TABLES OF SOME INDEFINITE INTEGRALS OF BESSEL FUNCTIONS

Werner Rosenheinrich 1604015 Ernst - Abbe - Hochschule Jena First variant: 40900 University of Applied Sciences Germany TABLES OF SOME INDEFINITE INTEGRALS OF BESSEL FUNCTIONS Integrals of the type J 0 d or J 0 aj 0 b d are well-known Most of the following integrals are not found in the widely used tables of Gradstein/Ryshik, Bateman/Erdélyi, Abramowitz/ Stegun, Prudnikov/Brychkov/Marichev or Jahnke/Emde/Lösch The goal of this table was to get tables for practicians So the integrals should be epressed by Bessel and Struve functions Indeed, there occured some eceptions Generally, integrals of the type µ J ν d may be written with Lommel functions, see 8], 10-74, or ], III In many cases reccurence relations define more integrals in a simple way Partially the integrals may be found by MAPLE as well In some cases MAPLE gives results with hypergeometric functions, see also ], 96, or 4] Some known integrals are included for completeness Here Z ν denotes some Bessel function or modified Bessel function of the first kind Partially the functions Y ν sometimes called Neumann s functions or Weber s functions and denoted by N ν ] and the Hankel functions H ν 1 and H ν are also considered The same holds for the modified Bessel function of the second kind K ν When a formula is continued in the net line, then the last sign + or - is repeated in the beginning of the new line On page 41 the used special functions and defined functions are described *E* - This sign marks formulas, that were incorrect in previous editions The pages with corrected errors are listed in the errata in the end I wish to epress my thanks to B Eckstein, S O Zafra, Yao Sun and F Nouguier for their remarks 1

References: 1] M Abramowitz, I Stegun: Handbook of Mathematical Functions, Dover Publications, NY, 1970 ] Y L Luke: Mathematical Functions and their Approimations, Academic Press, NY, 1975 ] Y L Luke: Integrals of Bessel Functions, MacGraw-Hill, NY, 196 4] A P Prudnikov, A Bryqkov, O I Mariqev: Integraly i r dy, t : Specialьnye funkcii, Nauka, Moskva, 00; FIZMATLIT, 00 5] E Jahnke, F Emde, F Lösch: Tafeln höherer Funktionen, 6 Auflage, B G Teubner, Stuttgart, 1960 6] I S Gradstein, I M Ryshik: Summen-, Produkt- und Integraltafeln / Tables of Series, Products, and Integrals, Band 1 / Volume 1, Verlag Harri Deutsch, Thun Frankfurt/M, 1981 7] I S Gradstein, I M Ryshik: Summen-, Produkt- und Integraltafeln / Tables of Series, Products, and Integrals, Band / Volume, Verlag Harri Deutsch, Thun Frankfurt/M, 1981 8] G N Watson: A Treatise on the Theory of Bessel Functions, Cambridge, University Press, 19 / 1995 9] P Humbert: Bessel-integral functions, Proceedings of the Edinburgh Mathematical Society Series, 19, :76-85 10] B A Peavy, Indefinite Integrals Involving Bessel Functions JOURNAL OF RESEARCH of the National Bureau of Standards - B, vol 718, Nos and, April - September 1967, pp 11-141 11] B G Korenev: Vvedenie v teori besselevyh funkci i, Nauka, Moskva, 1971 1] S K H Auluck, Some integral identities involving products of general solutions of Bessel s equation of integral order, arivorg/abs/10064471

1 1 n Z ν with integer values of n Contents 1 Integrals with one Bessel function 111 n Z 0 7 11 n+1 Z 0 11 11 n 1 Z 0 1 114 n Z 1 15 115 n Z 1 17 116 n+1 Z 1 19 117 n Z ν, ν > 1 118 Higher Antiderivatives 44 1 Elementary Function and Bessel Function 11 n+1/ J ν 46 a Z ν d 46 b Integrals 51 c Recurrence { Formulas } 56 1 n e ± Iν 57 K ν a Integrals with e 57 b Integrals with e 59 { } sinh 1 n I ν 6 14 n cosh { sin cos } J ν 65 15 n e a J ν 68 a General facts 68 b The Case a > 0 69 c The Case a < 0 77 d Integrals 8 e Special Cases 87 16 n 1/ { sin cos } J ν 90 17 n 1/ e ± I ν 9 a n 1/ e I ν 9 b n 1/ e I ν 94 c General formulas 95 18 n+1 ln Z 0 100 19 n ln Z 1 10 110 n+ν ln Z ν 104 a The Functions Λ k and Λ k 104 b Basic Integrals 108 c Integrals of n ln Z 0 111 c Integrals of n+1 ln Z 1 114 111 n e ± ln Z ν 117 11 Some Cases of n e ± Z 0 α 11 1 Special Function and Bessel Function 11 Orthogonal Polynomials 1 a Legendre Polynomials P n 1 b Chebyshev Polynomials T n 18

c Chebyshev Polynomials U n 14 d Laguerre Polynomials L n 146 e Hermite Polynomials H n 150 1 Eponential Integral 155 1 Sine and Cosine Integral 157 Products of two Bessel Functions 1 Bessel Functions with the the same Argument : 11 n+1 Zν 160 1 n Zν 165 1 n Zν 168 a The Functions Θ and Ω 168 b Integrals 174 14 n Z 0 Z 1 179 15 n+1 Z 0 Z 1 181 16 n+1 Z 0 Z 1 184 17 n+1 J 0 I 0 186 18 n J 0 I 1 187 19 n J 1 I 0 188 110 n+1 J 1 I 1 189 111 n+1] J µ Y ν 190 a n+1 J 0 Y 0 190 b n J 0 Y 0 191 c n J 0 Y 1 19 d n 1 J 0 Y 1 19 e n J 1 Y 0 194 f n 1 J 0 Y 0 195 g n+1 J 1 Y 1 196 h n J 1 Y 1 197 Bessel Functions with different Arguments α and β : 1 n+1 Z ν αz ν β 198 a ν = 0 198 b ν = 1 08 n Z 0 αz 1 β 19 n Z ν αz ν β 4 a Basic Integrals 4 b Integrals 44 4 n+1 Z 0 αz 1 β 51 5 n+1 J 0 αi 0 β 55 6 n J 0 αi 1 β 57 7 n J 1 αi 0 β 59 8 n+1 J 1 αi 1 β 61 9 n+1 J ν αy ν β 6 Bessel Functions with different Arguments and + α 1 1 Z ν Z 1 + α and + α] 1 Z 1 + αz 1 d 64 4 Elementary Function and two Bessel Functions 41 n+1 ln Z ν d and n ln Z 0 Z 1 d 65 4 n ln Z ν Z ν d 7 a Integrals with 4n+ ln J 0 Z 0 7 4

a Integrals with 4n+1 ln J 1 Z 1 7 c Integrals with n+1 ln I ν K ν 74 d Integrals with n+ ln I ν K 1 ν 76 4 Some Cases of n ln Z ν Zν α d 80 44 1 ep/ sin / cos Z ν Z 1 d 88 45 Some Cases of n e α Z ν Z 1 d 89 46 Some Cases of { } sin / cos n α Z sinh / cosh µ Zν β d 9 a { } sin n α Z µ Z ν β d 9 cos b { sinh n cosh } α Z µ Z ν β d 98 Products of three Bessel Functions 1 n Z0 m Z1 m 0 a Basic Integral Z0 0 b Basic Integral Z 0 Z1 06 c Basic Integral Z1 09 d n Z0 17 e n Z0 Z 1 0 f n Z 0 Z1 g n Z1 5 h Recurrence Relations 9 n Z κ α Z µ β Z ν γ 0 a n Z κ Z µ Z ν 0 b n Z κ α Z µ β Z ν α + β 41 c n Z κ α Z µ β Z ν α ± β 67 4 Products of four Bessel Functions 41 m Z0 n Z1 4 n 70 a Eplicit Integrals 70 b Basic Integral Z0 4 71 c Basic Integral Z0 Z1 70 d Basic Integral Z1 4 70 e Integrals of m Z0 4 8 f Integrals of m Z0 Z 1 85 g Integrals of m Z0 Z1 89 h Integrals of m Z 0 Z1 9 i Integrals of m Z1 4 95 j Recurrence relations 95 5 Quotients 411 51 Denominator p Z 0 + q Z 1 401 a Typ f Z µ /p Z 0 + q Z 1 ] 401 5 Denominator p Z 0 + q Z 1 ] 401 a Typ f Z µ /p Z 0 + q Z 1 ] 401 b Typ f Z0 n Z1 n /p Z 0 + q Z 1 ], n = 0, 1, 404 5 Denominator p Z 0 + q Z 1 ] 405 a Typ f Z µ /p Z 0 + q Z 1 ] 405 54 Denominator p Z 0 + q Z 1 ] 4 406 5

a Typ f Z µ /p Z 0 + q Z 1 ] 4 406 55 Denominator p Z0 + q Z1 408 a Typ f Z0 n Z1 n /p Z0 + q Z1 ], n = 0, 1, 408 b Typ f Z0 n Z1 n /p Z0 + q Z 0 Z 1 + rz1 ], n = 0, 1, 408 56 Denominator a Z 0 + b Z 1 + p Z0 0 Z 1 + r Z1 408 6 Miscellaneous 411 7 Used special functions and defined functions 41 8 Errata 41 6

1 Integrals with one Bessel Function: See also 10], 11 n Z ν with integer values of n 111 Integrals of the type n Z 0 d Let Φ = π J 1 H 0 J 0 H 1 ], where H ν denotes the Struve function, see 1], chapter 1117, 1118 and 1 And let Ψ = π I 0 L 1 I 1 L 0 ] be defined with the modified Struve function L ν Furthermore, let Φ Y = π Y 1 H 0 Y 0 H 1 ], Φ 1 π ] H = H 1 1 H 0 H 1 0 H 1, Φ π ] H = H 1 H 0 H 0 H 1 and Ψ K = π K 0 L 1 + K 1 L 0 ] In the following formulas J ν may be substituted by Y ν and simultaneously Φ by Φ Y or H ν p, p = 1, and Φ p H Well-known integrals: J 0 d = J 0 + Φ = Λ 0 I 0 d = I 0 + Ψ = Λ 0 K 0 d = K 0 + Ψ K The new-defined function Λ 0 is discussed in 110 a on page 104 and so is Λ 0 on page 106 See also 1], 111 H p 0 Y 0 d = Y 0 + Φ Y d = Hp 0 + Φp H, p = 1, J 0 d = J 1 Φ I 0 d = I 1 + Ψ K 0 d = K 1 + Ψ K 4 J 0 d = 4 9 J 1 + J 0 + 9Φ 4 I 0 d = 4 + 9 I 1 I 0 + 9Ψ 4 K 0 d = 4 + 9 K 1 K 0 + 9Ψ K E 7

6 J 0 d = 6 5 4 + 5 J 1 + 5 5 75 J 0 5Φ 6 I 0 d = 6 + 5 4 + 5 I 1 5 5 + 75 I 0 + 5Ψ 6 K 0 d = 6 + 5 4 + 5 K 1 5 5 + 75 K 0 + 5Ψ K and so on 8 J 0 d = 8 49 6 + 1 5 4 11 05 J 1 + 7 7 45 5 + 675 J 0 + 11 05Φ 8 I 0 d = 8 + 49 6 + 1 5 4 + 11 05 I 1 7 7 + 45 5 + 675 I 0 + 11 05Ψ 10 J 0 d = 10 81 8 + 969 6 99 5 4 + 89 05J 1 + +9 9 567 7 + 19 845 5 97 675 J 0 89 05Φ 10 I 0 d = 10 + 81 8 + 969 6 + 99 5 4 + 89 05I 1 9 9 + 567 7 + 19 845 5 + 97 675 I 0 + 89 05Ψ 1 J 0 d = 11 11 1 089 9 + 68 607 7 401 45 5 + 6 018 675 J 0 + Let + 1 11 10 + 9 801 8 480 49 6 + 1 006 5 4 108 056 05 J 1 + 108 056 05Φ 1 I 0 d = 1 + 11 10 + 9 801 8 + 480 49 6 + 1 006 5 4 + 108 056 05 I 1 11 11 + 1 089 9 + 68 607 7 + 401 45 5 + 6 018 675 I 0 + 108 056 05Ψ n!! = and n!! = 1 in the case n 0 General formulas: and + + n J 0 d = n 1 k=0 n { 4 n n, n = m 1 5 n n, n = m + 1 n 1 k n 1!!] n k 1 J 0 + n 1 k!!] n k!!] k=0 ] 1 k n 1!! n k J 1 + 1 n n 1!!] Φ = n 1 k!! = n 1 k=0 k=0 1 k n!] n k! n k 1! n k 1 k+1 n! J 0 + n k! n k! ] 1 k n! n k! k n k n! n k! n k=0 n I 0 d = n 1 n 1!!] n k 1 n 1 k!!] n k!!] ] n! J 1 + 1 n n Φ n! ] n 1!! n k I 1 n 1 k!! k=0 I 0 + n 1!!] Ψ = 8

n k=0 Recurrence formulas: = n 1 ] n! n k! k n k I 1 n! n k! k=0 n!] n k! n k 1! n k 1 k+1 n! n k! n k! ] n! I 0 + n Ψ n! n+ J 0 d = n + 1 n+1 J 0 + n+ J 1 n + 1 n+ I 0 d = n + 1 n+1 I 0 + n+ I 1 + n + 1 n+ K 0 d = n + 1 n+1 K 0 n+ K 1 + n + 1 n J 0 d n I 0 d n K 0 d In the case n < 0 the previous formulas give J0 d = J 1 + 1 J 0 Φ I0 d = 1 I 0 I 1 + Ψ K0 d = 1 K 0 + K 1 + Ψ K J0 4 d = 1 4 + ] 9 J 0 1 J 1 + Φ I0 4 d = 1 4 ] 9 I 0 + 1 I 1 + Ψ K0 4 d = 1 4 ] 9 K 0 + + 1 K 1 + Ψ K J0 6 d = 1 4 + 9 5 4 J 1 6 + 4 + 45 5 I0 6 d = 1 6 4 45 5 5 I 0 4 + + 9 4 6 4 45 5 K 0 + 4 + ] + 9 4 K 1 + Ψ K K0 6 d = 1 5 J0 8 d = I0 8 d = ] J 0 Φ ] I 1 + Ψ and so on 1 8 + 6 4 + 45 1 575 11 05 7 J 0 6 4 + 9 ] 5 6 J 1 + Φ 1 8 6 4 45 1 575 11 05 7 I 0 6 + 4 + 9 + 5 6 J0 1 8 6 + 9 4 5 + 11 05 10 d = 89 05 8 J 1 10 + 8 6 + 45 4 1 575 + 99 5 I0 10 d = 1 89 05 9 ] J 0 Φ 10 8 6 45 4 1 575 99 5 I 0 9 ] I 1 + Ψ E E 9

8 + 6 + 9 4 + 5 ] + 11 05 8 I 1 + Ψ J0 1 d = I0 1 d = 1 108 056 05 1 1 + 10 8 + 45 6 1 575 4 + 99 5 9 8 75 108 056 05 11 J 0 10 8 + 9 6 5 4 + 11 05 ] 89 05 J 1 + Φ 10 1 10 8 45 6 1 575 4 99 5 9 8 75 I 0 11 10 + 8 + 9 6 + 5 4 + 11 05 + 89 05 10 I 1 + Ψ General formula: With n!! as defined on page 8 holds J0 d 1 n n 1 n = n 1!!] + 1 k k + 1!! k 1!! k 1 J 0 k=0 ] n 1 1 k k + 1!!] k J 1 + Φ = k=0 = 1n n n! n! 1 n k=0 { n 1 + 1 k k +! k! k+1 k + 1! k! k+1 J 0 1 k k+ k=0 ] } k +! k+1 J 1 + Φ k + 1! With obviously modifications one gets the the formulas for the integrals n I 0 d and n K 0 d ] 10

11 Integrals of the type n+1 Z 0 d In the following formulas J ν may be substituted by Y ν or H p ν, p = 1, J 0 d = J 1 I 0 d = I 1 K 0 d = K 1 J 0 d = J 0 + 4 J 1 ] I 0 d = + 4 I 1 I 0 ] K 0 d = + 4 K 1 + K 0 ] 5 J 0 d = 4 J 0 + 4 16 + 64 J 1 ] 5 I 0 d = 4 + 16 + 64 I 1 4 + I 0 ] 5 K 0 d = 4 + 16 + 64 K 1 + 4 + K 0 ] 7 J 0 d = 6 5 144 + 1 15 J 0 + 6 6 4 + 576 04 J 1 ] 7 I 0 d = 6 + 6 4 + 576 + 04 I 1 6 5 + 144 + 1 15 I 0 ] 7 K 0 d = 6 + 6 4 + 576 + 04 K 1 + 6 5 + 144 + 1 15 K 0 ] 9 J 0 d = = 8 7 84 5 + 9 16 7 78 J 0 + 8 64 6 + 04 4 6 864 + 147 456 J 1 ] 9 I 0 d = = 8 + 64 6 + 04 4 + 6 864 + 147 456 I 1 8 7 + 84 5 + 9 16 + 7 78 I 0 ] 9 K 0 d = = 8 + 64 6 + 04 4 + 6 864 + 147 456 K 1 + 8 7 + 84 5 + 9 16 + 7 78 K 0 ] Let m J 0 d = P m J 0 + Q m J 1 ] and m I 0 d = Q mi 1 P mi 0 ], m K 0 d = Q mk 1 + P mk 0 ], then holds P 11 = 10 9 800 7 + 8400 5 91600 + 77800 Q 11 = 10 100 8 + 6400 6 0400 4 + 686400 14745600 P11 = 10 9 + 800 7 + 8400 5 + 91600 + 77800 Q 11 = 10 + 100 8 + 6400 6 + 0400 4 + 686400 + 14745600 *E* 11

P 1 = 1 11 1440 9 + 11500 7 559600 5 + 1710400 10616800 Q 1 = 1 144 10 + 14400 8 91600 6 + 177600 4 50841600 + 166400 P1 = 1 11 + 1440 9 + 11500 7 + 559600 5 + 1710400 + 10616800 Q 1 = 1 + 144 10 + 14400 8 + 91600 6 + 177600 4 + 50841600 + 166400 P 15 = 14 1 5 11 + 840 9 57900 7 + 108801600 5 60118400 + 0808990700 Q 15 = 14 196 1 + 84 10 8400 8 + 1806600 6 650809600 4 + 10404495600 416179814400 P 15 = = 14 1 5 11 + 840 9 + 57900 7 + 108801600 5 + 60118400 + 0808990700 *E* Q 15 = 14 + 196 1 + 84 10 + 8400 8 + 1806600 6 + 650809600 4 + 10404495600 + 416179814400 Recurrence formulas: n+1 J 0 d = n n J 0 + n+1 J 1 4n n+1 I 0 d = n n I 0 + n+1 I 1 + 4n n+1 K 0 d = n n K 0 n+1 K 1 + 4n k=0 n 1 J 0 d General formula: With n!! as defined on page 8 holds n 1 n+1 J 0 d = 1 k n!!] n k n k!!] n k!!] = n 1 1 k k=0 n ] + 1 k n!! n+1 k J 1 = n k!! k=0 n J 0 + k+1 n! n k n k! n k 1! k=0 n 1 I 0 d n 1 K 0 d J 0 + ] 1 k k n! n+1 k J 1 n k! With obviously modifications one gets the the formulas for the integrals n+1 I 0 d and n+1 K 0 d *E* 1

11 Integrals of the type n 1 Z 0 d The basic integral J0 d can be epressed by 0 1 J 0 t t J 0 t dt dt or = Ji 0, t see 1], equation 11119 and the following formulas There are given asymptotic epansions and polynomial approimations as well Tables of these functions may be found by 1], 111] or 11] The function Ji 0 is introduced and discussed in 9] For fast computations of this integrals one should use approimations with Chebyshev polynomials, see ], tables 9 I got the information from F Nouguier, that there is an error in a formula in 9], p 78 The true formula is The power series in Ji 0 ln = sin π π I0 d can be used without numerical problems γ ln + sin π π = ln + k=1 s=1 1 s 1 s Ji 0s ln s] 1 k k k! E In the following formulas J ν may be substituted by Y ν or H ν p, p = 1, J0 d = J 0 + J 1 4 1 J0 d 4 I0 d = I 0 I 1 4 + 1 I0 d 4 J0 d 1 5 = 1 4 4 J 0 + 1 64 + 1 16 J 1 + 1 J0 d 64 I0 d 1 5 = + 1 1 4 4 I 0 64 + 1 16 I 1 + 1 I0 d 64 J0 d 7 = 4 + 8 19 115 6 J 0 + 4 4 + 64 04 5 J 1 1 J0 d 04 I0 d 7 = 4 + 8 + 19 115 6 I 0 4 + 4 + 64 04 5 I 1 + 1 I0 d 04 J0 d 9 = = 6 8 4 + 19 916 778 8 J 0 + 6 + 4 4 64 + 04 147456 7 J 1 + I0 d 9 = = 6 + 8 4 + 19 + 916 778 8 I 0 6 + 4 4 + 64 + 04 147456 7 I 1 + J0 d 11 = 8 + 8 6 19 4 + 916 7780 E 77800 10 J 0 + + 8 4 6 + 64 4 04 + 147456 1 J0 d 14745600 9 J 1 14745600 I0 d 11 = 8 + 8 6 + 19 4 + 916 + 7780 77800 10 I 0 8 + 4 6 + 64 4 + 04 + 147456 1 14745600 9 I 1 + 14745600 1 J0 d 147456 1 I0 d 147456 I0 d 1

Descending recurrence formulas: n 1 J 0 d = 1 4n n 1 I 0 d = 1 4n n+1 J 1 n n J 0 n+1 I 1 n n I 0 + General formula: With n!! as defined on page 8 holds J0 d n+1 = ] n+1 J 0 d ] n+1 I 0 d { n 1 n 1 = 1n k k +!! k!! } n!!] 1 k+ J 0 1 k k!!] J0 d k+1 J 1 + = E k=0 k=0 { n 1 n 1 = 1n n n! 1 k k+1 k + 1! k! k+ J 0 1 k k k! } J0 d k+1 J 1 + E k=0 With obviously modifications one gets the the formula for the integral n 1 I 0 d k=0 14

114 Integrals of the type n Z 1 d In the following formulas J ν may be substituted by Y ν or H p ν, p = 1, J 1 d = J 0 I 1 d = I 0 K 1 d = K 0 J 1 d = J 1 J 0 ] I 1 d = I 0 I 1 ] K 1 d = K 0 + K 1 ] 4 J 1 d = 4 16 J 1 8 J 0 ] 4 I 1 d = + 8 I 0 4 + 16 I 1 ] 4 K 1 d = + 8 K 0 + 4 + 16 K 1 ] 6 J 1 d = 6 4 96 + 84 J 1 5 4 + 19 J 0 ] 6 I 1 d = 5 + 4 + 19 I 0 6 4 + 96 + 84 I 1 ] 6 K 1 d = 5 + 4 + 19 K 0 + 6 4 + 96 + 84 K 1 ] 8 J 1 d = = 8 6 88 4 + 4 608 18 4 J 1 7 48 5 + 1 15 9 16 J 0 ] 8 I 1 d = = 7 + 48 5 + 1 15 + 9 16 I 0 8 6 + 88 4 + 4 608 + 18 4 I 1 ] 8 K 1 d = = 7 + 48 5 + 1 15 + 9 16 K 0 + 8 6 + 88 4 + 4 608 + 18 4 K 1 ] 10 J 1 d = 10 8 640 6 + 040 4 68 640 + 1 474 560 J 1 9 80 7 + 840 5 9 160 + 77 80 J 0 ] 10 I 1 d = 9 + 80 7 + 840 5 + 9 160 + 77 80 I 0 10 8 + 640 6 + 040 4 + 68 640 + 1 474 560 I 1 ] 10 K 1 d = 9 + 80 7 + 840 5 + 9 160 + 77 80 K 0 + 15

+10 8 + 640 6 + 040 4 + 68 640 + 1 474 560 K 1 ] Let m J 1 d = Q m J 1 P m J 0 ] and m I 1 d = P mi 0 Q mi 1 ], m K 1 d = P mi 0 + Q mi 1 ], then holds P 1 = 11 10 9 + 9600 7 460800 5 + 1105900 8847600 Q 1 = 1 10 100 8 + 76800 6 764800 4 + 446800 17694700 P 1 = 11 + 10 9 + 9600 7 + 460800 5 + 1105900 + 8847600 Q 1 = 1 10 + 100 8 + 76800 6 + 764800 4 + 446800 + 17694700 *E* P 14 = 1 168 11 + 0160 9 161800 7 + 77414400 5 1857945600 + 1486564800 Q 14 = 14 1 016 10 + 01600 8 190400 6 + 464486400 4 74178400 + 97719600 P14 = 1 + 168 11 + 0160 9 + 161800 7 + 77414400 5 + 1857945600 + 1486564800 Q 14 = 14 1 + 016 10 + 01600 8 + 190400 6 + 464486400 4 + 74178400 + 97719600 Recurrence formulas: n+ J 1 d = n+ J 0 + n + n+1 J 1 4nn + 1 n J 1 d n+ I 1 d = n+ I 0 n + n+1 I 1 + 4nn + 1 n+ K 1 d = n+ K 0 n + n+1 K 1 + 4nn + 1 n I 1 d n K 1 d General formula: With n!! as defined on page 8 holds n 1 n k n!!] n!!] n 1 k J 1 d = 1 n k!!] = n 1 k=0 1 k k=0 n 1 k=0 n 1 n!! n!! n k n k!!] n k!!] J 0 = 1 k k+1 n! n 1! n 1 k n 1 k!] J 1 k=0 1 k k n! n 1!! n k J 0 n k! n 1 k! J 1 With obviously modifications one gets the the formulas for the integrals n I 1 d and n K 1 d 16

115 Integrals of the type n Z 1 d About the integrals see 11, page 1 J0 d and I0 d In the following formulas J 0 may be substituted by Y 0 and simultaneously J 1 by Y 1 J1 d = 1 J 1 + 1 J0 d I1 d = 1 I 1 + 1 I0 d J1 d 4 = 1 8 J 0 + 4 16 J 1 1 J0 d 16 I1 d 4 = 1 8 I 0 + 4 16 I 1 + 1 I0 d 16 J1 d 6 = = 8 19 4 J 0 + 4 + 4 64 84 5 J 1 + 1 J0 d 84 I1 d 6 = + 8 19 4 I 0 4 + 4 + 64 84 5 I 1 + 1 I0 d 84 J1 d 8 = = 4 + 8 19 916 6 J 0 + 6 4 4 + 64 04 184 7 J 1 1 J0 d 184 I1 d 8 = 4 + 8 + 19 916 6 I 0 6 + 4 4 + 64 + 04 184 7 I 1 + 1 I0 d 184 J1 d 10 = Recurrence formulas: = 6 8 4 + 19 916 7780 8 J 0 + 8 + 4 6 64 4 + 04 147456 1474560 9 J 1 + 1 J0 d + 1474560 I1 d 10 = = 6 + 8 4 + 19 + 916 7780 8 I 0 8 + 4 6 + 64 4 + 04 + 147456 1474560 9 I 1 + 1 I0 d + 1474560 E E J1 d J 0 n+ = 4nn + 1 n I1 d I 0 n+ = 4nn + 1 n J 1 n + n+1 1 4nn + 1 I 1 n + n+1 + 1 4nn + 1 J1 d n I1 d n 17

General formula: With n!! as defined on page 8 holds J1 d 1 n+1 n = n!! n!! { n n 1 k k +!! k!! } 1 k+ J 0 1 k k!!] J0 d k+1 J 1 + = k=0 k=0 1 n+1 = n 1 n! n 1! n 1 1 k k+1 k + 1! k! k+ J 0 1 k k k! ] J0 d k+1 J 1 + n k=0 With obviously modifications one gets the the formula for the integral n I 1 d k=0 18

116 Integrals of the type n+1 Z 1 d Φ, Φ Y, Ψ and Ψ K are the same as in 111, page 7 In the following formulas J ν may be substituted by Y ν and simultaneously Φ by Φ Y or H ν p, p = 1, and Φ p H J 1 d = Φ I 1 d = Ψ K 1 d = Ψ K J 1 d = J 1 J 0 Φ I 1 d = I 1 + I 0 Ψ K 1 d = K 1 K 0 + Ψ K 5 J 1 d = 5 4 45 J 1 5 15 J 0 + 45Φ 5 I 1 d = 5 4 + 45 I 1 + 5 + 15 I 0 45Ψ 5 K 1 d = 5 4 + 45 K 1 5 + 15 K 0 + 45Ψ K 7 J 1 d = 7 6 175 4 + 1 575 J 1 7 5 5 + 55 J 0 1 575Φ E 7 I 1 d = 7 6 + 175 4 + 1 575 I 1 + 7 + 5 5 + 55 I 0 1 575Ψ 7 K 1 d = 7 6 + 175 4 + 1 575 K 1 7 + 5 5 + 55 K 0 + 1 575Ψ K 9 J 1 d = = 9 8 441 6 + 11 05 4 99 5 J 1 9 6 7 + 05 5 075 J 0 + 99 5 Φ 9 I 1 d = = 9 8 + 441 6 + 11 05 4 + 99 5 I 1 + 9 + 6 7 + 05 5 + 075 I 0 99 5 Ψ 9 K 1 d = = 9 8 + 441 6 + 11 05 4 + 99 5 K 1 9 + 6 7 + 05 5 + 075 K 0 + 99 5 Ψ General formula: With n!! as defined on page 8 holds n 1 n+1 k n + 1!! n 1!! n k J 1 d = 1 n 1 k!!] J 1 n 1 k=0 k=0 k n + 1!! n 1!! n+1 k 1 J 0 + 1 n n + 1!! n 1!! Φ = n + 1 k!! n 1 k!! 19

n 1 k=0 n 1 = 1 k n +! n! n k!] n k k+1 n + 1! n! n k!] J 1 k=0 k n +! n! n + 1 k! n k! n+1 k 1 k J 0 + n + 1! n! n + k! n k! + 1 n n +! n! n+1 n + 1! n! Φ With obviously modifications one gets the the formulas for the integrals n+1 I 1 d and n+1 K 1 d Recurrence formulas: n+1 J 1 d = n+1 J 0 + n + 1 n J 1 n 1n + 1 n 1 J 1 d n+1 I 1 d = n+1 I 0 n + 1 n I 1 + n 1n + 1 n+1 K 1 d = n+1 K 0 n + 1 n K 1 + n 1n + 1 n 1 I 1 d n 1 K 1 d Descending: J1 d J 0 n+1 = 4n 1 n 1 J 1 n + 1 n 1 J1 d 4n 1 n 1 I1 d I 0 n+1 = 4n 1 n 1 I 1 n + 1 n + 1 I1 d 4n 1 n 1 K1 d K 0 n+1 = 4n 1 n 1 K 1 n + 1 n + 1 K1 d 4n 1 n 1 J1 d = J 0 J 1 + Φ I1 d = I 0 I 1 + Ψ K1 d = K 0 K 1 Ψ K J1 d = 1 ] 1 J 1 + 1 J 0 Φ I1 d = 1 ] + 1 I 1 + 1 I 0 + Ψ K1 d = 1 ] + 1 K 1 1 K 0 Ψ K J1 5 d = 1 4 + 45 J 0 4 ] + 9 4 J 1 + Φ I1 5 d = 1 4 45 I 0 4 + ] + 9 4 I 1 + Ψ K1 5 d = 1 4 45 K 0 4 + ] + 9 4 K 1 Ψ K J1 7 d = 1 6 4 + 9 5 1 575 6 J 1 6 + 4 ] + 45 5 J 0 Φ 0

I1 7 d = 1 1 575 K1 7 d = 1 1 575 6 + 4 + 9 + 5 6 + 4 + 9 + 5 = 1 8 + 6 4 + 45 1 575 99 5 = 1 8 6 4 45 1 575 99 5 = 1 8 6 4 45 1 575 99 5 J1 11 d = 6 I 1 + 6 4 45 5 6 K 1 6 4 45 5 J1 9 d = 7 J 0 8 6 + 9 4 5 + 11 05 8 I1 9 d = 7 I 0 8 + 6 + 9 4 + 5 + 11 05 8 K1 9 d = ] I 0 + Ψ ] K 0 Ψ k ] J 1 + Φ ] I 1 + Ψ 7 K 0 8 + 6 + 9 4 + 5 + 11 05 8 1 10 8 + 9 6 5 4 + 11 05 89 05 9 8 75 10 J 1 10 + 8 6 + 45 4 1 575 + 99 5 I1 11 d = 1 9 8 75 K1 11 d = 9 ] J 0 Φ 10 + 8 + 9 6 + 5 4 + 11 05 + 89 05 I 1 + 10 + 10 8 6 45 4 1 575 99 5 9 ] I 0 + Ψ 1 10 + 8 + 9 6 + 5 4 + 11 05 + 89 05 9 8 75 10 10 8 6 45 4 1 575 99 5 9 K 0 + Ψ K General formula: With n!! as defined on page 8 holds { J1 d 1 n n+1 = + n + 1!! n 1!! n 1 k=0 ] ] I 1 Ψ K K 1 1 k k + 1!! k 1!! k+1 J 0 } n 1 k k + 1!!] 1 1 J 1 + Φ = k=0 k+ { = n+1 n 1 n + 1! n! 1 k k +! k! n +! n! k+1 k + 1! k! k+1 J 0 k=0 } n 1 1 1 k k +!] k+ k + 1!] k+ J 1 + Φ k=0 With obviously modifications one gets the the formulas for the integrals n 1 I 1 d and n 1 K 1 d 1

117 Integrals of the type n Z ν d, ν > 1 : From the well-known recurrence relations one gets immadiately J ν+1 d = J ν + J ν 1 d and I ν+1 d = I ν I ν 1 d With this formulas follows J ν t dt = Λ 0 0 κ=1 n J κ 1, J ν+1 t dt = 1 J 0 0 κ=1 n J κ 0 I ν t dt = 1 n Λ 0+ n 1 n+κ I κ 1, κ=1 0 I ν+1 t dt = 1 n I 0 1]+ n 1 n+κ I κ The integrals Λ 0 and Λ 0 are defined on page 7 and discussed on page 104 and 106 Holds n n Y ν d = Y 0 + Φ Y Y κ 1, Y ν+1 d = Y 0 Y κ 1 H 1 ν H ν d = H1 0 + Φ1 d = H 0 + Φ κ=1 H n κ=1 H n κ=1 H 1 κ 1, H κ 1, H 1 ν+1 d = H1 H ν+1 d = H { K ν d = 1 n K 0 + π } K 0L 1 + K 1 L 0 ] + K ν+1 d = 1 n+1 K 0 + About the functions Φ Y, Φ 1 H, Φ H see page 7 Further on, holds 0 0 0 0 t J ν+1 t dt = ν + 1Λ 0 J 0 + ] ν 1 t J ν t dt = J 1 + J κ+1 κ=1 t I ν+1 t dt = 1 ν+1 ν + 1Λ 0 I 0 κ=1 κ=1 κ=1 0 n κ=1 0 n κ=1 H 1 κ 1 H κ 1 n 1 n+κ K κ 1, κ=1 n 1 n+κ+1 K κ κ=1 ] ν J κ κ=1 ν 1 4 κ=0 ν 1 + ν1 J 0 ] 4 κ=1 ν κj κ+1 ν κj κ ] ν ν 1 1 κ I κ 4 1 κ ν κi κ+1 ] ν 1 ν 1 t I ν t dt = 1 I ν+1 1 + 1 κ I κ+1 + ν1 I 0 ] 4 1 κ ν κi κ Some of the previous sums may cause numerical problems, if is located near 0 For instance, the sum 0 gives with = 0 t I 6 t dt = J 1 J + J 5 + 6 6J 0 + 8J 4J 4 0045 508 15 001 0000 9 40 714 + 0000 000 81 114 + 6 615 761 76 110 + 0090 676 901 88 0000 084 755 400 = 616 185 44 40 616 185 44 4 = 0000 000 000 179, κ=1 κ=0 κ=1

which means the loss of 10 decimal digits For that reason the value of such integrals should be computed by the power series or other formulas See also the following remark In the following the integrals are epressed by Z 0 and Z 1 Integrals with n 4 are written eplicitely: at first n = 0, 1,,, 4, after them n = 1, In the other cases the functions P ν n, Q n ν and the coefficients R n ν, S ν n describe the integral n J ν d = P ν n J 0 + Q n ν J 1 + R n ν Λ 0 + S ν n J0 d Furthermore, let n I ν d = P n, ν I 0 + Q n, ν I 1 + R n, ν Λ 0 + S ν n, I0 d Concerning 1 Z 0 d see 11, page 1 Simple recurrence formula: n J ν+1 d = ν n 1 J ν d n J ν 1 d n I ν+1 d = ν n 1 J ν d + n J ν 1 d The integrals of n Z 0 and n Z 1 to start this recurrences are already described Remark: Let F ν m denote the antiderivative of m Z ν as given in the following tables They do not eist in the point = 0 in the case ν + m < 0 However, even if ν + m 0 the value of F ν m 0 sometimes turns out to be a limit of the type For instance, holds J d = J 0 J 1 = F with lim F = 1 0 8 With L ν,m = lim 0 F ν m for the Bessel functions J ν and L ν,m for the modified Bessel functions I ν one has the following limits in the tables of integrals The values L ν,m = 0 are omitted: L, 1 = 1/, L, 1 = 1/ L,0 = 1, L, = 1/8; L,0 = 1, L, = 1/8 L 4,1 = 4, L 4, 1 = 1/4, L 4, = 1/48; L 4,1 = 4, L 4, 1 = 1/4, L 4, = 1/48 L 5, = 4, L 5,0 = 1, L 5, = 1/4, L 5, 4 = 1/84; L 5, = 4, L 5,0 = 1, L 5, = 1/4, L 5, 4 = 1/84 L 6, = 19, L 6,1 = 6, L 6, 1 = 1/6, L 6, = 1/19, L 6, 5 = 1/840; L 6, = 19, L 6,1 = 6, L 6, 1 = 1/6, L 6, = 1/19, L 6, 5 = 1/840 L 7,4 = 190, L 7, = 48, L 7,0 = 1, L 7, = 1/48, L 7, 4 = 1/190, L 7, 6 = 1/46080; L 7,4 = 190, L 7, = 48, L 7,0 = 1, L 7, = 1/48, L 7, 4 = 1/190, L 7, 6 = 1/46080 L 8,5 = 040, L 8, = 480, L 8,1 = 8, L 8, 1 = 1/8, L 8, = 1/480, L 8, 5 = 1/040; L 8,5 = 040, L 8, = 480, L 8,1 = 8, L 8, 1 = 1/8, L 8, = 1/480, L 8, 5 = 1/040 L 9,6 = 560, L 9,4 = 5760, L 9, = 80, L 9,0 = 1, L 9, = 1/80, L 9, 4 = 1/5760, L 9, 6 = 1/560; L 9,6 = 560, L 9,4 = 5760, L 9, = 80, L 9,0 = 1, L 9, = 1/80, L 9, 4 = 1/5760, L 9, 6 = 1/560 L 10,7 = 5160960, L 10,5 = 80640, L 10, = 960, L 10,1 = 10, L 10, 1 = 1/10, L 10, = 1/960, L 10, 5 = 1/80640; L 10,7 = 5160960, L 10,5 = 80640, L 10, = 960, L 10,1 = 10, L 10, 1 = 1/10, L 10, = 1/960, L 10, 5 = 1/80640

In the described cases of limits of the type the numerical computation of F ν m causes difficulties, if 0 < << 1 Then it is preferable to use the power series, which has a fast convergengence for such values of With m + ν 0 holds 0 t m J ν t dt = m+ν+1 ν 1 k k k! ν + k! 4 k m + ν + 1 + k k=0 and From this one has For instance, 0 t m I ν t dt = m+ν+1 ν F m ν = L ν,m + 0 000 k=0 k k! ν + k! 4 k m + ν + 1 + k t m J ν t dt and F,m ν = L ν,m + J d = J 0 J 1 000 = 0 t m I ν t dt = 00889466165577080 0051154784098666 4999975000006500 499998750000084 = = 00540101400650990086 0149999584 = 007098981869 It was a loss of seven decimal digits at = 000 This value may be found without problems by the power series: F 000 = = 1 8 +5 10 7 0085 104166666666666667 10 8 +694444444444444445 10 16 ] = = 1 8 + 416666614586806 10 8 = 014999958854 In the previous value, signed by *, the last digit should be instead of 4 and the result had to finish with 8 The integrals with I ν may be computed in the same way This method can be used even if ν + m < 0 For instance, 000 000 J 4 1 J 4 J 4 7 d = 000 7 d + 1 7 d and the second integral is given in the following tables For the first one holds with the power series of the function J 4 1 J 4 000 7 d = 1 1 1 = 000 7 84 4 1 1 1 1 1 7680 6 + 68640 8 0965760 10 + 9661780 1 71451110400 14 + d = 1 1 = 84 1 7680 + 1 68640 1 1 1 0965760 + 9661780 5 71451110400 7 + d = = 1 768 ln 7680 + 1 7780 1 186040 4 + 1 78170680 6 1 57076088800 8 + = 0001008 00 + 156 10 6 8074 10 9 + 40491 10 11 01750 10 1 + 5508+0000809197681589549+54547 10 1 1917 10 19 +06911 10 7 = = 00010075080871678 550041956509 = 5518740457006 Here are no differences of nearly the same values 1 000 = 4

Z : J d = J 1 + Λ 0 I d = I 1 Λ 0 J d = J 0 J 1 I d = I 0 + I 1 J d = J 0 J 1 + Λ 0 I d = I 0 + I 1 + Λ 0 J d = 4 J 0 8 J 1 I d = 4 I 0 + + 8 I 1 4 J d = 5 J 0 15 J 1 15Λ 0 4 I d = 5 + I 0 + + 15 I 1 + 15Λ 0 J d = J 1 I d = I 1 J d = 1 J 0 + J 1 + 1 Λ 0 I d = 1 I 0 I 1 + 1 Λ 0 P 5 = 6 8, Q 5 = 4 4 + 96, R 5 = 0, S 5 = 0 P 5, = 6 + 8, Q 5, = 5 + 4 + 96, R 5, = 0, S 5, = 0 P 6 = 7 4 15 + 45, Q 6 = 4 5 + 15, R 6 = 15, S 6 = 0 P 6, = 7 5 + 15 + 45, Q 6, = 6 + 5 4 + 15, R 6, = 15, S 6, = 0 P 7 = 8 4 4 + 19, Q 7 = 6 48 4 + 768 07, R 7 = 0, S 7 = 0 P 7, = 8 4 +19 +156, Q 7, = 7 +48 5 +768 +07, R 7, = 0, S 7, = 0 P 8 = 9 6 5 4 + 55 1575, Q 8 = 6 6 4 + 1575 14175, R 8 = 14175, S 8 = 0 P 8, = 9 7 + 15 5 + 475 + 14175, Q 8, = 8 + 6 6 + 1575 4 + 14175, R 8, = 14175, S 8, = 0 P 9 = 10 6 48 4 + 115 916, Q 9 = 8 80 6 + 880 4 46080 + 1840, R 9 = 0, S 9 = 0 5

P 9, = 10 8 +480 6 +1150 4 +9160, Q 9, = 9 +80 7 +880 5 +46080 +1840, R 9, = 0, S 9, = 0 P 10 = 11 8 6 6 + 05 4 075 + 995, Q 10 = 8 99 6 + 4851 4 1175 + 1091475, R 10 = 1091475, S 10 = 0 P 10, = 11 9 + 69 7 + 455 5 + 685 + 1091475, Q 10, = 10 + 99 8 + 4851 6 + 1175 4 + 1091475, R 10, = 1091475, S 10, = 0 P = 1 4, Q = + 4 8, R = 0, S = 1 8 P, = 1 4, Q, = 4 8, R, = 0, S, = 1 8 P 4 = 15, Q 4 = 4 6 15 4, R 4 = 1 15, S 4 = 0 P 4, = + 15, Q 4, = 4 + 6 15 4, R 4, = 1 15, S 4, = 0 P 5 = 8 48 4, Q 5 = 4 4 96 5, R 5 = 0, S 5 = 1 96 P 5, = + 8 48 4, Q 5, = 4 + 4 96 5, R 5, = 0, S 5, = 1 96 P 6 = 4 + 45 15 5, Q 6 = 6 4 + 9 + 90 15 6, R 6 = 1 15, S 6 = 0 P 6, = 4 + + 45 15 5, Q 6, = 6 + 4 + 9 90 15 6, R 6, = 1 15, S 6, = 0 Z : J d = J 0 4 J 1 I d = I 0 4 I 1 J d = J 0 8J 1 + Λ 0 I d = I 0 8I 1 + Λ 0 J d = 8J 0 6J 1 I d = + 8I 0 6I 1 J d = 15 J 0 7 J 1 + 15Λ 0 I d = + 15 I 0 7 I 1 15Λ 0 4 J d = 4 J 0 8 6 J 1 4 I d = + 4 I 0 8 + 6 I 1 J d = 4 J 0 + 8 J 1 + 1 Λ 0 6

I d = 4 I 0 + 8 I 1 1 Λ 0 J d = J 0 J 1 I d = I 0 I 1 P 5 = 5 5 + 105, Q 5 = 9 4 105, R 5 = 105, S 5 = 0 P 5, = 5 + 5 + 105, Q 5, = 9 4 + 105, R 5, = 105, S 5, = 0 P 6 = 6 48 4 + 84, Q 6 = 10 5 19 + 768, R 6 = 0, S 6 = 0 P 6, = 6 + 48 4 + 84, Q 6, = 10 5 + 19 + 768, R 6, = 0, S 6, = 0 P 7 = 7 6 5 +945 85, Q 7 = 11 6 15 4 +85, R 7 = 85, S 7 = 0 P 7, = 7 +6 5 +945 +85, Q 7, = 11 6 +15 4 +85, R 7, = 85, S 7, = 0 P 8 = 8 80 6 + 190 4 1560, Q 8 = 1 7 480 5 + 7680 070, R 8 = 0, S 8 = 0 P 8, = 8 + 80 6 + 190 4 + 1560, Q 8, = 1 7 + 480 5 + 7680 + 070, R 8, = 0, S 8, = 0 P 9 = 9 99 7 + 465 5 51975 + 15595, Q 9 = 1 8 69 6 + 175 4 15595, R 9 = 15595, S 9 = 0 P 9, = 9 + 99 7 + 465 5 + 51975 + 15595, Q 9, = 1 8 + 69 6 + 175 4 + 15595, R 9, = 15595, S 9, = 0 P 10 = 10 10 8 + 5760 6 1840 4 + 110590, Q 10 = 14 9 960 7 + 4560 5 55960 + 11840, R 10 = 0, S 10 = 0 P 10, = 10 + 10 8 + 5760 6 + 1840 4 + 110590, Q 10, = 14 9 + 960 7 + 4560 5 + 55960 + 11840, R 10, = 0, S 10, = 0 P = + 1 15, Q = 4 + 4 15 4, R = 1 15, S = 0 P, = 1 15, Q, = 4 + + 4 15 4, R, = 1 15, S, = 0 P 4 = + 16 4 4, Q 4 = 4 4 + 64 48 5, R 4 = 0, S 4 = 1 48 P 4, = 16 4 4, Q 4, = 4 + 4 + 64 48 5, R 4, = 0, S 4, = 1 48 P 5 = 4 60 105 5, Q 5 = 6 4 + 9 10 105 6, R 5 = 1 105, S 5 = 0 P 5, = 4 + 60 105 5, Q 5, = 6 + 4 + 9 + 10 105 6, R 5, = 1 105, S 5, = 0 P 6 = 4 8 19 84 6, Q 6 = 6 4 4 + 64 768 768 7, R 6 = 0, S 6 = 1 768 P 6, = 4 + 8 19 84 6, Q 6, = 6 + 4 4 + 64 + 768 768 7, R 6, = 0, S 6, = 1 768 7

Z 4 : J 4 d = 8J 0 16J 1 + Λ 0 I 4 d = 8I 0 + 16I 1 + Λ 0 J 4 d = 8J 0 + 4 J 1 I 4 d = 8I 0 + + 4 I 1 J 4 d = 9J 0 + 48J 1 + 15Λ 0 I 4 d = 9I 0 + + 48I 1 15Λ 0 J 4 d = 10 48J 0 + 44 J 1 I 4 d = 10 + 48I 0 + + 44 I 1 4 J 4 d = 11 105 J 0 + 57 J 1 + 105Λ 0 4 I 4 d = 11 + 105 I 0 + + 57 I 1 + 105Λ 0 J4 d = 6J 0 + 1 J 1 I4 d = 6J 0 + + 1 J 1 J4 d = + 7 15 J 0 4 16 + 144 15 4 J 1 + 1 15 Λ 0 I4 d = 7 15 I 0 + 4 + 16 + 144 15 4 I 1 1 15 Λ 0 P 5 4 = 1 4 19, Q 5 4 = 5 7 + 84, R 5 4 = 0, S 5 4 = 0 P 5, 4 = 1 4 + 19, Q 5, 4 = 5 + 7 + 84, R 5, 4 = 0, S 5, 4 = 0 P 6 4 = 1 5 15 + 945, Q 6 4 = 6 89 4 + 945, R 6 4 = 945, S 6 4 = 0 P 6, 4 = 1 5 + 15 + 945, Q 6, 4 = 6 + 89 4 + 945, R 6, 4 = 945, S 6, 4 = 0 P 7 4 = 14 6 480 4 + 840, Q 7 4 = 7 108 5 + 190 7680, R 7 4 = 0, S 7 4 = 0 P 7, 4 = 14 6 + 480 4 + 840, Q 7, 4 = 7 + 108 5 + 190 + 7680, R 7, 4 = 0, S 7, 4 = 0 P 8 4 = 15 7 69 5 + 1095 1185, Q 8 4 = 8 19 6 + 465 4 1185, R 8 4 = 1185, S 8 4 = 0 8