TABLES OF SOME INDEFINITE INTEGRALS OF BESSEL FUNCTIONS


 Χλωρίς Χρηστόπουλος
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1 Werner Rosenheinrich Ernst  Abbe  Hochschule Jena First variant: 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 wellknown 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], 1074, 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
2 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, MacGrawHill, 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, ] 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, ] I S Gradstein, I M Ryshik: Summen, Produkt und Integraltafeln / Tables of Series, Products, and Integrals, Band / Volume, Verlag Harri Deutsch, Thun Frankfurt/M, ] G N Watson: A Treatise on the Theory of Bessel Functions, Cambridge, University Press, 19 / ] P Humbert: Besselintegral functions, Proceedings of the Edinburgh Mathematical Society Series, 19, : ] 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 ] B G Korenev: Vvedenie v teori besselevyh funkci i, Nauka, Moskva, ] S K H Auluck, Some integral identities involving products of general solutions of Bessel s equation of integral order, arivorg/abs/
3 1 1 n Z ν with integer values of n Contents 1 Integrals with one Bessel function 111 n Z n+1 Z n 1 Z n Z n Z n+1 Z n Z ν, ν > 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 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 n 1/ { sin cos } J ν n 1/ e ± I ν 9 a n 1/ e I ν 9 b n 1/ e I ν 94 c General formulas n+1 ln Z n ln Z 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 c Integrals of n+1 ln Z n e ± ln Z ν 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
4 c Chebyshev Polynomials U n 14 d Laguerre Polynomials L n 146 e Hermite Polynomials H n Eponential Integral Sine and Cosine Integral 157 Products of two Bessel Functions 1 Bessel Functions with the the same Argument : 11 n+1 Zν n Zν n Zν 168 a The Functions Θ and Ω 168 b Integrals n Z 0 Z n+1 Z 0 Z n+1 Z 0 Z n+1 J 0 I n J 0 I n J 1 I n+1 J 1 I n+1] J µ Y ν 190 a n+1 J 0 Y b n J 0 Y c n J 0 Y 1 19 d n 1 J 0 Y 1 19 e n J 1 Y f n 1 J 0 Y g n+1 J 1 Y h n J 1 Y Bessel Functions with different Arguments α and β : 1 n+1 Z ν αz ν β 198 a ν = 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
5 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 ep/ sin / cos Z ν Z 1 d Some Cases of n e α Z ν Z 1 d 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 Z c Basic Integral Z0 Z1 70 d Basic Integral Z 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 Z j Recurrence relations 95 5 Quotients Denominator p Z 0 + q Z a Typ f Z µ /p Z 0 + q Z 1 ] 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, Denominator p Z 0 + q Z 1 ] 405 a Typ f Z µ /p Z 0 + q Z 1 ] Denominator p Z 0 + q Z 1 ]
6 a Typ f Z µ /p Z 0 + q Z 1 ] 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, Denominator a Z 0 + b Z 1 + p Z0 0 Z 1 + r Z Miscellaneous Used special functions and defined functions 41 8 Errata 41 6
7 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 Wellknown 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 newdefined 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 = I 1 I 0 + 9Ψ 4 K 0 d = K 1 K 0 + 9Ψ K E 7
8 6 J 0 d = J J 0 5Φ 6 I 0 d = I I 0 + 5Ψ 6 K 0 d = K K 0 + 5Ψ K and so on 8 J 0 d = J J Φ 8 I 0 d = I I Ψ 10 J 0 d = J J Φ 10 I 0 d = I I Ψ 1 J 0 d = J 0 + Let J Φ 1 I 0 d = I I Ψ 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 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 n n Φ n! ] n 1!! n k I 1 n 1 k!! k=0 I 0 + n 1!!] Ψ = 8
9 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 J 0 Φ I0 d = 1 I 0 I 1 + Ψ K0 d = 1 K 0 + K 1 + Ψ K J0 4 d = ] 9 J 0 1 J 1 + Φ I0 4 d = 1 4 ] 9 I I 1 + Ψ K0 4 d = 1 4 ] 9 K K 1 + Ψ K J0 6 d = J I0 6 d = I K ] K 1 + Ψ K K0 6 d = 1 5 J0 8 d = I0 8 d = ] J 0 Φ ] I 1 + Ψ and so on J ] 5 6 J 1 + Φ I J d = J I0 10 d = ] J 0 Φ I 0 9 ] I 1 + Ψ E E 9
10 ] I 1 + Ψ J0 1 d = I0 1 d = J ] J 1 + Φ I 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 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 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 J 1 ] I 0 d = + 4 I 1 I 0 ] K 0 d = + 4 K 1 + K 0 ] 5 J 0 d = 4 J J 1 ] 5 I 0 d = I I 0 ] 5 K 0 d = K K 0 ] 7 J 0 d = J J 1 ] 7 I 0 d = I I 0 ] 7 K 0 d = K K 0 ] 9 J 0 d = = J J 1 ] 9 I 0 d = = I I 0 ] 9 K 0 d = = K 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 = Q 11 = P11 = Q 11 = *E* 11
12 P 1 = Q 1 = P1 = Q 1 = P 15 = Q 15 = P 15 = = *E* Q 15 = 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
13 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 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 J0 d 4 I0 d = I 0 I I0 d 4 J0 d 1 5 = J J J0 d 64 I0 d 1 5 = I I I0 d 64 J0 d 7 = J J 1 1 J0 d 04 I0 d 7 = I I I0 d 04 J0 d 9 = = J J 1 + I0 d 9 = = I I 1 + J0 d 11 = E J J0 d J I0 d 11 = I I J0 d I0 d I0 d 1
14 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
15 114 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 I 1 ] 4 K 1 d = + 8 K K 1 ] 6 J 1 d = J J 0 ] 6 I 1 d = I I 1 ] 6 K 1 d = K K 1 ] 8 J 1 d = = J J 0 ] 8 I 1 d = = I I 1 ] 8 K 1 d = = K K 1 ] 10 J 1 d = J J 0 ] 10 I 1 d = I I 1 ] 10 K 1 d = K
16 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 = Q 1 = P 1 = Q 1 = *E* P 14 = Q 14 = P14 = Q 14 = 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
17 115 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 J0 d I1 d = 1 I I0 d J1 d 4 = 1 8 J J 1 1 J0 d 16 I1 d 4 = 1 8 I I I0 d 16 J1 d 6 = = J J J0 d 84 I1 d 6 = I I I0 d 84 J1 d 8 = = J J 1 1 J0 d 184 I1 d 8 = I I I0 d 184 J1 d 10 = Recurrence formulas: = J J J0 d I1 d 10 = = I I I0 d 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 nn + 1 J1 d n I1 d n 17
18 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
19 116 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 = J J Φ 5 I 1 d = I I 0 45Ψ 5 K 1 d = K K Ψ K 7 J 1 d = J J Φ E 7 I 1 d = I I Ψ 7 K 1 d = K K Ψ K 9 J 1 d = = J J Φ 9 I 1 d = = I I Ψ 9 K 1 d = = K K 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 n n + 1!! n 1!! Φ = n + 1 k!! n 1 k!! 19
20 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 J 0 Φ I1 d = 1 ] + 1 I I 0 + Ψ K1 d = 1 ] + 1 K 1 1 K 0 Ψ K J1 5 d = J 0 4 ] J 1 + Φ I1 5 d = I ] I 1 + Ψ K1 5 d = K ] K 1 Ψ K J1 7 d = J ] J 0 Φ 0
21 I1 7 d = K1 7 d = = = = J1 11 d = 6 I K J1 9 d = 7 J I1 9 d = 7 I K1 9 d = ] I 0 + Ψ ] K 0 Ψ k ] J 1 + Φ ] I 1 + Ψ 7 K J I1 11 d = K1 11 d = 9 ] J 0 Φ I ] I 0 + Ψ 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 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
22 117 Integrals of the type n Z ν d, ν > 1 : From the wellknown 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 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 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 J 0 + 8J 4J = = , κ=1 κ=0 κ=1
23 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 = 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 = , 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 = , 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
24 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 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 k=0 k k! ν + k! 4 k m + ν k t m J ν t dt and F,m ν = L ν,m + J d = J 0 J = 0 t m I ν t dt = = = = It was a loss of seven decimal digits at = 000 This value may be found without problems by the power series: F 000 = = ] = = = 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, J 4 1 J 4 J 4 7 d = d 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 d = = d = 1 1 = d = = ln = = = = Here are no differences of nearly the same values = 4
25 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 I 1 4 J d = 5 J 0 15 J 1 15Λ 0 4 I d = 5 + I I Λ 0 J d = J 1 I d = I 1 J d = 1 J 0 + J Λ 0 I d = 1 I 0 I Λ 0 P 5 = 6 8, Q 5 = , R 5 = 0, S 5 = 0 P 5, = 6 + 8, Q 5, = , R 5, = 0, S 5, = 0 P 6 = , Q 6 = , R 6 = 15, S 6 = 0 P 6, = , Q 6, = , R 6, = 15, S 6, = 0 P 7 = , Q 7 = , R 7 = 0, S 7 = 0 P 7, = , Q 7, = , R 7, = 0, S 7, = 0 P 8 = , Q 8 = , R 8 = 14175, S 8 = 0 P 8, = , Q 8, = , R 8, = 14175, S 8, = 0 P 9 = , Q 9 = , R 9 = 0, S 9 = 0 5
26 P 9, = , Q 9, = , R 9, = 0, S 9, = 0 P 10 = , Q 10 = , R 10 = , S 10 = 0 P 10, = , Q 10, = , R 10, = , 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 = , R 4 = 1 15, S 4 = 0 P 4, = + 15, Q 4, = , R 4, = 1 15, S 4, = 0 P 5 = , Q 5 = , R 5 = 0, S 5 = 1 96 P 5, = , Q 5, = , R 5, = 0, S 5, = 1 96 P 6 = , Q 6 = , R 6 = 1 15, S 6 = 0 P 6, = , Q 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 Λ 0 I d = + 15 I 0 7 I 1 15Λ 0 4 J d = 4 J J 1 4 I d = + 4 I I 1 J d = 4 J J Λ 0 6
27 I d = 4 I I 1 1 Λ 0 J d = J 0 J 1 I d = I 0 I 1 P 5 = , Q 5 = , R 5 = 105, S 5 = 0 P 5, = , Q 5, = , R 5, = 105, S 5, = 0 P 6 = , Q 6 = , R 6 = 0, S 6 = 0 P 6, = , Q 6, = , R 6, = 0, S 6, = 0 P 7 = , Q 7 = , R 7 = 85, S 7 = 0 P 7, = , Q 7, = , R 7, = 85, S 7, = 0 P 8 = , Q 8 = , R 8 = 0, S 8 = 0 P 8, = , Q 8, = , R 8, = 0, S 8, = 0 P 9 = , Q 9 = , R 9 = 15595, S 9 = 0 P 9, = , Q 9, = , R 9, = 15595, S 9, = 0 P 10 = , Q 10 = , R 10 = 0, S 10 = 0 P 10, = , Q 10, = , R 10, = 0, S 10, = 0 P = , Q = , R = 1 15, S = 0 P, = 1 15, Q, = , R, = 1 15, S, = 0 P 4 = , Q 4 = , R 4 = 0, S 4 = 1 48 P 4, = , Q 4, = , R 4, = 0, S 4, = 1 48 P 5 = , Q 5 = , R 5 = 1 105, S 5 = 0 P 5, = , Q 5, = , R 5, = 1 105, S 5, = 0 P 6 = , Q 6 = , R 6 = 0, S 6 = P 6, = , Q 6, = , R 6, = 0, S 6, =
28 Z 4 : J 4 d = 8J 0 16J 1 + Λ 0 I 4 d = 8I I 1 + Λ 0 J 4 d = 8J J 1 I 4 d = 8I I 1 J 4 d = 9J J Λ 0 I 4 d = 9I I 1 15Λ 0 J 4 d = 10 48J J 1 I 4 d = I I 1 4 J 4 d = J J Λ 0 4 I 4 d = I I Λ 0 J4 d = 6J J 1 I4 d = 6J J 1 J4 d = J J Λ 0 I4 d = 7 15 I I Λ 0 P 5 4 = , Q 5 4 = , R 5 4 = 0, S 5 4 = 0 P 5, 4 = , Q 5, 4 = , R 5, 4 = 0, S 5, 4 = 0 P 6 4 = , Q 6 4 = , R 6 4 = 945, S 6 4 = 0 P 6, 4 = , Q 6, 4 = , R 6, 4 = 945, S 6, 4 = 0 P 7 4 = , Q 7 4 = , R 7 4 = 0, S 7 4 = 0 P 7, 4 = , Q 7, 4 = , R 7, 4 = 0, S 7, 4 = 0 P 8 4 = , Q 8 4 = , R 8 4 = 1185, S 8 4 = 0 8
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