Handbook of special functions: derivatives, integrals, series and other formulas
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, Fla. [u.a.]
CRC Press
2008
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| Series: | A Chapman & Hall book
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 669 - 672 |
| Physical Description: | XIX, 680 S. |
| ISBN: | 9781584889564 |
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| 100 | 1 | |a Bryčkov, Jurij A. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Handbook of special functions |b derivatives, integrals, series and other formulas |c Yuri A. Brychkov |
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| adam_text | Titel: Handbook of special functions
Autor: Bryčkov, Jurij A.
Jahr: 2008
Contents
Preface................................................ xix
Chapter 1. The Derivatives.............................. 1
1.1. Elementary Functions.............................. 1
1.1.1. General formulas................................... 1
1.1.2. Algebraic functions.................................. 1
1.1.3. The exponential function.............................. 4
1.1.4. Hyperbolic functions................................. 6
1.1.5. Trigonometric functions............................... 7
1.1.6. The logarithmic function.............................. 12
1.1.7. Inverse trigonometric functions.......................... 13
1.2. The Hurwitz Zeta Function C(i/, z).................... 15
1.2.1. Derivatives with respect to the argument................... 15
1.2.2. Derivatives with respect to the parameter................... 15
1.3. The Exponential Integral Ei (z)...................... 16
1.3.1. Derivatives with respect to the argument................... 16
1.4. The Sine si (z) and Cosine ci(z) Integrals............... 17
1.4.1. Derivatives with respect to the argument................... 17
1.5. The Error Functions erf (z) and erfc (z)................ 18
1.5.1. Derivatives with respect to the argument................... 18
1.6. The Fresnel Integrals S(z) and C{z)................... 20
1.6.1. Derivatives with respect to the argument................... 20
1.7. The Generalized Fresnel Integrals S{z,i ) and C(z,t/)..... 20
1.7.1. Derivatives with respect to the argument................... 20
1.8. The Incomplete Gamma Functions 7(1/, z) and T{v,z)..... 21
1.8.1. Derivatives with respect to the argument................... 21
1.8.2. Derivatives with respect to the parameter................... 22
1.9. The Parabolic Cylinder Function Dv{z)................ 22
1.9.1. Derivatives with respect to the argument................... 22
1.9.2. Derivatives with respect to the order...................... 24
Contents
1.10. The Bessel Function Jv{z).......................... 25
1.10.1. Derivatives with respect to the argument................... 25
1.10.2. Derivatives with respect to the order...................... 28
1.11. The Bessel Function Yv(z).......................... 29
1.11.1. Derivatives with respect to the argument................... 29
1.11.2. Derivatives with respect to the order...................... 31
1.12. The Hankel Functions h11}(z) and HJ)2)(z)............. 31
1.12.1. Derivatives with respect to the argument................... 31
1.12.2. Derivatives with respect to the order...................... 32
1.13. The Modified Bessel Function Iv{z)................... 32
1.13.1. Derivatives with respect to the argument................... 32
1.13.2. Derivatives with respect to the order...................... 35
1.14. The Macdonald Function Kv{z)...................... 36
1.14.1. Derivatives with respect to the argument................... 36
1.14.2. Derivatives with respect to the order...................... 40
1.15. The Struve Functions H?(z) and L?(z)................. 41
1.15.1. Derivatives with respect to the argument................... 41
1.15.2. Derivatives with respect to the order...................... 42
1.16. The Anger J?(z) and Weber E?(z) Functions............ 45
1.16.1. Derivatives with respect to the argument................... 45
1.16.2. Derivatives with respect to the order...................... 47
1.17. The Kelvin Functions ber?(z), bei?(z), ker?(z) and keiv(z) 48
1.17.1. Derivatives with respect to the argument................... 48
1.17.2. Derivatives with respect to the order...................... 51
1.18. The Legendre Polynomials Pn(z)..................... 56
1.18.1. Derivatives with respect to the argument................... 56
1.19. The Chebyshev Polynomials Tn(z) and Un(z)........... 59
1.19.1. Derivatives with respect to the argument................... 59
1.20. The Hermite Polynomials Hn(z)..................... 62
1.20.1. Derivatives with respect to the argument................... 62
1.21. The Laguerre Polynomials £*(z)..................... 63
1.21.1. Derivatives with respect to the argument................... 63
1.21.2. Derivatives with respect to the parameter................... 64
1.22. The Gegenbauer Polynomials c£(z)................... 64
1.22.1. Derivatives with respect to the argument................... 64
1.22.2. Derivatives with respect to the parameter................... 66
1.23. The Jacobi Polynomials PiP,cr)(z)..................... 66
1.23.1. Derivatives with respect to the argument................... 66
1.23.2. Derivatives with respect to parameters..................... 69
vi
Contents
1.24. The Complete Elliptic Integrals K(z), E(z) and D(z)..... 69
1.24.1. Derivatives with respect to the argument................... 69
1.25. The Legendre Function P?(z)........................ 70
1.25.1. Derivatives with respect to the argument................... 70
1.25.2. Derivatives with respect to parameters..................... 71
1.26. The Kummer Confluent Hypergeometric Function
iFi(a;b;z)....................................... 73
1.26.1. Derivatives with respect to the argument................... 73
1.26.2. Derivatives with respect to parameters..................... 75
1.27. The Tricomi Confluent Hypergeometric Function 9(a; b; z) 76
1.27.1. Derivatives with respect to the argument................... 76
1.27.2. Derivatives with respect to parameters..................... 77
1.28. The Whittaker Functions M^,?(«) and W^,?(z).......... 80
1.28.1. Derivatives with respect to the argument................... 80
1.29. The Gauss Hypergeometric Function 2^1(0, b; c;z)........ 80
1.29.1. Derivatives with respect to the argument................... 80
1.29.2. Derivatives with respect to parameters..................... 85
1.30. The Generalized Hypergeometric Function pFg((ap);(bq);z) 86
1.30.1. Derivatives with respect to the argument................... 86
1.30.2. Derivatives with respect to parameters..................... 87
Chapter 2. Limits...................................... 95
2.1. Special Functions................................. 95
2.1.1. The Bessel functions Ju(z), Yv(z), I?{z) and Kv{z)............ 95
2.1.2. The Struve functions H?(z) and ~L?(z)..................... 95
2.1.3. The Kelvin functions ber?(z), bei?(z), ker?(z) and kei?(z)....... 95
2.1.4. The Legendre polynomials Pn(z)......................... 96
2.1.5. The Chebyshev polynomials Tn(z) and Un{z)................ 96
2.1.6. The Hermite polynomials H?(z)......................... 97
2.1.7. The Laguerre polynomials Ln(z)......................... 97
2.1.8. The Gegenbauer polynomials Crl(z)....................... 98
2.1.9. The Jacobi polynomials P^, r)(z)........................ 98
2.1.10. Hypergeometric functions.............................. 99
Chapter 3. Indefinite Integrals............................ 101
3.1. Elementary Functions.............................. 101
3.1.1. The logarithmic function.............................. 101
3.2. Special Functions................................. 102
3.2.1. The Bessel functions J?(x), Yu{x), Iv{x) and K?(x)........... 102
vii
Contents
3.2.2. The Struve functions H?(z) and L?(z)..................... 105
3.2.3. The Airy functions Ai (z) and Bi (z)...................... 105
3.2.4. Various functions................................... 109
Chapter 4. Definite Integrals............................. Ill
4.1. Elementary Functions.............................. Ill
4.1.1. Algebraic functions.................................. Ill
4.1.2. The exponential function.............................. 116
4.1.3. Hyperbolic functions................................. 117
4.1.4. Trigonometric functions............................... 121
4.1.5. The logarithmic function.............................. 132
4.1.6. Inverse trigonometric functions.......................... 155
4.2. The Dilogarithm Li2(z)............................. 178
4.2.1. Integrals containing Li2(z) and algebraic functions............. 178
4.2.2. Integrals containing Li2(z) and trigonometric functions.......... 179
4.2.3. Integrals containing Li2(z) and the logarithmic function......... 180
4.2.4. Integrals containing Li2(z) and inverse trigonometric functions..... 180
4.3. The Sine Si (z) and Cosine ci (z) Integrals.............. 181
4.3.1. Integrals containing Si(z) and algebraic functions.............. 181
4.3.2. Integrals containing Si (z) and trigonometric functions........... 181
4.3.3. Integrals containing Si (z) and the logarithmic function.......... 183
4.3.4. Integrals containing Si (z) and inverse trigonometric functions..... 183
4.3.5. Integrals containing products of Si (z) and ci (z)............... 184
4.4. The Error Functions erf (z), erfi (z) and erfc (z).......... 184
4.4.1. Integrals containing erf (z) and algebraic functions............. 184
4.4.2. Integrals containing erf (z), erfc(z) and the exponential function . . . 185
4.4.3. Integrals containing erf (z) and trigonometric functions.......... 186
4.4.4. Integrals containing erf (z) and the logarithmic function......... 187
4.4.5. Integrals containing erf (z), erfi (z) and inverse trigonometric
functions......................................... 187
4.4.6. Integrals containing products of erf (z), erfc(z) and erfi(z)....... 188
4.5. The Fresnel Integrals S(z) and C(z)................... 189
4.5.1. Integrals containing S(z) and algebraic functions.............. 189
4.5.2. Integrals containing S(z) and trigonometric functions........... 190
4.5.3. Integrals containing S(z) and the logarithmic function.......... 191
4.5.4. Integrals containing C(z) and algebraic functions.............. 191
4.5.5. Integrals containing C(z) and trigonometric functions........... 191
4.5.6. Integrals containing C(z) and the logarithmic function.......... 192
4.6. The Incomplete Gamma Function 7(1/, z)............... 192
4.6.1. Integrals containing -y(v, z) and algebraic functions............. 192
4.6.2. Integrals containing 7(1/, z) and the exponential function......... 193
4.6.3. Integrals containing 7(1/, z) and trigonometric functions......... 193
viii
Contents
4.6.4. Integrals containing 7(1/, z) and the logarithmic function......... 195
4.6.5. Integrals containing 7(1/, z), erf (z) and erfi (z)................ 195
4.6.6. Integrals containing products of 7(1/, z).................... 195
4.7. The Bessel Function Jv(z).......................... 196
4.7.1. Integrals containing Jv{z) and algebraic functions............. 196
4.7.2. Integrals containing Jv(z) and the exponential function.......... 197
4.7.3. Integrals containing Jv(z) and trigonometric functions.......... 197
4.7.4. Integrals containing Jv(z) and the logarithmic function.......... 199
4.7.5. Integrals containing Jv{z) and inverse trigonometric functions..... 199
4.7.6. Integrals containing Ju{z), Si (z) and ci (z).................. 200
4.7.7. Integrals containing products of Jv(z)..................... 201
4.8. The Bessel Function Yv(z).......................... 204
4.8.1. Integrals containing Yu z) and algebraic functions............. 204
4.8.2. Integrals containing Y?(z) and J?(z)...................... 204
4.9. The Modified Bessel Function Iv(z)................... 205
4.9.1. Integrals containing Iv(z) and algebraic functions.............. 205
4.9.2. Integrals containing Iv{z) and the exponential function.......... 206
4.9.3. Integrals containing Iv{z) and trigonometric functions........... 208
4.9.4. Integrals containing Iv{z) and the logarithmic function.......... 210
4.9.5. Integrals containing Iv{z) and inverse trigonometric functions..... 211
4.9.6. Integrals containing Iv{z) and special functions............... 212
4.9.7. Integrals containing products of Iv(z)..................... 213
4.10. The Macdonald Function Ku{z) ...................... 216
4.10.1. Integrals containing Kv(z), Jv(z), Y?(z) and 7?(z)............ 216
4.10.2. Integrals containing products of K?(z)..................... 216
4.11. The Struve Functions H?(z) and L?(z)................. 217
4.11.1. Integrals containing Hv{z), L?(z) and algebraic functions........ 217
4.11.2. Integrals containing H,,(z) and hyperbolic functions............ 219
4.11.3. Integrals containing Mv(z), L?(z) and trigonometric functions..... 219
4.11.4. Integrals containing Hi,(z), L?(z) and the logarithmic function..... 220
4.11.5. Integrals containing H?(z), L?(z) and inverse trigonometric functions 221
4.12. The Kelvin Functions berI/(z), bei?(z), ker?(z) and kei?(z) 221
4.12.1. Integrals containing ber*,(z), beij,(z), ker?(z), keL(z) and algebraic
functions......................................... 221
4.13. The Airy Functions Ai (z) and Bi (z).................. 222
4.13.1. Integrals containing products of Ai(z) and Bi(z).............. 222
4.14. The Legendre Polynomials Pn{z)..................... 223
4.14.1. Integrals containing Pn(z) and algebraic functions............. 223
4.14.2. Integrals containing Pn(z) and trigonometric functions.......... 226
4.14.3. Integrals containing Pu{z) and the logarithmic function.......... 231
ix
Contents
4.14.4. Integrals containing Pn{z), Ju(z), Iv(z) and Kv{z)............ 232
4.14.5. Integrals containing products of P?(z)..................... 232
4.15. The Chebyshev Polynomials Tn(z).................... 234
4.15.1. Integrals containing Tn(z) and algebraic functions............. 234
4.15.2. Integrals containing Tn(z) and trigonometric functions.......... 236
4.15.3. Integrals containing Tn(z) and special functions............... 241
4.16. The Chebyshev Polynomials Un(z).................... 241
4.16.1. Integrals containing Un{z) and algebraic functions............. 241
4.16.2. Integrals containing U?(z) and trigonometric functions.......... 242
4.16.3. Integrals containing Un(z) and Kv(z)..................... 246
4.16.4. Integrals containing products of Un(z)..................... 246
4.17. The Hermite Polynomials Hn(z)..................... 247
4.17.1. Integrals containing Hn{z) and algebraic functions............. 247
4.17.2. Integrals containing Hn(z) and the exponential function......... 248
4.17.3. Integrals containing Hn(z) and trigonometric functions.......... 249
4.17.4. Integrals containing Hn(z), erf (z) and erfc (z)............... 251
4.17.5. Integrals containing Hn(z) and Kv(z)..................... 251
4.17.6. Integrals containing products of H?(z)..................... 252
4.18. The Laguerre Polynomials £*(z)..................... 254
4.18.1. Integrals containing Ln(z) and algebraic functions............. 254
4.18.2. Integrals containing Ln{z) and trigonometric functions.......... 255
4.18.3. Integrals containing Ln(z) and erfc(z)..................... 256
4.18.4. Integrals containing products of L?(z)..................... 256
4.19. The Gegenbauer Polynomials C^ z)................... 257
4.19.1. Integrals containing Cn (z) and algebraic functions............. 257
4.19.2. Integrals containing Cn(z) and trigonometric functions.......... 257
4.19.3. Integrals containing products of C^(z)..................... 261
4.20. The Jacobi Polynomials PJf ^iz)..................... 262
4.20.1. Integrals containing P% T {z) and algebraic functions........... 262
4.20.2. Integrals containing Pff^{z) and trigonometric functions........ 263
4.20.3. Integrals containing P^ r){z) and J?(z).................... 263
4.20.4. Integrals containing products of P^ (z)................... 264
4.21. The Complete Elliptic Integral K(z).................. 265
4.21.1. Integrals containing K(z) and algebraic functions.............. 265
4.21.2. Integrals containing K(z), the exponential, hyperbolic and
trigonometric functions............................... 267
4.21.3. Integrals containing K(z) and the logarithmic function.......... 268
4.21.4. Integrals containing K (z) and inverse trigonometric functions..... 271
4.21.5. Integrals containing K(z) and Lia(z)...................... 274
Contents
4.21.6. Integrals containing K (z), shi (z) and Si (z)................. 274
4.21.7. Integrals containing K (z) and erf (z)...................... 275
4.21.8. Integrals containing K(z), S(z) and C(z)................... 275
4.21.9. Integrals containing K (z) and 7(1/, z)..................... 275
4.21.10. Integrals containing K(z), Jv(z) and Iv(z).................. 276
4.21.11. Integrals containing K(z), H?(z) and L?(z)................. 277
4.21.12. Integrals containing K(z) and L {z)...................... 278
4.21.13. Integrals containing products of K(z)...................... 278
4.22. The Complete Elliptic Integral E(z)................... 279
4.22.1. Integrals containing E(z) and algebraic functions.............. 279
4.22.2. Integrals containing E(z), the exponential, hyperbolic and
trigonometric functions............................... 283
4.22.3. Integrals containing E(z) and the logarithmic function.......... 286
4.22.4. Integrals containing E(z) and inverse trigonometric functions...... 289
4.22.5. Integrals containing E(z) and Li2(z)...................... 292
4.22.6. Integrals containing E(z), shi(z) and Si (z)................. 293
4.22.7. Integrals containing E(z) and erf (z)...................... 293
4.22.8. Integrals containing E(z), S(z) and C(z)................... 293
4.22.9. Integrals containing E(z) and 7(1/, z)...................... 294
4.22.10. Integrals containing E(z), J?(z) and I?(z).................. 294
4.22.11. Integrals containing E(z), H?(z) and L?(z)................. 296
4.22.12. Integrals containing E(z) and L*(z)...................... 296
4.22.13. Integrals containing products of E(z) and K(z)............... 296
4.22.14. Integrals containing products of E(z)...................... 299
4.23. The Complete Elliptic Integral D(z)................... 300
4.23.1. Integrals containing D(z) and elementary functions............ 300
4.23.2. Integrals containing products of D(z), K(z) and E(z).......... 302
4.24. The Generalized Hypergeometric Function pFq({av);{bq);z) 303
4.24.1. Integrals containing pF9((op);(6g);z) and algebraic functions...... 303
4.24.2. Integrals containing pFq((ap); (6,); z) and trigonometric functions. . . 304
4.24.3. Integrals containing pFq((ap); (bq); z) and the logarithmic function . . 306
4.24.4. Integrals containing pF,((op); (6,);z), K(z) and E(z).......... 306
4.24.5. Integrals containing products of PFg((ap); (bq);z).............. 306
Chapter 5. Finite Sums.................................. 309
5.1. The Psi Function ip(z)............................. 309
5.1.1. Sums containing -4 (k + a)............................. 309
5.1.2. Sums containing products of ij}(k + a)..................... 311
5.1.3. Sums containing if (k + a.z)........................... 311
xi
Contents
5.2. The Incomplete Gamma Functions -y(v,z) and T(v,z)..... 312
5.2.1. Sums containing ^(nk + v, z)........................... 312
5.2.2. Sums containing products of j(i/ ± k, z).................... 313
5.2.3. Sums containing T(u ±k,z)............................ 313
5.3. The Bessel Function Jv(z).......................... 314
5.3.1. Sums containing Jv±nk{z)............................. 314
5.3.2. Sums containing products of Jv±nk{z)..................... 314
5.4. The Modified Bessel Function Iv(z)................... 315
5.4.1. Sums containing Iv±nk{z)............................. 315
5.4.2. Sums containing products of J?±nj.(z) and Iv±nk{z)............ 315
5.4.3. Sums containing products of I?±nk(z)..................... 317
5.5. The Macdonald Function Kv(z)...................... 318
5.5.1. Sums containing Kv±n t{z)............................. 318
5.5.2. Sums containing K?±nk(z) and special functions.............. 318
5.5.3. Sums containing products of Kv±nk{z)..................... 319
5.6. The Struve Functions H?(z) and L?(z)................. 319
5.6.1. Sums containing Hfc+?(z) and Lfc+?(z)..................... 319
5.7. The Legendre Polynomials Pn(z)..................... 320
5.7.1. Sums containing Pm±nk(z)............................. 320
5.7.2. Sums containing Pn(z) and special functions................. 325
5.7.3. Sums containing products of Pm±nk{z)..................... 325
5.7.4. Sums containing Pm( p(k, z))........................... 326
5.7.5. Sums containing Pk(ip(k,z))........................... 328
5.7.6. Sums containing products of Pm{ip{k, z))................... 328
5.8. The Chebyshev Polynomials Tn(z) and Un{z)........... 329
5.8.1. Sums containing Tm+?j.(z)............................. 329
5.8.2. Sums containing products of Tm+nk(z)..................... 331
5.8.3. Sums containing Tn(ip(k, z))........................... 331
5.8.4. Sums containing Um+nk(z)............................. 332
5.8.5. Sums containing products of Un(z)....................... 335
5.8.6. Sums containing Un((p(k,z))........................... 335
5.9. The Hermite Polynomials Hn{z)..................... 337
5.9.1. Sums containing i?m±?fc(z)............................ 337
5.9.2. Sums containing Hm±?k(z) and special functions.............. 340
5.9.3. Sums containing products of Hm±nk(z).................... 341
5.9.4. Sums containing Hn(cp(k, z))........................... 342
5.9.5. Sums containing Hm±?k(ip(k, z)) ........................ 343
5.9.6. Sums containing products of Hm±nk{tp(k, z))................ 344
5.10. The Laguerre Polynomials L?(z)..................... 344
5.10.1. Sums containing i,^±tt*(z)............................. 344
Contents
5.10.2. Sums containing Lm±?ft(z)............................. 346
5.10.3. Sums containing L^kk(z)............................. 348
5.10.4. Sums containing Lm±pk(z) and special functions.............. 355
5.10.5. Sums containing products of L^J±pft(z)..................... 357
5.10.6. Sums containing L^kk(tp(k, z))......................... 359
5.10.7. Sums containing Lm±pk(ip(k, z)) and special functions.......... 361
5.10.8. Sums containing products of L^±pkk{tp k,z))................. 362
5.11. The Gegenbauer Polynomials C*(z)................... 363
5.11.1. Sums containing C^nh(z)............................. 363
5.11.2. Sums containing Cm±pk{z)............................. 365
5.11.3. Sums containing C£±pkk(z)............................. 365
5.11.4. Sums containing Cm± j,(z) and special functions.............. 373
5.11.5. Sums containing products of C^±pk(z)..................... 378
5.11.6. Sums containing C%£fn(tp(k, z))......................... 381
5.11.7. Sums containing Cmk+^n(ip(k,z)) and special functions.......... 388
5.11.8. Sums containing products of C**l£( ( ,*))................ 390
5.12. The Jacobi Polynomials P^p )(z)..................... 391
5.12.1. Sums containing p6 ±p*.«-±?*)(2.)......................... 391
5.12.2. Sums containing P%±Zk,cr±gk)(z)......................... 392
5.12.3. Sums containing Pi±Zk ±qk)(z) and special functions.......... 402
5.12.4. Sums containing products of P^t ^ ^ {z)................ 405
5.12.5. Sums containing PiP±^,° ±**)(v(*. *))..................... 405
5.12.6. Sums containing Pi±^fc T±, )(¥ (fe, z)) and special functions...... 408
5.12.7. Sums containing products of fi ^ fe^z))............. 410
5.13. The Legendre Function P£(z)........................ 411
5.13.1. Sums containing P^k(z).............................. 411
5.14. The Kurnmer Confluent Hypergeometric Function
iFi(o;6;z)....................................... 411
5.14.1. Sums containing tFi(a;b;z)............................ 411
5.14.2. Sums containing iFi(o;6;z) and special functions............. 412
5.14.3. Sums containing products of i F (a: 6; z).................... 413
5.15. The Tricomi Confluent Hypergeometric Function P (a; 6; z) 413
5.15.1. Sums containing #{a; b;z)............................. 413
5.15.2. Sums containing ®(a; b; z) and special functions............... 414
5.16. The Gauss Hypergeometric function ^Fi(a,b;c;z)........ 414
5.16.1. Sums containing vFi(a, b;c;z).......................... 414
xiii
Contents
5.16.2. Sums containing 2Fi(a, b; c; z) and special functions............ 415
5.16.3. Sums containing products of 2F%(a, b; c;z).................. 417
5.17. The Generalized Hypergeometric Function pFg((ap); (bq);z) 418
5.17.1. Sums containing PFq((ap) ± mk; (bq) ± nk; z)................ 418
5.17.2. Sums containing PFq((ap) ± mk; (bg) ± nk; z) and special functions 421
5.17.3. Sums containing pFg((ap) ± mk; (bg) ± nk; p(k, z))............ 422
5.17.4. Sums containing pFg((ap) ± mk; (bg) ± nk; p(k, z)) and special
functions......................................... 424
5.17.5. Sums containing products of pFq((ap) ± mk; (bg) ± nk; ip(k, z)). . . . 425
5.17.6. Various sums containing pFq((ap) +mk;(bq) +nk;z)............ 425
5.18. Multiple Sums................................... 426
5.18.1. Sums containing Bessel functions......................... 426
5.18.2. Sums containing orthogonal polynomials.................... 427
Chapter 6. Infinite Series................................ 429
6.1. Elementary Functions.............................. 429
6.1.1. Series containing algebraic functions....................... 429
6.1.2. Series containing the exponential function................... 429
6.1.3. Series containing hyperbolic functions...................... 430
6.1.4. Series containing trigonometric functions.................... 431
6.2. The Psi Function ip(z)............................. 431
6.2.1. Series containing ij (ka + b)............................. 431
6.2.2. Series containing i/i(ka + b) and trigonometric functions......... 447
6.2.3. Series containing products of ip(ka + b).................... 447
6.2.4. Series containing il (ka + b)............................ 449
6.3. The Hurwitz Zeta Function C(« *0.................... 451
6.3.1. Series containing ((k, z).............................. 451
6.4. The Sine Si (z) and Cosine ci (z) Integrals.............. 451
6.4.1. Series containing Si ( p(k)x)............................ 451
6.4.2. Series containing ci (ip(k)x)............................ 452
6.4.3. Series containing Si {kx) and trigonometric functions........... 453
6.4.4. Series containing products of Si (kx)....................... 453
6.5. The Fresnel Integrals S(x) and C(x)................... 453
6.5.1. Series containing S(ip(k)x), C((p(k)x) and algebraic functions..... 453
6.5.2. Series containing S( p(k)x), C(ip(k)x) and trigonometric functions . . 455
6.5.3. Series containing S(kx), C(kx) and Si (kx)................. 458
6.5.4. Series containing products of S(kx) and C(kx)............... 459
6.6. The Incomplete Gamma Function 7(1/, z)............... 459
6.6.1. Series containing 7(4 ± k, z)............................ 459
6.6.2. Series containing products of f(t/ + k,z).................... 460
xiv
Contents
6.7. The Parabolic Cylinder Function Dv(z)................ 460
6.7.1. Series containing Dv±nk(z) and elementary functions........... 460
6.8. The Bessel Functions Jv(z) and Yv(z)................. 461
6.8.1. Series containing Jnk+v(z)............................. 461
6.8.2. Series containing two Bessel functions Jnk+v(z)............... 462
6.8.3. Series containing three Bessel functions Jnk+v(z).............. 465
6.8.4. Series containing four Bessel functions Jnk+v(z)............... 466
6.8.5. Series containing Jk+V(z) and ip(z)....................... 467
6.8.6. Series containing J?(ip(k, x))........................... 467
6.8.7. Series containing J?(kx) and trigonometric functions........... 470
6.8.8. Series containing products of Jv(ip(k, x))................... 470
6.8.9. Series containing products of Ju(kx) and trigonometric functions . . . 471
6.8.10. Series containing Jv(kx) and Si(fea;)...................... 472
6.8.11. Series containing Jv(kx), S(kx) and C(kx)................. 472
6.8.12. Series containing Jktj,+u(lf(k, z))......................... 473
6.8.13. Various series containing J?(z).......................... 474
6.8.14. Series containing Yk+V(z).............................. 475
6.9. The Modified Bessel Function Iv(z)................... 475
6.9.1. Series containing I?k+v (z)............................. 475
6.9.2. Series containing is,+I/(z) and i[ (z)....................... 476
6.9.3. Series containing products of Ink+v (z)..................... 477
6.9.4. Series containing Ink+n((nk + v)z)....................... 480
6.9.5. Series containing products of Ink+i,((nk + v)z)............... 480
6.10. The Struve Functions H?(z) and L?(z)................. 481
6.10.1. Series containing Hft+?(z) and Ljt+?(z).................... 481
6.10.2. Series containing Hv(ip(k)x)............................ 482
6.10.3. Series containing H?(kx) and trigonometric functions........... 483
6.10.4. Series containing Hj,( a;) and Si (kx)...................... 484
6.10.5. Series containing Hv(kx), S(kx) and C(kx)................. 484
6.10.6. Series containing H?( ^(fe)a;) and JM(fcx)................... 485
6.10.7. Series containing product of H?(ftsc)....................... 485
6.11. The Legendre Polynomials Pn(z)..................... 486
6.11.1. Series containing Pnk+m(z)............................ 486
6.11.2. Series containing Pnk+m(z) and Bessel functions.............. 486
6.11.3. Series containing products of Pnk+m(z).................... 488
6.11.4. Series containing Pnk+m( p(k, z))........................ 488
6.12. The Chebyshev Polynomials Th(z) and Uk(z)............ 489
6.12.1. Series containing jTBfc+m (¥ ( ,*))........................ 489
6.12.2. Series containing Tnk+m(z) and Bessel functions.............. 489
Contents
6.12.3. Series containing Unk+m( p(k, z))........................ 490
6.12.4. Series containing Unk+m(z) and Bessel functions.............. 490
6.13. Hermite Polynomials Hn(z)......................... 492
6.13.1. Series containing Hnk+m.(z) and Bessel functions.............. 492
6.13.2. Series containing products of Hnk+m(z).................... 493
6.13.3. Series containing Hnk+m( p(k, z))........................ 493
6.13.4. Series containing Hnk+m( p(k, z)) and special functions.......... 494
6.13.5. Series containing products of if?fc+m(^(fc, z))................ 494
6.14. The Laguerre Polynomials L*(z)..................... 495
6.14.1. Series containing L^J| m(z)............................. 495
6.14.2. Series containing L^^m(z) and special functions.............. 496
6.14.3. Series containing products of L^l+m(z).................... 497
6.14.4. Series containing products of Ln(kx)...................... 497
6.14.5. Series containing L%j;+m(ip(k, z))........................ 497
6.14.6. Series containing L^+m( p(k, z)) and special functions.......... 498
6.14.7. Series containing products of L ^klkn(np(k, z))................ 498
6.15. The Gegenbauer Polynomials C*(z)................... 499
6.15.1. Series containing C^+^z)............................ 499
6.15.2. Series containing Cnk+m(z) and special functions.............. 500
6.15.3. Series containing products of C*klkm(z).................... 502
6.15.4. Series containing C?k+m( p(k, z))......................... 503
6.15.5. Series containing Cnk+m((p(k, z)) and special functions.......... 504
6.16. The Jacobi Polynomials PiP CT)(z)..................... 505
6.16.1. Series containing P^t^^Hz)-........................ 505
6.16.2. Series containing P^tnt ^^ (z) and special functions.......... 506
6.16.3. Series containing products of P^,±^ ° ±?fe)(z)................ 507
6.16.4. Series containing /S ^ ^fJt, z))..................... 507
6.16.5. Series containing Pr£±£k r q ( p(k, z)) and special functions...... 507
6.17. The Generalized Hypergeometric Function pFq((ap); (6,);z) 508
Series containing pFq((ap(k)); (bq(k)); z)................... 508
6.17.1
6.17.2
6.17.3
6.17.4
6.17.5
6.17.6
6.17.7.
6.17.8
Series containing PFq((ap(k)); (bq(k)); z) and trigonometric functions 521
Series containing pFq((ap(k)); (bg(k)); z) and special functions..... 524
Series containing products of PFg((ap(k)); (bg(k));z)........... 527
Series containing PFp+i((ap); (bp+i);ip(k, x))................ 527
Series containing pFp+1((ap(k));(bp+1(k)); p(k)z)............. 531
Series containing pFq((ap(k)); (bq(k)); (p(k)z) and special functions. . 533
Series containing products of PFg((ap(k));(bg(k)); p(k)z)........ 536
Contents
Chapter 7. The Connection Formulas...................... 537
7.1. Elementary Functions.............................. 537
7.1.1. Trigonometric functions............................... 537
7.2. Special Functions................................. 537
7.2.1. The psi function V(z)................................ 537
7.2.2. The incomplete gamma functions T(v, z) and 7(1/, z)............ 538
7.2.3. The parabolic cylinder function Dv(z)..................... 538
7.2.4. The Bessel functions Jv(z), H^(z), H^ z), I?(z) and K?(z) .... 539
7.2.5. The Struve functions H?(z) and L?(z)..................... 542
7.2.6. The Anger J?(z) and Weber E?(z) functions................. 543
7.2.7. The Airy functions Ai (z) and Bi (z)...................... 545
7.2.8. The Kelvin functions ber?(z), bei?(z), ker?(z) and kei?(z)....... 546
7.2.9. The Legendre polynomials Pn(z)......................... 549
7.2.10. The Chebyshev polynomials Tn(z) and Un(z)................ 549
7.2.11. The Hermite polynomials Hn(z)......................... 551
7.2.12. The Laguerre polynomials Ln(z)......................... 552
7.2.13. The Gegenbauer polynomials C*(z)....................... 553
7.2.14. The Jacobi polynomials P^ a)(z)........................ 555
7.2.15. The polynomials of the imaginary argument................. 558
7.2.16. The complete elliptic integral K (z)....................... 559
7.2.17. The complete elliptic integral E(z)........................ 560
7.2.18. The Legendre function Pjf(z)........................... 561
Chapter 8. Representations of Hypergeometric Functions
and of the Meijer G Function.................. 563
8.1. The Hypergeometric Functions....................... 563
8.1.1. The Gauss hypergeometric function 2F (a,b;c;z)............. 563
8.1.2. The hypergeometric function sF^ai, 02, 03; 61, 2;z)........... 589
8.1.3. The hypergeometric function 4Fa(ai, as, ag, 04:61, 62, 63; z)....... 601
8.1.4. The hypergeometric function sFjf^ai,... , as); ( i,. .., 64); z)..... 612
8.1.5. The hypergeometric function e F5(a ...., a%; 61,..., b;z)....... 618
8.1.6. The hypergeometric function yF^ ai,. .. ,ai;b ,... ,b^;z)....... 619
8.1.7. The hypergeometric function sFr(ai,... ,ag;6i,... ,br;z)....... 621
8.1.8. The hypergeometric function ioF9(ai,..., am; 61,.. ?, 69; z)...... 621
8.1.9. The Rummer confluent hypergeometric function F (a;b;z)...... 622
8.1.10. The Tricomi confluent hypergeometric function 9(a; b; z)........ 623
8.1.11. The hypergeometric function iF2(oi;6i, 62; z)................ 625
8.1.12. The hypergeometric function 2F2(oi, 02; 61, 62; z)............. 629
8.1.13. The hypergeometric function 2F3(oi,02:61, 62,63;z)........... 629
8.1.14. The hypergeometric function 3Fo(ai,02, 03; z)............... 631
8.1.15. The hypergeometric function sF0(ai,a2,... ,og;z)............ 631
8.1.16. The hypergeometric function 4Fi(oi,..., an; h; z)............. 631
Contents
8.1.17. The hypergeometric function 6Fi(ai,... ,06;6i;z)............. 632
8.1.18. The hypergeometric function 8F3(ai,... ,fl8;61,62,63;z)........ 632
8.1.19. The hypergeometric function (^3(61,62, 63; z)................ 633
8.1.20. The hypergeometric function oFr(6i,. .., 67; z)............... 634
8.1.21. The hypergeometric function 2Fs(ai, 02; 61,..., 65; z).......... 635
8.1.22. The hypergeometric function 4F7(ai,..., 04; 61,..., 67; z)....... 636
8.1.23. The generalized hypergeometric function pFq((ap);(bq);z)........ 637
8.2. The Meijer Function G
¦m,n I Kap)
P q (bq)J
638
8.2.1. General formulas................................... 638
8.2.2. Various Meijer G functions............................. 639
8.3. Representation in Terms of Hypergeometric Functions .... 652
8.3.1. Elementary functions................................. 652
8.3.2. Special functions.................................... 655
References............................................. 669
Index of Notations for Functions and Constants..................... 673
Index of Notations for Symbols................................ 679
|
| any_adam_object | 1 |
| author | Bryčkov, Jurij A. |
| author_facet | Bryčkov, Jurij A. |
| author_role | aut |
| author_sort | Bryčkov, Jurij A. |
| author_variant | j a b ja jab |
| building | Verbundindex |
| bvnumber | BV025531782 |
| classification_rvk | SK 680 |
| ctrlnum | (OCoLC)166358507 (DE-599)BVBBV025531782 |
| dewey-full | 515/.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.5 |
| dewey-search | 515/.5 |
| dewey-sort | 3515 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:35:59Z |
| institution | BVB |
| isbn | 9781584889564 |
| language | English |
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| spelling | Bryčkov, Jurij A. Verfasser aut Handbook of special functions derivatives, integrals, series and other formulas Yuri A. Brychkov Boca Raton, Fla. [u.a.] CRC Press 2008 XIX, 680 S. txt rdacontent n rdamedia nc rdacarrier A Chapman & Hall book Literaturverz. S. 669 - 672 Spezielle Funktion (DE-588)4182213-4 gnd rswk-swf (DE-588)4155008-0 Formelsammlung gnd-content Spezielle Funktion (DE-588)4182213-4 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020134851&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bryčkov, Jurij A. Handbook of special functions derivatives, integrals, series and other formulas Spezielle Funktion (DE-588)4182213-4 gnd |
| subject_GND | (DE-588)4182213-4 (DE-588)4155008-0 |
| title | Handbook of special functions derivatives, integrals, series and other formulas |
| title_auth | Handbook of special functions derivatives, integrals, series and other formulas |
| title_exact_search | Handbook of special functions derivatives, integrals, series and other formulas |
| title_full | Handbook of special functions derivatives, integrals, series and other formulas Yuri A. Brychkov |
| title_fullStr | Handbook of special functions derivatives, integrals, series and other formulas Yuri A. Brychkov |
| title_full_unstemmed | Handbook of special functions derivatives, integrals, series and other formulas Yuri A. Brychkov |
| title_short | Handbook of special functions |
| title_sort | handbook of special functions derivatives integrals series and other formulas |
| title_sub | derivatives, integrals, series and other formulas |
| topic | Spezielle Funktion (DE-588)4182213-4 gnd |
| topic_facet | Spezielle Funktion Formelsammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020134851&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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