it is always possible to satisfy this constraint by an appropriate choice of the c j. Then, applying eq. 5, one may conclue that I l,,...,m N = l N π N mj s j [ Rj mj iπ + l+ N cj +i c j i ] sj Rj s j s j + s j mj s j + l +. 3 As a particular check, it is easy to erive eq. 6 from eq. 3 by putting N =, m = m =, R = R D, R = R S an l =, an multiplying by R D RS. Inthecase where N +l+ an N + l +arepositive numbers, we recognize in eq. 3 the MB representation of the multiple Lauricella function F N moulo an overall factor [Exton 976]. We therefore have I l,,...,m N = l π N F N N m j + l +, N N N + + l+ + l + N mj Rj + l +,+m,...,+m N ; R,..., R N Lauricella functions are the generalizations of Appell functions an F is, obviously, nothing but the Appell F 4 function. The integral representation in eq. 3 has been obtaine with the initial constraint that the an R j are strictly positive for all j {,...,N}. By analytic continuation, it is however possible to inclue other values for these parameters an, among others, the important case where some of the are equal to zero since eq. 3 is efine as long as the quantities 4. 5

6 N + l + l+ an N + are not negative integers. In fact, in all particular situations consiere in [onway 6] these two quantities are positive numbers. Therefore one may irectly use eq. 4 to compute them an this is one in a subsection to follow. Using the multiple series representation of the multiple Lauricella function F N [Exton 976], one obtains: I l,,...,m N = l π... k = k N = N + +l N N m j + l + N + + N l+ N k j N +mj kj k j! N + l + mj Rj N k j k kn N k R RN j... 5 where as before a m is the Pochhammer symbol. Thismultipleseriesconvergesintheregion N R j < [Exton976]. An analytic continuation formula for F N, generalizing eq. 9, is [Exton 976]: F N a,b,c,...,c N ;x,...,x N = c Nb a c N ab x N a F N a,a+ c N,c,...,c N,a+ b; x,..., x N, x N x N c N a b + F N c N ba x N b b+ c N,b,c,...,c N,b+ a; x,..., x N, x N x N x N x N.6 From this equation, one can obtain a series representation vali in the region x x N x N <. As in the case of F4, symmetric expressions an their corresponing convergence regions are straightforwar to obtain. 3.. Particular cases To ease the use of previous equations, we give explicit expressions for the cases where N = an N = 3. We suppose that the constraints iscusse 6

7 above are checke. I l,,m = k k l e k J kr J m kr = l m R R c +i π iπ c i s s R R s s = π l c +i c i s s +m s +m s s s + l+ s s + l + m R R m + m + l+ + m + l + +m +m F 4 + l +, m + l +,+m,+m ; R, R l m R R = π k =k = R m + m + l+ +k +k + m + l ++k +k +m +k +m +k k!k! k R k. 7 I l,,m,m 3 = = π iπ R k k l e k J kr J m kr J m3 kr 3 l m R R3 3 c +i c i c +i c3 +i c i c 3 i m3 s s s 3 R s s s 3 +m s +m s +m 3 s 3 + m 3 s s s 3 + l+ + m 3 s s s 3 + l + 7 s s R R3 s3

8 = l π R m + m + m 3 + l+ m R R3 m3 + m + m 3 + l + +m +m +m 3 F + m 3 + l+, m + m 3 + l +,+m,+m, +m 3 ; R, R, R 3 8 = l m m3 R R R3 π m + m + m 3 + l+ +k +k +k 3 +m +k +m +k +m 3 +k 3 k!k!k 3! k = k = k 3 = + m 3 + l k k k3 ++k +k +k 3 R R R Applications We will now consier several particular cases stuie in [onway 6]. First, the non-coaxial situation where the source isk has a constant surface emissivity. In this case G = R D R S k e k k J kaj kr S J kr D, 9 where a is the offset istance of the isks axes. It is clear from the iscussion above that putting N = 3, l =, m = an m = m 3 = in eq. 4 or in eq. 8 we get RD G = F, 3 a,,,,; RS RD,. The corresponing triple series representation, converging in the region a + R D + R S <, may be obtaine from eq. 5. It matches with the triple series of [Pommé-Paepen 7]. Other triple series, converging in other regions, may easily be foun from eq. 6. An example of non-constant surface emissivity parabolic raial istribution has also been consiere in [onway 6], the corresponing formula 8

9 being G = 4R D k e k RS k J kaj kr S J kr D. This gives N = 3, l =, m =, m = an m 3 = an then RD G = F, 3 a,,,3,; RS RD,. Now, let us consier the non-coaxial ring source case, where G = R D k e k J kaj kr S J kr D. 3 In this case, we have N = 3, l =, m = m = an m 3 =. This leas again to an expression very similar to eq. : RD G = F, 3 a,,,,; RS RD, onclusions In this note, we showe that by using the MB representation metho, it has been possible to compute a family of integrals, see eq., which inclue Ruby s soli angle formula as a particular case. Notice that integrals of this family involving three Bessel functions also appear in other contexts, see for instance [onway, onway 3] an some of the references therein by the same author. In these references, it is claime that these particular integrals have not yet been calculate in close-form. By computing the whole family in this paper, we therefore fill this gap. We also woul like to a the following remark. We have been able to recognize the close-form expression for Ruby s formula, eq. 7, irectly from the MB representation, eq. 6. Then, analytical continuations coul be easily obtaine from the known analytical continuation formulas of the Appell F 4 function. However, let us imagine now that we woul have been in a less simple case where we coul not recognize any known function from the MB representation, or even from a ouble series representation similar to eq. 8, compute from the MB representation or by another metho. 9

10 The real power of the MB metho rests on the fact that, for a large class of MB integrals, we still woul have been able to obtain several series representations, analytic continuations of each other, following the metho presente in [Friot-Greynat ]. References References L. Ruby an J. B. Rechen, Nucl. Instr. an Meth J. T. onway, Nucl. Instr. an Meth. in Phys. Res. A L. Ruby, Nucl. Instr. an Meth. in Phys. Res. A S. Pommé, Nucl. Instr. an Meth. in Phys. Res. A R. B. Paris an D. Kaminski, Encyclopeia Math. Appl. 85, ambrige University Press. S. Friot an D. Greynat, J. Math. Phys P. Appell an J. Kampé e Fériet, Fonctions hypergéométriques et hypersphériques - Polynômes Hermite, Gautiers-Villars et ie 96. H. Exton, J. Phys. A: Math. Gen H. Exton, Multiple hypergeometric functions an applications, New-York, Wiley 976. S. Pommé an J. Paepen, Nucl. Instr. an Meth. in Phys. Res. A J. T. onway, IEEE Trans. Magn J. T. onway, IEEE Trans. Magn