On certain special functions that arise in the solution of ordinary differential equations by asymptotic methods

M. K. Kerimov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 36 (1996) no. 11, pp. 80-102

Bibliographie

[1] Amer. Math. Soc. Translations, 88, 1953 | MR

[2] Dorodnitsyn A. A., “Asimptoticheskie zakony raspredeleniya sobstvennykh znachenii dlya nekotorykh osobykh vidov differentsialnykh uravnenii vtorogo poryadka”, Uspekhi matem. nauk, 7:6 (1952), 3–96

[3] Kerimov M. K., “Spetsialnye funktsii, voznikayuschie v teorii nelineinykh kolebanii”, Zh. vychisl. matem. i matem. fiz., 36:8 (1996), 57–72 | MR | Zbl

[4] Kerimov M. K., “Pamyati Anatoliya Alekseevicha Dorodnitsyna”, Zh. vychisl. matem. i matem. fiz., 35:6 (1995), 819–842 | MR | Zbl

[5] Horn J., “Über eine lineare Differentialgleichung zweiter Ordnung mit einem willkürlichen Parameter”, Math. Ann., 52 (1899), 271–292 | DOI | MR | Zbl

[6] Jeffreys H., “On certain approximate solutions of linear differential equations of the second order”, Proc. London Math. Soc., 23 (1923), 428–436 | DOI

[7] Jeffreys H., “Asymptotic solutions of linear differential equations”, Philos. Mag., 33 (1942), 451–456 | MR | Zbl

[8] Longer R., “The asymptotic solutions of ordinary linear differential equations of the second order, with special reference to a turning point”, Trans. Amer. Math. Soc., 67 (1949), 461–490 | DOI | MR

[9] Petrashen M. I., “O poluklassicheskikh metodakh resheniya volnovogo uravneniya”, Uch. zap. LGU, 1949, no. 7, 59–78

[10] Olver F. W. J., Asymptotics and special functions, Acad. Press, London–New York, 1974 ; Olver F., Asimptotika i spetsialnye funktsii, Nauka, M., 1990 | MR | MR | Zbl

[11] Fedoryuk M. V., Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1983 | MR | Zbl

[12] Fletcher A., Miller J. C. P., Rosenhead L., Comrie L. J., An index of mathematical tables, v. 1, 2, Sec. ed., Blackwell Sci. Publ. Ltd., Oxford, 1962

[13] Kerimov M. K., “O metodakh vychisleniya dzeta-funktsii Rimana i nekotorykh ee obobschenii”, Zh. vychisl. matem. i matem. fiz., 20:6 (1980), 1580–1597 | MR | Zbl

[14] Eremin A. Yu., Kaporin I. E., Kerimov M. K., “O vychislenii dzeta-funktsii Rimana v kompleksnoi oblasti”, Zh. vychisl. matem. i matem. fiz., 25:4 (1985), 500–511 | MR | Zbl

[15] Eremin A. Yu., Kaporin I. E., Kerimov M. K., “O vychislenii proizvodnykh dzeta-funktsii Rimana v kompleksnoi oblasti”, Zh. vychisl. matem. i matem. fiz., 28:8 (1988), 1135–1148 | MR | Zbl

[16] Smirnov A. D., Tablitsy funktsii Eiri i spetsialnykh vyrozhdennykh gipergeometricheskikh funktsii dlya asimptoticheskikh reshenii differentsialnykh uravnenii vtorogo poryadka, Izd-vo AN SSSR, M., 1955

[17] Fedoryuk M. V., Metod perevala, Nauka, M., 1977 | MR

[18] Fedoryuk M. V., Asimptotika, integraly i ryady, Nauka, M., 1987 | MR

[19] Maslov V. P., Teoriya vozmuschenii i asimptoticheskie metody, Izd-vo MGU, M., 1965

[20] Maslov V. P., Fedoryuk M. V., Kvaziklassicheskie priblizheniya dlya uravnenii kvantovoi mekhaniki, Nauka, M., 1976 | MR

[21] Vasileva A. B., Butuzov V. F., Asimptoticheskie razlozheniya reshenii singulyarno vozmuschennykh uravnenii, Nauka, M., 1978 | MR

[22] Vasileva A. B., Butuzov V. F., Asimptoticheskie metody v teorii singulyarnykh vozmuschenii, Vyssh. shkola, M., 1990 | MR

[23] Butuzov V. F., Vasileva A. B., Fedoryuk M. V., “Asimptoticheskie metody v teorii obyknovennykh differentsialnykh uravnenii”, Itogi nauki. Matem. analiz, VINITI AN SSSR, M., 1969, 5–73

[24] Lomov S. A., Vvedenie v obschuyu teoriyu singulyarnykh vozmuschenii, Nauka, M., 1981 | MR

[25] Rapoport I. M., O nekotorykh asimptoticheskikh metodakh v teorii differentsialnykh uravnenii, Izd-vo AN USSR, Kiev, 1954

[26] Ilin A. M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989 | MR

[27] Feschenko S. F., Shkil N. I., Nikolenko L. D., Asimptoticheskie metody v teorii lineinykh differentsialnykh uravnenii, Nauk. dumka, Kiev, 1966 | MR

[28] Khedding Dzh., Vvedenie v metod fazovykh integralov (metod VKB), Perev. s angl. M. V. Fedoryuka, Mir, M., 1965

[29] Freman N., Freman P. U., VKB-priblizhenie, Mir, M., 1967

[30] Babich V. M., Buldyrev V. S., Asimptoticheskie metody v zadachakh difraktsii korotkikh voln, Nauka, M., 1972 | MR

[31] Erdeii A., Asimptoticheskie razlozheniya, Fizmatgiz, M., 1966

[32] Kopson E. T., Asimptoticheskie razlozheniya, Mir, M., 1966

[33] Vazov V., Asimptoticheskie razlozheniya reshenii obyknovennykh differentsialnykh uravnenii, Mir, M., 1968

[34] Jeffreys H., Asymptotic approximations, Oxford Univ. Press, London etc., 1962 | MR | Zbl

[35] Riekstynsh E. Ya., Asimptoticheskie razlozheniya integralov, v. 1, Zinatne, Riga, 1974; т. 2, 1977; т. 3, 1981; Оценки остатков в асимптотических разложениях, 1986

[36] Olver F. W. J., “Uniform asymptotic expansions for Weber parabolic cylinder functions of large order”, J. Res. Nat. Bur. Standards. Ser. B, 63:2 (1959), 131–169 | MR | Zbl

[37] Kerimov M. K., Certain new results of the theory of Weber functions, Bell Lab. Rept. TR 71-86, U.S.A., 1986

[38] Temme N. M., “Uniform asymptotic expansions of confluent hypergeometric functions”, J. Inst. Math. and Appl., 20 (1978), 215–223 | DOI | MR

[39] Kostomarov D. P., “Ob asimptoticheskom povedenii reshenii nekotorykh lineinykh differentsialnykh uravnenii vtorogo poryadka, soderzhaschikh bolshoi parametr”, Matem. sb., 45(87):1 (1958), 17–30 | MR | Zbl

[40] Dnestrovskii Yu. N., Kostomarov D. P., “Ob asimptotike sobstvennykh znachenii dlya nesamosopryazhennykh kraevykh zadach”, Zh. vychisl. matem. i matem. fiz., 4:2 (1964), 267–277 | MR

[41] Evgrafov M. A., Fedoryuk M. V., “Asimptotika reshenii uravneniya $w''(z)-p(z, \lambda)w(z)=0$ pri $\lambda\to\infty$ v kompleksnoi ploskosti $z$”, Uspekhi matem. nauk, 21:1 (1966), 3–50 | MR | Zbl

[42] Stokes G. G., “On the discontinuity of arbitrary constants which appear in divergent developments”, Trans. Cambridge Philos. Soc., 10 (1857), 105–128

[43] Ditkin B. A., Karpov K. A., Kerimov M. K., “O vychislenii spetsialnykh funktsii”, Zh. vychisl. matem. i matem. fiz., 20:5 (1980), 1093–1104 | MR | Zbl

[44] Lozier D. W., Olver F. W. J., “Numerical evaluation of special functions”, Math. Comput. 1943–1993: a Halfcentury of Comput. Math., Proc. Symp. Appl. Math., 48, ed. W. Gautschi, Amer. Math. Soc., Providence, 1994, 79–125 | MR | Zbl

[45] Van der Laan C. G., Temme N. M., Calculation of special functions: the gamma function, the exponential integrals and error-like functions, CWI Tract., 10, Sec. ed., Centre Math. and Comput. Sci., Amsterdam, 1986

[46] Lebedev A. V., Fedorova P. M., Spravochnik po matematicheskim tablitsam, Izd-vo AN SSSR, M., 1956

[47] Fok V. A., Tablitsy funktsii Eiri, Izdanie informatsionnogo otdela NII-108, M., 1946

[48] Hartree D. R., “Some calculations of transients in an electronic valve”, Appl. Sci. Res. Ser. B, 1:5 (1950), 379–390 | DOI | MR

[49] Woodward P. M., Woodward A. M., “Four-figure tables of the Airey functions in the complexe plane”, Philos. Mag., (7)37:267 (1946), 236–261 | MR

[50] Miller J. C. P., The Airey integral giving tables of solutions of the differential equation $y''=xy$, Cambridge Univ. Press, Cambridge, 1946 ; Math. Tables, v. B, British Assoc. Advancement Sci. | MR

[51] Yakovleva G. D., Tablitsy funktsii Eiri i ikh proizvodnykh, Nauka, M., 1969 | MR

[52] Nosova (Osipova) L. N., Tumarkin S. A., Tablitsy obobschennykh funktsii Eiri dlya asimptoticheskogo resheniya differentsialnykh uravnenii $\varepsilon(py')'+(q+\varepsilon r)y=f$, VTs AN SSSR, M., 1961 | MR

[53] Schulten Z., Anderson D. G. M., Gordon R. G., “An algorithm for the evaluation of the complex Airey functions”, J. Comput. Phys., 31 (1979), 60–75 | DOI | MR | Zbl

[54] Moon W., “Airey functions with complex arguments”, Comput. Phys. Communs., 22 (1981), 411–417 | DOI

[55] Razaz M., Schonfelder J. L., “Remark on algorithm 498. Airey functions using Chebyshev series approximations”, ACM Trans. Math. Software, 7 (1981), 404–405 | DOI

[56] M. Abramowitz, I. A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Nat. Bur. Standards, Appl. Math. Ser., 55, V.S. Government Printing Office, Washington, D.C. ; M. Abramovits, I. Stigan (red.), Spravochnik po spetsialnym funktsiyam, Nauka, M., 1979 | MR | MR

[57] Watson G. N., A treatise on the theory of Bessel functions, Sec. ed., Cambridge Univ. Press, Cambridge, 1966 ; Vatson G. N., Teoriya besselevykh funktsii, Perev. s angl. B. C. Bermana, v. 1, 2, Izd-vo inostr. lit., M., 1949 | MR

[58] Luke Y. L., Algorithms for the computation of mathematical functions, Acad. Press, New York etc., 1977 | MR

[59] Gautschi W., “Computational methods in special functions — a survey”, Theory and Applic. Special Functions, Proc. Advanced Sem. Math. Res. Center. Univ. Wisconsin, Madison, Wis., Acad. Press, New York etc., 1975, 1–98 | MR

[60] Temme N. M., Some aspects of applied analysis: asymptotics, special functions and their numerical computation, Math. Centrum, Amsterdam, 1978

[61] SpanierJ., Oldham K. B., An atlas of functions, Hemisphere Publ. Corp., Washington, D.C., 1987

[62] Sprott J. C., Numerical recipes: routines and examples in BASIC, Cambridge Univ. Press, Cambridge, 1991 | MR | Zbl

[63] Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P., Numerical recipes: the art scientific computing, Sec. ed., Cambridge Univ. Press, Cambridge, 1992 | MR

[64] Du Toit C. F., “Bessel functions $J_n(z)$ and $Y_n(z)$ of integer order and complex argument”, Comput. Phys. Communs., 78 (1993), 181–189 | DOI | Zbl

[65] Goldstein M., Thaler R. M., “Recurrence techniques for the calculation of Bessel functions”, Nat. Tables and Other Aids Comput., 13 (1959), 102–108 | DOI | MR | Zbl

[66] Sookne D. J., “Bessel functions $J$ and $I$ of complex argument and integer order”, J. Res. Nat. Bur. Standards. Ser. B, 77:3–4 (1973), 111–114 | MR | Zbl

[67] Sookne D. J., “Bessel functions of real argument and integer order”, J. Res. Nat. Bur. Standards. Ser. B, 77:3–4 (1973), 125–132 | MR | Zbl

[68] Sookne D. J., “Certification of an algorithm for Bessel functions of complex argument”, J. Res. Nat. Bur. Standards. Ser. B, 77:3–4 (1973), 133–136 | MR | Zbl

[69] Sookne D. J., “Certification of an algorithm for Bessel functions of real argument”, J. Res. Nat. Bur. Standards. Ser. B, 77:3–4 (1973), 115–124 | MR | Zbl

[70] Cody W. J., Mothley R. M. Fullerton L. W., “The computation of real fractional order Bessel functions of the second kind”, ACM Trans. Math. Software, 3 (1977), 232–239 | DOI | Zbl

[71] Campbell J. B., “Bessel functions $J_v(x)$ and $Y_v(x)$ of real order and real argument”, Comput. Phys. Communs., 18 (1979), 133–142 | DOI

[72] Campbell J. B., “On Temme's algorithm for the modified Bessel function of the third kind”, ACM Trans. Math. Software, 6 (1980), 581–586 | DOI | MR

[73] Temme N. M., “On the numerical evaluation of the ordinary Bessel function of the second kind”, J. Comput. Phys., 21 (1976), 343–350 | DOI | MR | Zbl

[74] Amos D. E., “Algorithm 644. A portable package for Bessel function of a complex argument and nonnegative order”, ACM Trans. Math. Software, 12:3 (1986), 265–273 | DOI | MR | Zbl

[75] Amos D. E., “A remark on algorithm 644. A portable package for Bessel functions of a complex argument and nonnegative order”, ACM Trans. Math. Software, 21:4 (1995), 388–393 | DOI | MR | Zbl

[76] Thompson I. J., Barnett A. R., “Coulomb and Bessel functions of complex arguments and order”, J. Comput. Phys., 64 (1986), 490–509 | DOI | MR

[77] Thompson I. J., Barnett A. R., “Modified Bessel functions $I_v(z)$ and $K_v(z)$ of real order and complex argument, to selected accuracy”, Comput. Phys. Communs., 47 (1987), 245–257 | DOI | MR

[78] Kerimov M. K., Skorokhodov S. L., “O vychislenii modifitsirovannykh funktsii Besselya v kompleksnoi oblasti”, Zh. vychisl. matem. i matem. fiz., 24:5 (1984), 650–654 | MR

[79] Vlasov V. K., Glukhova M. N., Korolev L. I. i dr., “O vychislenii funktsii Besselya s kompleksnym indeksom i kompleksnym argumentom”, Vestn. MGU. Ser. XV. Vychisl. matem. i kibernetika, 1991, no. 3, 46–51 ; 64 | MR | Zbl

[80] Kolbig K. S., “A program for computing the conical functions of the first kind $P^m_{-1/2\in i\tau}(x)$ for $m=0$ and $m=1$”, Comput. Phys. Communs., 23 (1981), 51–61 | DOI

[81] Olver F. W. J., Smith J. M., “Associated Legendre functions on the cut”, J. Comput. Phys., 51 (1983), 502–518 | DOI | MR | Zbl

[82] Babushkina L. V., Kerimov M. K., Nikitin A. I., “Algoritmy vychisleniya funktsii Besselya polutselykh indeksov s kompleksnymi argumentami”, Zh. vychisl. matem. i matem. fiz., 28:10 (1988), 1449–1460 | MR

[83] Babushkina L. V., Kerimov M. K., Nikitin A. I., “Algoritmy vychisleniya sfericheskikh funktsii Besselya v kompleksnoi oblasti”, Zh. vychisl. matem. i matem. fiz., 28:12 (1988), 1779–1788 | MR

[84] Schulten Z., Gordon R. G., Anderson D. G. M., “A numerical algorithm for the evaluation of Weber parabolic cylinder functions $U(a, x)$, $V(a, x)$ and $W(a, \pm x)$”, J. Comput. Phys., 42 (1981), 213–237 | DOI | MR | Zbl

[85] Taubmann G., “Parabolic cylinder functions $U(n, x)$ for natural $n$ and positive $x$”, Comput. Phys. Communs., 69 (1992), 413–419

[86] Temme N. M., “The numerical computation of the confluent hypergeometric function $U(a, b, z)$”, Numer. Math., 41 (1983), 63–83 | DOI | MR

[87] Slater L. J., Confluent hypergeometric functions, Cambridge Univ. Press, Cambridge, 1960 ; Sleiter L. Dzh., Vyrozhdennye gipergeometricheskie funktsii, Perev. s angl. M. K. Kerimova, Izd-vo VTs AN SSSR, M., 1968 | MR | Zbl | MR

[88] Elizalde E., “Zeta-function regularization is uniquely defined and well”, J. Phys. A, 27:9 (1994), L299–L304 | DOI | MR | Zbl