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An historical survey of ordinary linear differential equations with a large parameter and turning points. (English) Zbl 0227.34009


MSC:

34-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to ordinary differential equations
34A30 Linear ordinary differential equations and systems
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
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References:

[1] Abramowitz, M., & Stegun, I. A., editors [1964], Handbook of Mathematical Functions, Formulas, Graphs, and Mathematical Tables, Appl. Math. Series 55, U.S. National Bureau of Standards, Washington, D.C. · Zbl 0171.38503
[2] Airy, G. B. [1838], ?On the Intensity of Light in the Neighborhood of a Caustic?, Trans. Camb. Philos. Soc., 6, 379-402.
[3] Aliev, N. A. [1966], ?Asymptotic Representation of Fundamental Solutions of a System of First Order Equations?, Azerbaidzan Hos. Univ. Ucen. Zap. Ser. Fiz.?Mat. Nauk. (Russian; Azerbaijani summary), No. 5, 3-13.
[4] Bailey, V. A. [1954], ?Reflection of Waves by an Inhomogeneous Medium?, Sci. Rep. 67, Ionospheric Res. Lab., Penn. State Univ. · Zbl 0058.23007
[5] Bailey, V. A. [1954], ?Reflections of Waves by an Inhomogeneous Medium?, Phys. Rev. 96, 865-868. · Zbl 0058.23007 · doi:10.1103/PhysRev.96.865
[6] Bell, R. P. [1944], ?Eigenvalues and Eigenfunctions for the Operator D xx - ?x??, Phil. Mag., 35, 582-588. · Zbl 0061.18305
[7] Bell, R. P. [1945], ?The Occurrence and Properties of Molecular Vibrations with V (x = a x 4)?, Proc. Roy. Soc. London Ser. A, 183, 328-337. · Zbl 0063.00293 · doi:10.1098/rspa.1945.0006
[8] Bellman, R., & Kalaba, R. [1958], ?Invariant Imbedding, Wave Propagation, and the WKB Approximation?, Proc. Nat. Acad. Sci., 44, 317-319. · Zbl 0080.18301 · doi:10.1073/pnas.44.4.317
[9] Bellman, R., & Kalaba, R. [1959], ?Functional Equations, Wave Propagation, and Invariant Imbedding?, J. Math. Mech., 8, 683-704. · Zbl 0090.45301
[10] Birkhoff, G. D. [1908], ?On the Asymptotic Character of the Solutions of Certain Differential Equations Containing a Parameter?, Trans. Am. Math. Soc., 9, 219-231. · JFM 39.0386.01 · doi:10.1090/S0002-9947-1908-1500810-1
[11] Birkhoff, G. D. [1909], ?Singular Points of Linear Ordinary Differential Equations?, Trans. Am. Math. Soc. 10, 436-470. · JFM 40.0352.02 · doi:10.1090/S0002-9947-1909-1500848-5
[12] Birkhoff, G. D. [1933], ?Quantum Mechanics and Asymptotic Series?, Bull. Am. Math. Soc., 39, 681-700. · Zbl 0008.08902 · doi:10.1090/S0002-9904-1933-05716-6
[13] Birkhoff, G. D., & Langer, R. E. [1923], ?The Boundary Problems and Developments Associated with a System of Ordinary Differential Equations of the First Order?, Proc. Amer. Acad. Arts. Sci., 58, 51-128. · JFM 49.0723.01 · doi:10.2307/20025975
[14] Bohnenblust, H. F., et al. [1953], ?Asymptotic Solutions of Differential Equations with Turning Points, Review of the Literature?, Tech. Rep. 1, NR 043-121, Dept. Math., Calif. Inst. Tech., Pasadena, Calif. · Zbl 0052.09001
[15] Booker, H. G., & Walkinshaw, W. [1946], ?Meteorological Factors in Radio-Wave Propagation?, Joint Conf. Phys. Soc. and Roy. Meteor. Soc., 80-127.
[16] Boris, J. P., & Greene, J. M. [1969], ?Determination of Subdominant Solutions Using a Partial Wronskian?, J. Comp. Phys., 4, 30-42. · Zbl 0197.25302 · doi:10.1016/0021-9991(69)90038-2
[17] Bragg, R. E. [1958], ?Fundamental Solutions of a Linear Ordinary Differential Equation of the Third Order in the Neighborhood of a Single Second Order Turning Point?, Duke Math. J., 25, 239-264. · Zbl 0082.07103 · doi:10.1215/S0012-7094-58-02522-5
[18] Brekhovskikh, L. M. [1960], Waves in Layered Media, Academic Press, New York. · Zbl 0558.73018
[19] Bremmer, H. [1949], Terrestrial Radio Waves, Elsevier Pub. Co., New York.
[20] Bremmer, H. [1951], ?The WKB Approximation as the First Term of a Geometric-Optical Series?, Comm. Pure and Appl. Math., 4, 105-115. · Zbl 0043.20301 · doi:10.1002/cpa.3160040111
[21] Brillouin, L. [1926], ?Rémarques sur la méchaniques ondulatoire?, J. Phys. Radium, 7, 353-368. · doi:10.1051/jphysrad:01926007012035300
[22] Budden, K. G. [1961], Radio Waves in the Ionosphere, Cambridge Univ. Press. · Zbl 0094.22802
[23] Carrier, G. F., Krook, M., & Pearson, C. E. [1966], Functions of a Complex Variable, McGraw-Hill, New York. · Zbl 0146.29801
[24] Cashwell, E. D. [1951], ?The Asymptotic Solutions of an Ordinary Differential Equation in which the Coefficient of the Parameter is Singular?, Pacific J. Math., 1, 337-353. · Zbl 0043.09101
[25] Cherry, T. M. [1949], ?Uniform Asymptotic Expansions?, J. London Math. Soc., 24, 121-130. · Zbl 0045.34402 · doi:10.1112/jlms/s1-24.2.121
[26] Cherry, T. M. [1950], ?Asyptotic Expansions for the Hypergeometric Functions Occurring in Gas Flow Theory?, Proc. Roy. Soc. London Ser. A, 202, 507-522. · Zbl 0037.33102 · doi:10.1098/rspa.1950.0116
[27] Cherry, T. M. [1950], ?Uniform Asymptotic Formulas for Functions with Transition Points?, Trans. Am. Math. Soc., 68, 224-257. · Zbl 0036.06102 · doi:10.1090/S0002-9947-1950-0034494-3
[28] Clark, R. A. [1963], ?Asymptotic Solution of a Nonhomogeneous Differential Equation with a Turning Point?, Arch. Rational Mech. Anal., 12, 34-51. · Zbl 0111.08402 · doi:10.1007/BF00281218
[29] Copson, E. [1965], Asyptotic Expansions, Cambridge Univ. Press. · Zbl 0123.26001
[30] van der Corput, J. A. [1956], ?Asymptotic Developments I: Fundamental Theorems of Asymptotics?, J. Anal. Mat., 4, 341-418. · Zbl 0075.04901
[31] Dingle, R. B. [1958], ?Asymptotic Expansions and Converging Factors I, II, III?, Proc. Roy. Soc. London Ser. A, 244, 456-475, 476-483, 484-490. · Zbl 0080.04302 · doi:10.1098/rspa.1958.0054
[32] Doetsch, G. [1943], Theorie und Anwendung der Laplace-Transformation, Dover Pub., New York. · Zbl 0060.24709
[33] Dunham, J. L. [1932], ?The WKB Method of Solving the Wave Equation?, Phys. Rev., 41, 713-720. · Zbl 0005.27402 · doi:10.1103/PhysRev.41.713
[34] Dorr, F. W. [1969], ?The Asymptotic Behaviour and Numerical Solution of Singular Perturbation Problems with Turning Points?, Thesis, Univ. Wis., Madison, Wisconsin. · Zbl 0185.42102
[35] Dorr, F. W. [1970], ?Some Examples of Singular Perturbation Problems with Turning Points?, SIAM J. Math. Anal., 1, 141-146. · Zbl 0211.12005 · doi:10.1137/0501014
[36] Dorr, F. W., & Parter, S. V. [1969], ?Extensions of Some Results on Singular Perturbation Problems with Turning Points?, LA-4290-MS Los Alamos Scientific Laboratory of Univ. Calif., Los Alamos, New Mexico.
[37] Dorr, F. W., & Parter, S. V. [1970], ?Singular Perturbations of Nonlinear Boundary Value Problems with Turning Points?, J. Math. Anal. Appl., 29, 273-293. · Zbl 0183.36301 · doi:10.1016/0022-247X(70)90079-X
[38] van Dyke, M. [1964], Perturbation Methods in Fluid Dynamics, Academic Press, New York. · Zbl 0136.45001
[39] Eagles, P. M. [1969], ?Composite Series in the Orr-Sommerfeld Problem for Symmetric Channel Flow?, Q. J. Mech. Appl. Math., 22, 129-182. · Zbl 0184.53203 · doi:10.1093/qjmam/22.2.129
[40] van Engen, H. [1939], ?Concerning Gamma Function Expansions?, Tôhoku Math. J., 45, 124-129. · Zbl 0020.02101
[41] Erdélyi, A. [1954], ?Asymptotic Solutions of Differential Equations with Transition Points?, Proc. Intern. Congr. of Math. Amsterdam 1954, 3, 92-101.
[42] Erdélyi, A. [1955], ?Differential Equations with Transition Points I: The First Approximation?, Tech. Rep. 6, Dept. of Math., Calif. Inst. Tech., Pasadena, Calif. · Zbl 0072.11702
[43] Erdélyi, A. [1956], ?Asymptotic Factorization of Ordinary Linear Differential Operators Containing a Large Parameter?, Tech. Rep. 8, Dept. of Math., Calif. Inst. Techn., Pasadena, Calif. · Zbl 0073.30904
[44] Erdélyi, A. [1960] ?Asymptotic Solutions of Differential Equations with Transition Points or Singularities?, J. Math. Phys., 1, 16-26. · Zbl 0125.04802 · doi:10.1063/1.1703631
[45] Erdélyi, A. [1961], ?An Expansion Procedure for Singular Perturbations?, Atti della Accad. Sci. Torino I, Classe Sci. Fis., Mat., e Nat., 95, 651-672. · Zbl 0107.31104
[46] Erdélyi, A. [1964] ?The Integral Equations of Asymptotic Theory?, appearing in Proc. Sympos., Asymptotic Solutions of Differential Equations and Their Applications, C. H. Wilcox, ed., John Wiley, New York, 211-230. · Zbl 0178.14502
[47] Erdélyi, A., Kennedy, M., & McGregor, J. [1954], ?Parabolic Cylinder Functions of Large Order?, J. Rational Mech. Anal., 3, 459-485. · Zbl 0057.05502
[48] Erdélyi, A., Magnus, W., Oberhettinger, F., & Tricomi, F. [1953], Higher Transcendental Functions I, II, III, McGraw-Hill, New York. · Zbl 0052.29502
[49] Evgrafov, M. A., & Fedorjuk, M. V. [1966], ?Asymptotic Behaviour of Solutions of the Equation w - p (z, ?) w = 0 as ? ? ? in the Complex z-Plane?, Usp. Mat. Nauk., 21, 3-50. · Zbl 0173.33801
[50] Evans, R. L. [1951], ?Asymptotic Solutions in the Neighborhood of a Turning Point for Linear Ordinary Differential Equations Containing a Parameter?, Thesis, Univ. Minn., April, 1951.
[51] Evans, R. L. [1953], ?Solution of Linear Ordinary Differential Equations Containing a Parameter?, Proc. Am. Math. Soc., 4, 92-94. · Zbl 0050.08904 · doi:10.1090/S0002-9939-1953-0052608-4
[52] Evans, R. L. [1953], ?Solution of Linear Ordinary Differential Equations Containing a Large Parameter?, OOR Rep., Contract DA-11-022-ORD-489, Univ. Minn.
[53] Fabry, E. [1885], ?Sur les intégrales des équations différentielles linéaires à coéfficients rationnels?, Thèse, Paris.
[54] Fedorjuk, M. [1965a], ?Asymptotics of the Discrete Spectrum of the Operator w? - ? 2 p (x) w = 0?, Mat. Sb. (Russian), 68, 81-110. · Zbl 0238.34032
[55] Fedorjuk, M. V. [1965b], ?Topology of Stokes Curves for the Equation of the Second Order?, Izv. Akad. Nauk. SSSR Ser. Mat. (Russian), 29, 645-656.
[56] Fedorjuk, M. V. [1965c], ?Asymptotic Behaviour in a One-Dimensional Scattering Problem?, Dokl. Akad. Nauk. SSSR (Russian), 162, 287-289.
[57] Fedorjuk, M. V. [1966], ?Asymptotic Methods in the Theory of One-Dimensional Singular Differential Operators?, Tr. Mosk. Matem. Obsc. (Russian), 15, 296-345. · Zbl 0163.32402
[58] Fedorjuk, M. V. [1969], ?Asymptotic Expansions of Solutions of Differential Linear Equations of the Second Order in a Complex Domain?, MRC Tech. Sum. Rep., No. 993, trans. (from Russian) by F. Czyzewski, Math. Res. Centr., Univ. of Wis., Madison, Wisconsin. · Zbl 0176.05502
[59] Feschenko, S. F., Shkil’, N. I., & Nikolenko, L. D. [1967], Asymptotic Methods in the Theory of Linear Differential Equations, trans. (from Russian) by Scripta Technica, American Elsevier Pub. Co., New York.
[60] Ford, W. [1936], ?The Asymptotic Developments of Functions Defined by MacLaurin Series?, Univ. of Mich. Science Series, No. 11. · JFM 62.1203.03
[61] Fowler, R. H. [1920] ?The Aerodynamics of a Spinning Shell?, Phil. Trans. Roy. Soc. London Ser. A, 221, 295-387.
[62] Fraenkel, L. E. [1969], ?On the Method of Matched Asymptotic Expansions I, II, III?, Proc. Camb. Phil. Soc., 65, 209-231, 233-261, 263-284. · Zbl 0187.24104 · doi:10.1017/S0305004100044212
[63] Friedrichs, K. O. [1953], Special Topics in Analysis, Part B (Lecture Notes), New York University.
[64] Friedrichs, K. O. [1955], ?Asymptotic Phenomena in Mathematical Physics?, Bull. Am. Math. Soc. 61, 485-504. · Zbl 0068.16406 · doi:10.1090/S0002-9904-1955-09976-2
[65] Fröman, N. [1966], ?Detailed Analysis of Some Properties of the JWKB Approximation?, Ark. Fys., 31, 381-408. · Zbl 0143.23104
[66] Fröman, N, [1966], ?A Method for Handling Approximate Solutions of Ordinary Linear Differential Equations?, Ark. Fys., 31, 445-451. · Zbl 0138.32303
[67] Fröman, N. [1966], ?Outline of a General Theory for Higher Order Approximations of the JWKB-Type?, Ark. Fys., 32, 541-548.
[68] Fröman, N. [1970], ?Connection Formulas for Certain Higher Order Phase-Integral Approximations?, Ann. Phys., 61, 451-464. · doi:10.1016/0003-4916(70)90292-7
[69] Fröman, N., & Fröman, P. O. [1965], JWKB Approximation: Contributions to the Theory, John Wiley, New York. · Zbl 0129.41907
[70] Fry, C. G., & Hughes, H. K. [1942], ?Asymptotic Developments of Certain Integral Functions?, Duke Math. J., 9, 791-802. · Zbl 0063.01463 · doi:10.1215/S0012-7094-42-00954-2
[71] Furry, W. H. [1947], ?Two Notes on Phase Integral Methods?, Phys. Rev., 71, 360-371. · Zbl 0032.23405 · doi:10.1103/PhysRev.71.360
[72] Gans, R. C. [1915], ?Fortpflanzung des Lichts durch ein inhomogenes Medium?, Ann. Phys.. (Leipzig), 47, 709-736. · JFM 45.0597.03 · doi:10.1002/andp.19153521402
[73] Gantacher, F. R. [1959], The Theory of Matrices, Vol. 2, trans. (from Russian) by K. A. Hirsch, Chelsea Pub. Co., New York.
[74] Gibbons, J. J., & Schrag, R. L. [1952], ?The Wave Equation in a Region of Rapidly Varying Complex Refractive Index?, J. Appl. Phys., 23, 1139-1142. · Zbl 0047.44302 · doi:10.1063/1.1701997
[75] Goetschel, R. H. [1966], ?Simplification of Certain Turning Point Problems for Systems of Order Four?, Thesis, University of Wisconsin.
[76] Gol’dman, I., I. [1964], Problems in Quantum Mechanics, trans., ed., and rev. by D. ter Haar, 2nd ed., Academic Press, New York.
[77] Goldstein, S. [1928], ?A Note on Certain Approximate Solutions of Linear Differential Equations of the Second Order with an Application to the Mathieu Equation?, Proc. London Math. Soc, (2), 8, 81-90. · JFM 54.0398.04 · doi:10.1112/plms/s2-28.1.81
[78] Goldstein, S. [1932], ?A Note on Certain Approximate Solutions of Linear Differential Equations of the Second Order?, Proc. London Math. Soc. (2), 33, 246-252. · Zbl 0003.34401 · doi:10.1112/plms/s2-33.1.246
[79] Green, G. [1837], ?On the Motion of Waves in a Variable Canal of Small Depth and Width?, Camb. Phil. Trans., 6, 457-462.
[80] Hanson, R. [1966], ?Reduction Theorem for Systems of Ordinary Differential Equations with Turning Points?, J. Math. Anal. and Appl., 16, 280-301. · Zbl 0144.09801 · doi:10.1016/0022-247X(66)90171-5
[81] Hanson, R. [1967], ?Analytic Linear Systems of Differential Equations in Implicit Form?, Funkcial. Ekvac., 10, 123-131. · Zbl 0166.34402
[82] Hanson, R. [1968], ?Simplification of Second Order Systems of Ordinary Differential Equations with Turning Points?, SIAM J. Appl. Math., 16, 1059-1080. · Zbl 0164.39403 · doi:10.1137/0116086
[83] Hanson, R., & Russel, D. [1967], ?Classification and Reduction of Second Order Systems at a Turning Point?, J. Math. and Phys., 46, 74-92. · Zbl 0207.38501
[84] Harper, E., & Chang, I. [1970], ?A Second Order JWKB Approximation with One Turning Point and Two Singular Points ? Stability of an Accelerating Liquid Sphere?, Internal Memorandum, Bell Telephone Laboratories.
[85] Harris, W. [1960], ?Singular Perturbation Problems?, Bol. Soc. Mat. Mex. (2), 5, 245-254. · Zbl 0112.31502
[86] Harris, W., & Turrittin, H. L. [1957], ?Simplification of Systems of Linear Differential Equations Involving a Turning Point?, Rep. 2, Inst. Tech., Univ. Minn. · Zbl 0104.06101
[87] Heading, J. [1957], ?The Stokes Phenomenon and Certain n th Order Differential Equations I, II?, Proc. Camb. Phil. Soc. 53, 399-418, 419-441. · Zbl 0078.27001 · doi:10.1017/S0305004100032400
[88] Heading, J. [1960], ?The Stokes Phenomenon and Certain n th Order Differential Equations?, Proc. Camb. Phil. Soc., 56, 329-341. · Zbl 0097.07201 · doi:10.1017/S0305004100034630
[89] Heading, J. [1961], ?The Nonsingular Imbedding of Transition Processes within a More General Framework of Coupled Variables?, J. Res. Nat. Bur. Stand. D, 65, 595-616. · Zbl 0100.40203
[90] Heading, J. [1962], An Introduction to Phase Integral Methods, Methuen and Co. Ltd., London. · Zbl 0115.07102
[91] Heading, J. [1962], ?The Stokes Phenomenon of the Whittaker Function?, J. Lond. Math. Soc., 37, 195-208. · Zbl 0117.29905 · doi:10.1112/jlms/s1-37.1.195
[92] Heading, J. [1962], ?Phase Integral Methods I?, Quart. J. Mech. Appl. Math., 15, 215-244. · Zbl 0125.32304 · doi:10.1093/qjmam/15.2.215
[93] Heading, J. [1963], ?Uniformly Approximate Solutions of Certain n th Order Differential Equations I?, Proc. Camb. Phil. Soc., 59, 95-110. · Zbl 0115.07103 · doi:10.1017/S0305004100002048
[94] Heading, J. [1964], ?Transition Point Values?, J. London Math. Soc., 39, 466-480. · Zbl 0142.34701 · doi:10.1112/jlms/s1-39.1.466
[95] Hines, C. O. [1953], ?Reflections of Waves from Varying Media?, Quart. Appl. Math., 11, 9-31. · Zbl 0051.16803
[96] Hinton, D. B., [1968], ?Asymptotic Behaviour of Solutions of (ry (m))(k){\(\pm\)} qy = 0?, J. Diff. Eqns., 4, 590-596. · Zbl 0175.38101 · doi:10.1016/0022-0396(68)90008-9
[97] Hochstadt, H. [1964], Differential Equations: A Modern Approach, Holt, Rhinehart, & Winston, New York. · Zbl 0128.30501
[98] Horn, J. [1896] and [1897], ?Über die Reihenentwicklung der Integrale eines Systems von Differentialgleichungen in der Umgebung gewisser singulärer Stellen?, J. Reine Angew. Math., 116, 265-306; 117, 104-128, 254-266. · JFM 27.0260.01 · doi:10.1515/crll.1896.116.265
[99] Horn, J. [1898], ?Über das Verhalten der Integrale von Differentialgleichungen bei der Annäherung der Veränderlichen an eine Unbestimmtheitstelle?, J. Reine Angew. Math., 119, 196-209, 267-290. · JFM 29.0276.01 · doi:10.1515/crll.1898.119.196
[100] Horn, J. [1899a], ?Über eine lineare Differentialgleichung Zweiter Ordnung mit einem willkürlichen Parameter?, Math. Ann., 52, 271-292. · JFM 30.0300.01 · doi:10.1007/BF01476159
[101] Horn, J. [1899b], ?Über lineare Differentialgleichungen Zweiter mit einem Veränderlichen Parameter?, Math. Ann., 52, 340-362. · JFM 30.0300.02 · doi:10.1007/BF01476164
[102] Horn, J. [1912], ?Fakultätenreihen in der Theorie der linearen Differentialgleichungen?, Math. Ann., 71, 510-532. · JFM 43.0397.02 · doi:10.1007/BF01456806
[103] Horn, J. [1915], ?Integration linearer Differentialgleichungen durch Laplacesche Integrale und Fakultätenreihen?, Jahresker, Deut. Math. Ver., 24, 309-325; 25, 74-83. · JFM 45.0487.01
[104] Horn, J. [1944], ?Integration von linearer Differentialgleichungen durch Laplacesche Integrale I, II?, Mat. Z., 49, 339-350, 684-701. · Zbl 0061.17305 · doi:10.1007/BF01174204
[105] Hsieh, P. [1965], ?A Turning Point Problem for a System of Linear Ordinary Differential Equations of the Third Order (of a Two-Dimensional Vector)?, Arch. Rational Mech. Anal., 19, 117-148. · Zbl 0135.13503 · doi:10.1007/BF00282278
[106] Hsieh, P. [1968], ?On an Analytic Simplification of a System of Linear Differential Equations Containing a Parameter?, Proc. Am. Math. Soc., 19, 1201-1206. · Zbl 0167.08203 · doi:10.1090/S0002-9939-1968-0232987-6
[107] Hsieh, P., & Sibuya, Y. [1966], ?On the Asymptotic Integration of Second Order Linear Ordinary Differential Equations with Polynomial Coefficients?, J. Math. Anal. Appl., 16, 84-103. · Zbl 0161.05803 · doi:10.1016/0022-247X(66)90188-0
[108] Hsieh, P., & Sibuya, Y. [1966], ?Regular Perturbations of Linear Differential Equations at an Irregular Singular Point?, Funkcial. Ekvac., 8, 99-108. · Zbl 0145.10404
[109] Hsieh, P., & Shouse, O. D. [1967], ?Reduction of Order of a System of Linear Nonhomogeneous Ordinary Differential Equations?, Bul. Inst. Politen. Bucuresti, 29, 21-24. · Zbl 0219.34015
[110] Hughes, H. [1943], ?On a Theorem of Newsom?, Bull. Am. Math. Soc. 49, 288-292. · Zbl 0063.02936 · doi:10.1090/S0002-9904-1943-07903-7
[111] Hughes, H. [1945], ?The Asymptotic Developments of a Class of Entire Functions?, Bull. Am. Math. Soc., 51, 456-461. · Zbl 0063.02938 · doi:10.1090/S0002-9904-1945-08376-1
[112] Hukuhara, M. [1937], ?Sur les propriétés asymptotiques des solutions d’un système d’équations différentielles linéaires contenant un paramètre?, Mem. Fac. Fngrg., Kyushu Imp. Univ., 8, 249-280.
[113] Hukuhara, M. [1942], ?Sur les points singuliers des équations différentielles formelles d’un systeme différentiel ordinaire linéaire III?, Mem. Fac. Sci. Kyushu Imp. Univ., 2, 125-137. · doi:10.2206/kyushumfs.2.125
[114] Hukuhara, M., & Iwano, M. [1959], ?Étude de la convergence des solutions formelles d’un systeme différentiel ordinaire linéaire?, Funkcial. Ekvac., 2, 1-18. · Zbl 0123.27404
[115] Hurd, C. C. [1938], ?Asymptotic Theory of Linear Differential Equations Singular in the Variable of Differentiation and in a Parameter?, Tôhoku Math. J., 44, 243-274. · Zbl 0019.16701
[116] Imai, I. [1948], ?On a Refinement of the WKB Method?, in a letter to Phys. Rev., 74, 113. · Zbl 0032.02002 · doi:10.1103/PhysRev.74.113.2
[117] Imai, I. [1956], ?A Refinement of the WKB Method ...?, IRE trans. A. P., 4, 233-239.
[118] Imai, I. [1958], ?On the Heat Transfer to Constant-Property Laminar Boundary Layer?, Quart. Appl. Math., 16, 33-45. · Zbl 0080.18803
[119] Iwano, M. [1963] and [1964], ?Asymptotic Solutions of a System of Linear Ordinary Differential Equations Containing a Small Parameter I, II?, Funkcial. Ekvac., 5, 71-134; 6, 89-141. · Zbl 0123.04903
[120] Iwano, M. [1964], ?On the Behaviour of the Solutions of a n th Order Ordinary Differential Equation?, Japan. J. Math., 34, 1-53. · Zbl 0143.11103
[121] Iwano, M. [1965], ?On the Study of Asymptotic Solutions of a System of Linear Ordinary Differential Equation Containing a Parameter with a Singular Point?, Japan. J. Math., 35, 1-30. · Zbl 0145.10501
[122] Iwano, M., & Sibuya, Y. [1963], ?Reduction of the Order of a Linear Ordinary Differential Equation containing a Parameter?, K?dai Math. Sem. Rep., 15, 1-28. · Zbl 0115.07001 · doi:10.2996/kmj/1138844728
[123] Jeffreys, B. [1956], ?The Use of the Airy Functions in a Potential Barrier Problem?, Proc. Camb. Phil. Soc., 52, 273-279. · Zbl 0073.44504 · doi:10.1017/S030500410003125X
[124] Jeffreys, H. [1924], ?On Certain Approximate Solutions of Linear Differential Equations of the Second Order?, Proc. London Math. Soc., (2), 23, 428-436. · JFM 50.0301.02 · doi:10.1112/plms/s2-23.1.428
[125] Jeffreys, H. [1942], ?Asymptotic Solutions of Linear Differential Equations?, Philos. Mag., 33, 451-456. · Zbl 0028.35802
[126] Jeffreys, H. [1953], ?On Approximate Solutions of Linear Differential Equations?, Proc. Camb. Phil. Soc., 49, 601-611. · Zbl 0053.38304 · doi:10.1017/S0305004100028802
[127] Jeffreys, H. [1956], ?On the Use of Asymptotic Approximations of Green’s Type when the Coefficient has a Zero?, Proc. Camb. Phil. Soc., 52, 61-66. · Zbl 0070.12703 · doi:10.1017/S030500410003098X
[128] Jeffreys, H. [1962], Asymptotic Approximation, Oxford University Press. · Zbl 0101.29002
[129] Jeffreys, H. & Jeffreys, B. [1956], Methods of Mathematical Physics, Cambridge University Press. · Zbl 0070.40402
[130] Jenssen, O. [1960], ?Asymptotic Integration of the Differential Equation for a Special Case of Symmetrically Loaded Toroidal Shells?, J. Math. Phys., 39, 1-17. · Zbl 0102.18208
[131] Jordan, P. F., & Shelley, P. E. [1965], ?Formal Solutions of a Nonhomogeneous Differential Equation with a Double Transition Point?, J. Math. Phys., 6, 118-135. · Zbl 0135.37703 · doi:10.1063/1.1704249
[132] Jorna, A. [1964a], ?Derivation of Green Type, Transitional, and Uniform Asymptotic Expansions from Differential Equations I. General Theory, and Application to Modified Bessel Functions of Large Order?, Proc. Roy. Soc. London Ser. A, 281, 99-110. · Zbl 0122.06501 · doi:10.1098/rspa.1964.0171
[133] Jorna, A. [1964b], ?Derivation of Green Type, Transitional, and Uniform Asymptotic Expansions from Differential Equations II. Whittaker Functions W k,m for Large k and Large |k 2?m 2|?, Proc. Roy. Soc. London Ser. A, 281, 111-129. · Zbl 0122.06501 · doi:10.1098/rspa.1964.0172
[134] Kazarinoff, N. [1955], ?Asymptotic Expansions for the Whittaker Functions of Large Complex Order?, Trans. Am. Math. Soc., 78, 305-328. · Zbl 0067.29603
[135] Kazarinoff, N. [1956], ?Asymptotic Expansions with Respect to a Parameter of a Differential Equation Having an Irregular Singular Point?, Proc. Am. Math. Soc., 7, 62-69. · Zbl 0071.08301 · doi:10.1090/S0002-9939-1956-0075371-2
[136] Kazarinoff, N. [1958a], ?Asymptotic Forms for the Whittaker Functions with Both Parameters Large?, J. Math. Mech., 6, 341-360. · Zbl 0077.28201
[137] Kazarinoff, N. [1958b], ?Asymptotic Theory of Second Order Differential Equations with Two Simple Turning Points?, Arch. Rat. Mech. Anal., 2, 129-150. · Zbl 0082.07102 · doi:10.1007/BF00277924
[138] Kazarinoff, N. [1964], ?Application of Langer’s Theory of Turning Points to Diffraction Problems?, appearing in Proc. Sympos., Asymptotic Solutions of Differential Equations and Their Applications, C. H. Wilcox, ed., John Wiley, New York, 231-244.
[139] Kazarinoff, H., & McKelvey, R. [1956], ?Asymptotic Solutions of Differential Equations in a Domain Containing a Regular Singular Point?, Can. J. Math., 8, 97-104. · Zbl 0071.08202 · doi:10.4153/CJM-1956-015-6
[140] Keller, H. B., & Keller, J. B. [1951], ?On Systems of Linear Differential Equations?, Rep. EM-33, New York University.
[141] Kemble, E. C. [1935], The Fundamental Principles of Quantum Mechanics, McGraw-Hill, New York. · Zbl 0012.32501
[142] Kemble, E. C. [1935], ?A Contribution to the Theory of the WKB Method?, Phys. Rev., 48, 549-561. · JFM 61.1573.04 · doi:10.1103/PhysRev.48.549
[143] Kemp, H., & Levinson, N. [1959], ?On y xx +(1+?g(x))y=0?, Proc. Am. Math. Soc., 10, 82-86. · Zbl 0090.05802
[144] Kiyek, K. [1967], ?Über eine spezielle Klasse linearer Differentialgleichungen mit einem kleinem Parameter?, Arch. Rational Mech. Anal., 25, 135-147. · Zbl 0189.37201 · doi:10.1007/BF00281293
[145] Kramers, H. A. [1926], ?Wellenmechanik und Halbzahlige Quantisierung?, Z. Physik., 39, 828-840. · doi:10.1007/BF01451751
[146] Kuhn, T. S. [1950], ?An Application of the WKB Method to the Cohesive Energy of Monovalent Metals?, Phys. Rev., 79, 515-519. · Zbl 0038.41701 · doi:10.1103/PhysRev.79.515
[147] Kurss, H. [1957], ?The Solution of Some Turning Point Problems?, Rep. IMM240, New York University.
[148] Labianca, F., & Mow, V. [1969], ?Radiation from a Point Source in a Bounded Ocean Characterized by Two Turning Points, Part I, The Formal Solution?, Internal Memorandum, Bell Telephone Laboratories.
[149] Lakin, W. D., & Reid, W. H. [1970], ?Stokes Multipliers for the Orr-Sommerfeld Equation?, Phil. Trans. Roy. Soc. London Ser. A, 268, 325-349. · doi:10.1098/rsta.1970.0077
[150] Lakin, W. D., & Sanchez, D. A. [1970], Topics in Ordinary Differential Equations: A Potpourri, Prindle, Weber & Schmidt Inc., Boston, Massachusetts. · Zbl 0267.34002
[151] Landau, L., & Lifshitz, E. [1958], Quantum Mechanics, Pergamon Press, London.
[152] Langer, R. [1931], ?On the Asymptotic Solutions of Ordinary Differential Equations with an Application to the Bessel Functions of Large Order?, Trans. Am. Math. Soc., 33, 23-64. · Zbl 0001.06003 · doi:10.1090/S0002-9947-1931-1501574-0
[153] Langer, R. [1932], ?On the Asymptotic Solutions of Differential Equations with an Applications to the Bessel Functions of Large Complex Order?, Trans. Am. Math. Soc., 34, 447-480. · Zbl 0005.15801 · doi:10.1090/S0002-9947-1932-1501648-5
[154] Langer, R. [1934a], ?The Asymptotic Solutions of Certain Linear Differential Equations of the Second Order?, Trans. Am. Math. Soc., 36, 90-106. · JFM 60.0386.02 · doi:10.1090/S0002-9947-1934-1501736-5
[155] Langer, R. [1934b], ?The Solutions of the Mathieu Equation with Complex Variables and at least One Parameter Large?, Trans. Am. Math. Soc., 36, 637-695. · JFM 60.0388.01 · doi:10.1090/S0002-9947-1934-1501760-2
[156] Langer, R. [1934c], ?The Asymptotic Solutions of Linear Ordinary Differential Equations with Reference to the Stokes Phenomenon?, Bull. Am. Math. Soc., 40, 545-582. · Zbl 0009.39703 · doi:10.1090/S0002-9904-1934-05913-5
[157] Langer, R. [1935], ?On the Asymptotic Solutions of Ordinary Differential Equations with Reference to the Stokes Phenomenon about a Singular Point?, Trans. Am. Math. Soc., 37, 397-416. · Zbl 0011.30101
[158] Langer, R. [1937], ?On the Connection Formulas and the Solution of the Wave Equation?, Phys. Rev., 51, 669-676. · Zbl 0017.01705 · doi:10.1103/PhysRev.51.669
[159] Langer, R. [1949], ?The Asymptotic Solution of Ordinary Linear Differential Equations with Special Reference to a Turning Point?, Trans. Am. Math. Soc., 67, 461-490. · Zbl 0041.05901 · doi:10.1090/S0002-9947-1949-0033420-2
[160] Langer, R. [1950], ?Asymptotic Solutions of a Differential Equation in the Theory of Microwave Propagation?, Comm. Pure Appl. Math., 3, 427-438. · Zbl 0041.06001 · doi:10.1002/cpa.3160030406
[161] Langer, R. [1955], ?On the Asymptotic Forms of Ordinary Differential Equations of the Third Order in a Region Containing a Turning Point?, Trans. Am. Math. Soc., 80, 93-123. · Zbl 0065.31703 · doi:10.1090/S0002-9947-1955-0073009-5
[162] Langer, R. [1955], ?The Solutions of the Differential Equation: y??2 z y?+3 ??2 y=0? Duke. Math. J., 22, 525-542. · Zbl 0068.06804 · doi:10.1215/S0012-7094-55-02259-6
[163] Langer, R. [1956a], ?The Solutions of a Class of Linear Ordinary Differential Equations of the Third Order in a Region Containing a Multiple Turning Point?, Duke Math. J., 23, 93-110. · Zbl 0073.30905 · doi:10.1215/S0012-7094-56-02310-9
[164] Langer, R. [1956b], ?On the Construction of a Related Differential Equation?, Trans. Am. Math. Soc., 81, 394-410. · Zbl 0071.08103 · doi:10.1090/S0002-9947-1956-0079159-2
[165] Langer, R. [1957], ?On the Asymptotic Solutions of a Class of Ordinary Differential Equations of the Fourth Order with a Special Reference to an Equation of Hydrodynamics?, Trans. Am. Math. Soc., 84, 144-191. · Zbl 0077.08801 · doi:10.1090/S0002-9947-1957-0083637-0
[166] Langer, R. [1959a], ?The Asymptotic Solutions of a Linear Differential Equation of the Second Order with Two Turning Points?, Trans. Am. Math. Soc., 90, 113-142. · Zbl 0102.07401 · doi:10.1090/S0002-9947-1959-0105530-9
[167] Langer, R. [1959b], ?Formal Solutions and a Related Equation for a Class of Fourth Order Differential Equations of Hydrodynamic Type?, Trans. Am. Math. Soc., 92, 371-410. · Zbl 0086.28603 · doi:10.1090/S0002-9947-1959-0109923-5
[168] Langer, R. [1959c], ?Asymptotic Theories for Linear Ordinary Differential Equations Depending Upon a Parameter?, SIAM J. Appl. Math., 7, 298-305. · Zbl 0093.08604 · doi:10.1137/0107023
[169] Langer, R. [1960], ?Turning Points in Linear Asymptotic Theory?, Bol. Soc. Mat. Mex. (2), 5, 1-12. · Zbl 0143.31401
[170] Leavitt, W. G. [1948], ?A Normal Form for Matrices whose Elements are Holomorphic Functions?, Duke Math. J., 15, 463-472. · Zbl 0030.25203 · doi:10.1215/S0012-7094-48-01545-2
[171] Leavitt, W. G. [1951], ?On Systems of Linear Differential Equations?, Am. J. Math., 73, 690-696. · Zbl 0054.04001 · doi:10.2307/2372319
[172] Lee, R. [1967], ?Uniform Reduction of a System of Ordinary Differential Equations at a Turning Point?, Math. Res. Centr., U.S. Army, Univ. Wis., Madison, Wisconsin.
[173] Lee, R. Y. [1968], ?Turning Point Problems of Almost Diagonal Systems?, J. Math. Anal. Appl., 24, 509-526. · Zbl 0193.05901 · doi:10.1016/0022-247X(68)90006-1
[174] Lee, R. Y. [1969], ?On Uniform Simplification of a Linear Differential Equation in a Full Neighborhood of a Turning Point?, J. Math. Anal. Appl., 27, 501-510. · Zbl 0182.11604 · doi:10.1016/0022-247X(69)90129-2
[175] Lin, C. C. [1945] and [1946], ?On the Stability of Two Dimensional Parallel Flows, I, II, III?, Quart. Appl. Math., 3, 117-142, 218-234, 277-301. · Zbl 0061.43503
[176] Lin, C. C. [1958a], ?On the Instability of Laminar Flow and its Transition to Turbulence?, appearing in Proc. Sympos. on Boundary Layer Theory (Freiburg), Springer-Verlag, New York, 144-160. · Zbl 0091.19201
[177] Lin, C. C. [1958b], ?On The Stability of the Laminar Boundary Layer?, appearing in Symposium Naval Hydro. Counc., Nat. Res. Counc. Pub., Washington, D.C., 353-371.
[178] Lin, C. C. [1964], ?Some Examples of Asymptotic Problems in Mathematical Physics?, appearing in Proc. Sympos., Asymptotic Solutions of Differential Equations and Their Applications, John Wiley, New York, 129-144. · Zbl 0275.34032
[179] Lin, C. C. [1966], The Theory of Hydrodynamic Stability, Cambridge University Press.
[180] Lin, C. C., & Rabenstein, A. L. [1960], ?On the Asymptotic Solutions of a Class of Ordinary Differential Equations of the Fourth Order?, Trans. Am. Math. Soc., 94, 24-57. · Zbl 0092.07804
[181] Lin, C. C., & Rabenstein, A. L. [1969], ?On the Asymptotic Theory of a Class of Ordinary Differential Equations of Fourth Order, II, Existence of Solutions which are Approximated by the Formal Solutions?, Studies in Appl. Math., 48, 311-340. · Zbl 0184.11804
[182] Linstone, H. A. [1954], ?Singular Peturbations of Linear Differential Equations in the Complex Domain?, Thesis, Univ. Calif., Los Angeles.
[183] Liouville, J. [1837], ?Sur les développement des fonctions ...?, J. Math. Pures Appl. (1), 2, 16-35.
[184] Lynn, R. [1968], ?Uniform Asymptotic Expansions of Second Order Differential Equations with Turning Points?, Thesis, New York University.
[185] Lynn, R., & Keller, J. B. [1970], ?Uniform Asymptotic Solutions of Second Order Linear Ordinary Differential Equations with Turning Points?, Comm. Pure Appl. Math., 23, 379-408. · Zbl 0194.12202 · doi:10.1002/cpa.3160230310
[186] Malmquist, J. [1941], ?Sur l’étude analytique ..., I, II, III?, Acta Math., 73, 87-129; 74, 1-64, 109-128. · Zbl 0025.32702 · doi:10.1007/BF02392228
[187] Malmquist, J. [1943], ?Sur les points singuliers des équations différentielles?, Ark. Math. Astr. Fys., 29A, No. 18. · Zbl 0028.40801
[188] McGuinness, D. L. [1965], ?Differential Equations with Second Order Turning Points?, Thesis, Case Inst. Tech.
[189] McGuiness, D. L. [1966], ?Nonhomogeneous Differential Equation with a Second Order Turning Point?, J. Math. Phys., 7, 1030-1037. · Zbl 0141.27801 · doi:10.1063/1.1704994
[190] McHugh, J. A. M. [1970], ?New Results in Turning Point Theory?, Thesis, New York University.
[191] McKelvey, R. [1955], ?The Solutions of Second Order Ordinary Differential Equations about a Turning Point of Order Two?, Trans. Am. Math. Soc., 79, 103-123. · Zbl 0065.31801 · doi:10.1090/S0002-9947-1955-0069344-7
[192] McKelvey, R. [1957], ?Solution About a Singular Point of a Linear Differential Equation Involving a Large Parameter?, Trans. Am. Math. Soc., 91, 410-424. · Zbl 0086.28602 · doi:10.1090/S0002-9947-1959-0113017-2
[193] McKelvey, & Kazarinoff, N. [1956], ?Asymptotic Solutions of Differential Equations in a Domain Containing a Regular Singular Point?, Can. J. Math., 8, 97-104. · Zbl 0071.08202 · doi:10.4153/CJM-1956-015-6
[194] McLeod, J. B. [1961], ?The Determination of the Transmission Coefficient?, Quart, J. Math. (2), 12, 153-158. · Zbl 0101.06501 · doi:10.1093/qmath/12.1.153
[195] Meksyn, D. [1947], ?Asymptotic Integration of a Fourth Order Differential Equation Containing a Large Parameter?, Proc. London Math. Soc. (2), 49, 436-457. · Zbl 0041.41902 · doi:10.1112/plms/s2-49.6.436
[196] Miller, J. C. P. [1946], ?The Airy Integral?, British Assoc. of Math. Tables, Cambridge Univ. Press. · Zbl 0061.30506
[197] Miller, J. C. P. [1955], ?Tables of Weber Parabolic Cylinder Functions?, H. M. Stationary Office, London (see also, [1968] Russ. trans. and suppl. by M. K. Kerimov, Vycisl. Centr. Akad. Nauk. SSSR, Moscow).
[198] Miller, S. G., & Good, R. H. [1953], ?A WKB Type Approximation to the Schroedinger Equation?, Phys. Rev., 91, 174-179. · Zbl 0050.22103 · doi:10.1103/PhysRev.91.174
[199] Millington, G. [1969], ?Stokes Phenomenon?, Radio Sci., 4, 95-115. · doi:10.1029/RS004i002p00095
[200] Moriguchi, H. [1959a], ?Connection Formulas for WKB Solutions with Two Turning Points?, J. Phys. Soc. Japan., 14, 968. · doi:10.1143/JPSJ.14.968
[201] Moriguchi, H. [1959b], ?An Improvement of the WBK Method...?, J. Phys. Soc. Japan. 14, 1771-1796. · doi:10.1143/JPSJ.14.1771
[202] Morse, P. M., & Feshback, H. [1953], Methods of Theoretical Physics, Part 2, McGraw-Hill Book Co., Inc., New York.
[203] Mullin, F. E. [1968], ?On the Regular Perturbation of the Subdominant Solution to Second Order Linear Ordinary Differential Equations?, Funkcial. Ekvac., 11, 1-38. · Zbl 0266.34052
[204] Murphy, E. L., & Good, R. H. [1964], ?WKB Connection Formulas?, J. Math. and Phys., 43, 251-254.
[205] Newsom, C. V., & Franck, A. [1940], ?On the Asymptotic Representation of Functions of the Bessel Type?, Bol. Mat., 13, 11-14. · Zbl 0061.14008
[206] Newsom, C. V. [1943], ?The Asymptotic Behaviour of a Class of Entire Functions?, Am. J. Math., 65, 450-454. · Zbl 0063.05942 · doi:10.2307/2371969
[207] Nishimoto, T. [1965], ?On Matching Problems for a Linear Ordinary Differential Equation Containing a Parameter, I, II, III?, K?dai Math. Sem. Rep., 17, 198-221, 307-328; 18, 61-86. · Zbl 0142.34404 · doi:10.2996/kmj/1138845081
[208] Nishimoto, T. [1967], ?On Matching Methods for a Linear Ordinary Differential Equation Containing a Parameter?, K?dai Math. Sem. Rep., 19, 80-94. · Zbl 0158.09803 · doi:10.2996/kmj/1138845344
[209] Nishimoto, T. [1968], ?A Turning Point Problem of an n th Order Differential Equation of Hydrodynamic Type?, K?dai Math. Sem. Rep., 20, 218-256. · Zbl 0159.11903 · doi:10.2996/kmj/1138845646
[210] Nishimoto, T. [1969], ?A Remark on a Turning Point Problem?, K?dai Math. Sem. Rep., 21, 58-63. · Zbl 0174.40002 · doi:10.2996/kmj/1138845830
[211] Nishimoto, T., & Okubo, K. [1966], ?A Connection Problem for a Nonhomogeneous System of Linear Ordinary Differential Equations?, Funkcial. Ekvac., 9, 291-298. · Zbl 0199.13903
[212] Noaillon, P. [1912], ?Dévelopments asymptotiques dans les équations linéaires à paramètre variable?, Mem. Soc. Roy. Sci. Liège IIIe Ser., 9. · JFM 43.0375.01
[213] Nörlund, N. [1926], ?Leçons sur les Séries d’Interpolation?, Gautiers-Villars, Paris.
[214] Okubo, K. [1961], ?On Certain Reduction Theorems for Systems of Differential Equation which Contain a Turning Point?, Proc. Japan Acad., 37, 544-549. · Zbl 0113.06904 · doi:10.3792/pja/1195523585
[215] Okubo, K. [1963], ?A Global Representation for a Fundamental Set of Solutions and a Stokes Phenomenon for a System of Linear Ordinary Differential Equations?, J. Math. Soc. Japan, 15, 268-288. · Zbl 0134.06902 · doi:10.2969/jmsj/01530268
[216] Okubo, K. [1965], ?A Connection Problem Involving a Logarithmic Function?, Publ. Res. Inst. Math. Sci. Univ. Kyoto, Ser. A., 1, 99-128. · Zbl 0196.35101 · doi:10.2977/prims/1195196437
[217] Olver, F. W. J. [1954a], ?The Asymptotic Solution of Linear Differential Equations of the Second Order for Large Values of a Parameter?, Phil. Trans. Roy. Soc. London Ser. A, 247, 307-327. · Zbl 0070.30801 · doi:10.1098/rsta.1954.0020
[218] Olver, F. W. J. [1954b], ?The Asymptotic Expansion of Bessel Functions of Large Order?, Phil. Trans. Roy. Soc. London Ser. A 247, 328-368. · Zbl 0070.30801 · doi:10.1098/rsta.1954.0021
[219] Olver, F. W. J. [1956], ?The Asymptotic Solutions of Linear Differential Equations of the Second Order in a Domain Containing One Transition Point?, Phil. Trans. Roy. Soc. London Ser. A, 248, 65-97. · Zbl 0070.30801
[220] Olver, F. W. J. [1958], ?Uniform Asymptotic Expansions of Linear Second Order Differential Equations for Large Values of a Parameter?, Phil. Trans. Roy. Soc. London Ser. A, 250, 479-517. · Zbl 0083.05701 · doi:10.1098/rsta.1958.0005
[221] Olver, F. W. J. [1959a], ?Linear Differential Equations of Second Order with a Large Parameter?, SIAM J. Appl. Math., 7, 306-310. · Zbl 0094.28404 · doi:10.1137/0107024
[222] Olver, F. W. J. [1959b], ?Uniform Asymptotic Expansions for Weber Parabolic Cylinder Functions of Large Order?, J. Res. Nat. Bur. Stand. Sect. B, 63, 131-169. · Zbl 0090.04602
[223] Olver, F. W. J. [1961], ?Error Bounds for the Liouville-Green (WKB) Approximations?, Proc. Camb. Phil. Soc., 57, 790-810. · Zbl 0168.14003 · doi:10.1017/S0305004100035945
[224] Olver, F. W. J. [1963], ?Error Bounds for First Approximations in Turning Point Problems?, SIAM J. Appl. Math., 11, 748-772. · Zbl 0118.32902 · doi:10.1137/0111057
[225] Olver, F. W. J. [1964], ?Error Bounds for Asymptotic Expansions with an Application to Cylinder Functions of Large Argument?, appearing in Proc. Sympos., Asymptotic Solutions of Differential Equations and Their Applications, C. H. Wilcox, ed., John Wiley, New York, pp 163-183. · Zbl 0137.26703
[226] Olver, F. W. J. [1965a], ?On the Asymptotic Solutions of Second Order Differential Equations Having an Irregular Singularity of Rank One, with an Application to Whittaker Functions?, SIAM J. Numer. Anal. 2, 225-243. · Zbl 0173.33901
[227] Olver, F. W. J. [1965b], ?Error Analysis of Phase Integral Methods, I: General Theory for Simple Turning Points?, J. Res. Nat. Bur. Stand. Sect. B, 69, 271-290. · Zbl 0138.32401
[228] Olver, F. W. J. [1965c], ?Error Analysis of Phase Integral Methods, II: Application to Wave Penetration Problems?, J. Res. Nat. Bur. Stand. Sect. B, 69, 291-300. · Zbl 0138.32401
[229] Olver, F. W. J., & Stenger, F. [1965], ?Error Bounds for Asymptotic Solutions of Second-Order Differential Equations Having an Irregular Singularity of Arbitrary Rank?, SIAM J. Numer. Anal., 2, 244-249. · Zbl 0173.34001
[230] O’Malley, R. E. [1970], ?On Boundary Value Problems for a Singularly Perturbed Differential Equation with a Turning Point?, SIAM J. Math. Anal., 1, 479-490. · Zbl 0208.12301 · doi:10.1137/0501041
[231] Perron, O. [1918], [1918], and [1919], ?Über die Abhängigkeit der Integrale eines Systems linearer Differentialgleichungen von einem Parameter?, Sitz. der Heidelberger Akad. der Wissen., Math. Naturw., Abh. 13, Abh. 15, Abh. 3.
[232] Philipson, L. L. [1954], ?The Asymptotic Character of the Solutions of a Class of Ordinary Linear Differential Equations Depending on a Parameter?, Thesis, Univ. Calif., Los Angeles.
[233] Pike, E. R. [1964], ?On the Related Equation Method of Asymptotic Approximation, I, II: Direct Solutions of Wave Penetration Problems?, Quart. J. Mech. Appl. Math., 17, 105-124, 369-379. · Zbl 0121.31002 · doi:10.1093/qjmam/17.1.105
[234] Pitts, C. G. [1966] ?y xx +(?-q(x))y=0 on x?0?, Quart. J. Math., (2), 17, 307-320. · Zbl 0142.06605 · doi:10.1093/qmath/17.1.307
[235] Poincaré, H. [1886], ?Sur les intégrales irrégulières des équations linéaires?, Acta. Math., 8, 295-344. · JFM 18.0273.02 · doi:10.1007/BF02417092
[236] Ráb, M. [1966], ?Note sur les formules asymptotiques pour les solutions d’un système d’équations différentielles linéaires?, Czechoslovak Math. J., 16, 127-129.
[237] Ráb, M. [1969], ?Asymptotic Formulas for the Solutions of a System of Linear Differential Equations y?=[A+B(x)]y?, Casopis Pest. Math. (Czech. summary), 94, 78-83, 107.
[238] Rabenstein, A. L. [1958], ?Asymptotic Solutions of u (4)+?2(zu?+?u?+?u)=0 for Large |?|?, Arch. Rational Mech. Anal., 1, 418-435. · Zbl 0083.07802 · doi:10.1007/BF00298019
[239] Rabenstein, A. L. [1959], ?The Determination of the Inverse Matrix for a Basic Reference Equation for the Theory of Hydrodynamic Stability?, Arch. Rational Mech. Anal., 2, 355-366. · Zbl 0086.28701 · doi:10.1007/BF00277935
[240] Rayleigh, J. W. (J. W. Strutt), [1912], ?On the Propagation of Waves Through a Stratified Medium with Special Reference to the Question of Reflection?, Proc. Roy. Soc. London Ser. A, 86, 207-226. · JFM 43.0962.01 · doi:10.1098/rspa.1912.0014
[241] Russell, D. L. [1967], ?Analytic Simplification of Second Order Systems with Combined Transition Point?Regular Singular Point?, Funkcial. Ekvac., 10, 15-34. · Zbl 0189.37602
[242] Russell, D., & Sibuya, Y. [1966], ?The Problem of Singular Perturbations of Linear Ordinary Differential Equations at Regular Singular Points, I?, Funkcial. Ekvac., 9, 207-218. · Zbl 0166.07703
[243] Russell, D. L., & Sibuya, Y. [1969], ?The Problem of Singular Perturbations of Linear Ordinary Differential Equations at Regular Singular Points, II?, Funkcial. Ekvac., 11, 175-184. · Zbl 0184.12203
[244] Saito, T. [1962], ?On a Singular Point of a Second Order Linear Differential Equation Containing a Parameter?, Funkcial. Ekvac., 5, 1-29. · Zbl 0114.29003
[245] Scheffé, H. [1936], ?Asymptotic Solutions of Certain Linear Differential Equations in Which the Coefficient of the Parameter may have a Zero?, Trans. Am. Math. Soc., 40, 127-154. · JFM 62.0525.01
[246] Scheffé, H. [1942], ?Linear Differential Equations with Two Term Recurrence Formulas?, J. Math. Phys., 21, 240-249. · Zbl 0061.13906
[247] Schlesinger, L. [1907], ?Über asymptotische Darstellungen der Lösungen linearer Differentialsysteme als Funktionen eines Parameters?, Math. Ann., 63, 277-300. · JFM 38.0353.01 · doi:10.1007/BF01449198
[248] Schwid, N. [1935], ?The Asymptotic Forms of the Hermite and Weber Functions?, Trans. Am. Math. Soc., 37, 339-362. · Zbl 0011.21402 · doi:10.1090/S0002-9947-1935-1501790-1
[249] Sibuya, Y. [1954], ?Sur un système des équations différentielles ordinaires linéaires à coéfficients periodiques et contenant des paramètres?, J. Fac. Sci., Univ. Tokyo, (1), 7, 229-241. · Zbl 0058.07801
[250] Sibuya, Y. [1958a], ?Second Order Linear Ordinary Differential Equations Containing a Large Parameter?, Proc. Japan. Acad., 34, 229-234. · Zbl 0122.32401 · doi:10.3792/pja/1195524671
[251] Sibuya, Y. [1958b] ?Sur réduction analytique d’un système d’équations différentielles ordinaires linéaires contenant un paramètre?, J. Fac. Sci., Univ. Tokyo, (1), 7, 527-540. · Zbl 0081.08103
[252] Sibuya, Y. [1959], ?On the Problem of Turning Points?, MRC Techn. Sum. Report, No. 105, Math. Res. Ctr. U.S. Army, Univ. of Wis., Madison, Wis.
[253] Sibuya, Y. [1960a], ?On Perturbations of Discontinuous Solutions of Ordinary Differential Equations?, Nat. Sci. Rep., Ochaniomizu Univ., 11, 1-18. · Zbl 0107.07301
[254] Sibuya, Y. [1960b], ?On Nonlinear Ordinary Differential Equations Containing a Parameter?, J. Math. Mech., 9, 369-398. · Zbl 0113.06804
[255] Sibuya, Y. [1962a], ?Simplification of a System of Linear Ordinary Differential Equations about a Singular Point?, Funkcial. Ekvac., 4, 29-56. · Zbl 0145.32305
[256] Sibuya, Y. [1962b], ?Asymptotic Solutions of a System of Linear Ordinary Differential Equations Containing a Parameter?, Funkcial. Ekvac., 4, 83-113. · Zbl 0123.04902
[257] Sibuya, Y. [1962c], ?Formal Solutions of a Linear Ordinary Differential Equation of the n th Order at a Turning Point?, Funkcial. Ekvac., 4, 115-139. · Zbl 0123.04801
[258] Sibuya, Y. [1963a], ?Simplification of a Linear Ordinary Differential Equation of the n th Order at a Turning Point?, Arch. Rational Mech. Analysis, 13, 206-221. · Zbl 0115.29903 · doi:10.1007/BF01262693
[259] Sibuya, Y. [1963b], ?Asymptotic Solutions of a Linear Ordinary Differential Equation of n th Order about a Simple Turning Point?, appearing in Internat. Symp. Nonlinear Differential Equations and Nonlinear Mechanics, 485-488, Academic Press, New York. · Zbl 0139.03703
[260] Sibuya, Y. [1964], ?On the Problem of Turning Points for Systems of Linear Ordinary Differential Equations of Higher Orders?, Proc. Sympos., Asymptotic Solutions of Differential Equations and Their Applications, Math. Res. Ctr., U.S. Army, Univ. Wisconsin, Madison, Wis., Wiley, New York, 145-162. · Zbl 0161.06003
[261] Sibuya, Y. [1967], ?Subdominant Solutions of the Differential Equation y?=?2 (x-a 1) ... (x-a m )y?, Acta. Math., 119, 235-272. · Zbl 0159.11601 · doi:10.1007/BF02392084
[262] Slavjanov, S. J. [1969], ?Asymptotic Behaviour of Singular Sturm-Liouville Problems for Large Parameter in the Neighborhood of Nearby Transition Points?, Differencial’nye Uravenija, 5, 313-325. · Zbl 0164.39001
[263] Slavjanov, S. J., & Buldgrev, V. S. [1968], ?Uniform Asymptotic Expansions for Solutions of an Equation of Schrödinger Type with Two Transition Points, I?, Vestnik Leningrad. Univ., 23, 70-84.
[264] Slepian, D. [1965], ?Some Asymptotic Expansions for Prolate Spheroidal Wave Functions?, J. Math. and Phys., 44, 99-140. · Zbl 0128.29601
[265] Stengle, G. [1961], ?A Construction for Solutions of an n th Order Linear Differential Equation in the Neighborhood of a Turning Point?, Thesis, Univ. Wis.
[266] Stengle, G. [1969], ?Uniform Asymptotic Solution of Second Order Linear Differential Equation without Turning Varieties?, Math. Comp., 23, 1-22. · Zbl 0176.38804 · doi:10.1090/S0025-5718-1969-0247197-5
[267] Stokes, G. G. [1857], ?On the Discontinuity of Arbitrary Constants which Appear in Divergent Developments?, Trans. Camb. Phil. Soc., 10, 105-128.
[268] Stokes, G. G. [1868], ?Supplement to a Paper on the Discontinuity of Arbitrary Constants, etc.?, Trans. Camb. Phil. Soc., 11, 412-425.
[269] Stokes, G. G. [1889], ?Note on the Determination of Arbitrary Constants which Appear as Multipliers of Semi-Convergent Series?, Proc. Camb. Phil. Soc., 6, 362-366. · JFM 22.0297.01
[270] Streifer, W. [1968], ?Uniform Asymptotic Expansions for Prolate Spheriodal Wave Functions?, J. Math. and Phys., 47, 407-415. · Zbl 0277.33009
[271] Swann, D. W. [1970a], ?Asymptotic Solutions of Second Order Linear Differential Equations with Two Simple Turning Points whose Locations Depend Upon a Parameter?, Internal Memorandum, Bell Telephone Laboratories.
[272] Swann, D. W. [1970b], ?Improved Solutions to an Underwater Sound Propagation Problem?, Internal Memorandum, Bell Telephone Laboratories.
[273] Swanson, C. A. [1956], ?Differential Equations with Singular Points?, Techn. Rep. 16, Contract Nonr. 220(11), Dept. of Math., Cal. Inst. Techn., Pasadena, Calif.
[274] Swanson, C. A., & Headley, V. B. [1967], ?An Extension of Airy’s Equation?, SIAM J. Appl. Math., 15, 1400-1412. · Zbl 0161.27705 · doi:10.1137/0115123
[275] Synge, J. [1938], ?Hydrodynamical Stability?, Semi-Centenial Publications of Am. Math. Soc., 2, 227-269.
[276] Tamarkin, J. [1927], ?Some General Problems of the Theory of Ordinary Linear Differential Equations and Expansions of an Arbitrary Function in Series of Fundamental Functions?, Math. Z., 37, 1-54. · JFM 53.0419.02
[277] Tamarkin, J., & Besikowitsch, A. [1924], ?Über die asymptotischen Ausdrücke für die Integrale eines Systems linearer Differentialgleichungen die von einem Parameter abhängen?. Math. Z., 21, 119-125. · JFM 50.0300.02 · doi:10.1007/BF01187456
[278] Thorne, R. C. [1956], ?The Asymptotic Expansion of Legendre Functions of Large Degree and Order?, Tech. Rep. 12, 13, ONR, NR-043-121, Dept. Math., Calif. Inst. Tech., Pasadena, Calif. · Zbl 0078.05803
[279] Thorne, R. C. [1957a], ?The Asymptotic Solution of Differential Equations with a Turning Point and Singularities?, Proc. Cam. Phil. Soc., 53, 382-398. · Zbl 0078.05801 · doi:10.1017/S0305004100032394
[280] Thorne, R. C. [1957b], ?The Asymptotic Solution of Linear Second Order Differential Equations in a Domain Containing a Turning Point and a Regular Singularity?, Phil. Trans. Roy. Soc. London Ser. A, 249, 585-596. · Zbl 0078.05802 · doi:10.1098/rsta.1957.0007
[281] Titchmarsh, E. C. [1946], Eigenfunctions Expansions, Vol. 1, Oxford. · Zbl 0061.13505
[282] Tollmien, W. [1929], ?Über die Entstehung der Turbulenz?, Nachr. Ges. Wiss. Göttingen, Math. Phys. Klasse, 21-44. · JFM 55.0474.01
[283] Tollmien, W. [1947], ?Asymptotische Integration der Störungsdifferentialgleichung ebener laminarer Strömungen bei hohen Reynoldschen Zahlen?, Z. Angew. Math. Mech., 25/27, 33-50, 70-83. · Zbl 0029.08801 · doi:10.1002/zamm.19470250201
[284] Tollmien, W. [1948], ?Laminare Grenzschichten?, appearing in Fiat Rev. German Sci. 1939-40, Hydro and Aero-Dynamics, Wiesbaden.
[285] Trjitzinsky, W. J. [1934], ?Analytic Theory of Linear Differential Equations?, Acta. Math., 62, 167-226. · JFM 60.1109.01 · doi:10.1007/BF02393604
[286] Trjitzinsky, W. J. [1935], ?Laplace Integrals and Factorial Series in the Theory of Linear Differential and Difference Equations?, Trans. Am. Math. Soc., 37, 80-146. · Zbl 0011.06901 · doi:10.1090/S0002-9947-1935-1501779-2
[287] Trjitzinsky, W. J. [1936], ?Theory of Linear Differential Equations Containing a Parameter?, Acta. Math., 67, 1-50. · Zbl 0014.34802 · doi:10.1007/BF02401737
[288] Turrittin, W. J. [1936], ?Asymptotic Solutions of Certain Ordinary Differential Equations Associated with Multiple Roots of the Characteristic Equation?, Am. J. Math., 58, 364-376. · Zbl 0013.40005 · doi:10.2307/2371046
[289] Turrittin, H. L. [1950], ?Stokes Multipliers for Asymptotic Solutions of a Certain Differential Equation?, Trans. Am. Math. Soc., 68, 304-329. · Zbl 0037.06505 · doi:10.1090/S0002-9947-1950-0034491-8
[290] Turrittin, H. L. [1952], ?Asymptotic Expansions of Solutions of Systems of Ordinary Differential Equations?, Contributions to the Theory of Nonlinear Oscillations II; Ann. of Math. Studies No. 29, 81-116, Princeton. · Zbl 0047.08602
[291] Turrittin, H. L. [1955], ?Convergent Solutions of Ordinary Linear Homogeneous Differential Equations in the Neighborhood of an Irregular Singular Point?, Acta Math., 93, 27-66. · Zbl 0064.33603 · doi:10.1007/BF02392519
[292] Turrittin, H. L. [1963], ?Reducing the Rank of Ordinary Differential Equations?, Duke Math. J., 30, 271-274. · Zbl 0116.29005 · doi:10.1215/S0012-7094-63-03030-8
[293] Turrittin, H. L. [1964], ?Solvable Related Equations Pertaining to Turning Point Problems?, Asymptotic Solutions of Differential Equations and Their Applications, edited by C. H. Wilcox, Wiley, New York, 27-52. · Zbl 0144.10604
[294] Turrittin, H. L. [1966], ?Stokes Multipliers for the Differential Equation x y (n)=y?, Funkcial. Ekvac., 9, 261-272. · Zbl 0166.07605
[295] Turrittin, H. L., & Harris, W. [1957], ?Simplification of Systems of Linear Differential Equations Involving a Turning Point?, Tech. Rep. 2, Inst. Tech., Univ. Minn. · Zbl 0104.06101
[296] Wasow, W. [1944], ?On the Asymptotic Solution of Boundary Value Problems for Ordinary Differential Equations Containing a Parameter?, J. Math. Phys., 32, 173-183. · Zbl 0061.18202
[297] Wasow, W. [1948], ?The Complex Asymptotic Theory of a Fourth Order Differential Equation of Hydrodynamics?, Ann. Math., 49, 852-871. · Zbl 0031.40202 · doi:10.2307/1969402
[298] Wasow, W. [1950a], ?On the Construction of Periodic Solutions of Singular Perturbation Problems?, Contributions to the Theory of Nonlinear Oscillations, Ann. of Math. Studies vol. 20, Princeton, 313-350. · Zbl 0038.24602
[299] Wasow, W. [1950b], ?A Study of the Solutions of the Differential Equation y (4)+?2(x y?+y)=0 for Large Values of ??, Ann. Math., (2), 52, 350-361. · Zbl 0038.24602 · doi:10.2307/1969474
[300] Wasow, W. [1953], ?Asymptotic Solution of the Differential Equation of Hydrodynamic Stability in a Domain Containing a Transition Point?, Ann. Math., 58, 222-252. · Zbl 0051.06602 · doi:10.2307/1969787
[301] Wasow, W. [1956], ?Singular Perturbations of Boundary Value Problems for Nonlinear Differential Equations of the Second Order?, Comm. Pure Appl. Math., 9, 93-113. · Zbl 0074.30502 · doi:10.1002/cpa.3160090107
[302] Wasow, W. [1959], ?Solution of Nonlinear Differential Equations with a Parameter by Asymptotic Series?, Ann. Math., 69, 486-509. · Zbl 0091.08401 · doi:10.2307/1970196
[303] Wasow, W. [1960], ?A Turning Point Problem for a System of Two Linear Differential Equations?, J. Math. Phys., 38, 257-278. · Zbl 0091.26003
[304] Wasow, W. [1961], ?Turning Point Problems for Systems of Linear Equations, I. ? The Formal Theory?, Comm. Pure Appl. Math., 14, 657-673. · Zbl 0106.29301 · doi:10.1002/cpa.3160140336
[305] Wasow, W. [1962a], ?Turning Point Problems for Systems of Linear Differential Equations, II. ? The Analytic Theory?, Comm. Pure Appl. Math., 15, 173-187. · Zbl 0142.34403 · doi:10.1002/cpa.3160150206
[306] Wasow, W. [1962b], ?On Holomorphically Similar Matrices?, J. Math. Anal. Appl., 4, 202-206. · Zbl 0108.01601 · doi:10.1016/0022-247X(62)90050-1
[307] Wasow, W. [1963a], ?Simplification of Turning Point Problems for System of Linear Differential Equations?, Trans. Am. Math. Soc., 106, 100-114. · Zbl 0109.31003 · doi:10.1090/S0002-9947-1963-0142836-2
[308] Wasow, W. [1964], ?Asymptotic Expansions for Ordinary Differential Equations: Trends and Problems?, appearing in Proc. Sympos., Asymptotic Solutions of Differential Equations and Their Applications, Math. Res. Centr., U.S. Army, Univ. Wis., Madison, Wis., 3-26, John Wiley, New York, 1964. · Zbl 0136.08101
[309] Wasow, W. [1965], Asymptotic Expansions for Ordinary Differential Equations, Interscience, New York. · Zbl 0133.35301
[310] Wasow, W. [1966a], ?Almost Diagonal Systems?, Funkcial. Ekvac., 8, 143-171. · Zbl 0151.12602
[311] Wasow, W. [1966b], ?On Analytic Validity of Formal Simplifications of Linear Differential Equations, I?, Funkcial. Ekvac., 9, 83-92. · Zbl 0156.09505
[312] Wasow, W. [1967], ?On the Analytic Validity of Formal Simplifications of Linear Differential Equations, II?, Funkcial. Ekvac., 10, 107-122. · Zbl 0155.42102
[313] Wasow, W. [1968], ?Connection Problems for Asymptotic Series?, Bull. Am. Math. Soc., 74, 831-853. · Zbl 0174.39601 · doi:10.1090/S0002-9904-1968-12055-5
[314] Wasow, W. [1970a], ?Simple Turning Point Problems in Unbounded Domains?, SIAM J. Math. Anal., 1, 153-170. · Zbl 0211.11002 · doi:10.1137/0501016
[315] Wasow, W. [1970b], ?The Central Connection Problem at Turning Points of Linear Differential Equations?, to be published. · Zbl 0211.11002
[316] Watson, G. N. [1948], A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge Univ. Press.
[317] Wentzel, G. [1926], ?Eine Verallgemeinerung der Quantenbedingung für die Zwecke der Wellenmechanik?, Z. Physik., 38, 518-529. · JFM 52.0969.03 · doi:10.1007/BF01397171
[318] Wilcox, C. H. ed. [1964], Proc. Sympos., Asymptotic Solutions of Differential Equations and Their Applications, Math. Res. Ctr., U.S. Army, Univ. Wis., Madison, Wis., John Wiley, New York.
[319] Wright, E. M. [1935], ?The Asymptotic Expansion of the Generalized Bessel Function?, Proc. London Math. Soc. (2), 38, 257-270. · Zbl 0010.21103 · doi:10.1112/plms/s2-38.1.257
[320] Wright, E. M. [1935], ?The Asymptotic Expansion of the Generalized Hypergeometric Function?, J. London Math. Soc., 10, 286-293. · Zbl 0013.02104 · doi:10.1112/jlms/s1-10.40.286
[321] Wright, E. M. [1940a], ?The Asymptotic Expansion of Integral Functions Defined by Taylor Series?, Phil. Trans. Roy. Soc. London Ser. A, 238, 423-451. · Zbl 0023.14002 · doi:10.1098/rsta.1940.0002
[322] Wright, E. M. [1940b], ?The Asymptotic Expansion of the Generalized Hypergeometric Function?, Proc. London Math. Soc., (2), 46, 389-408. · Zbl 0025.40402 · doi:10.1112/plms/s2-46.1.389
[323] Wright, E. M. [1940c], ?The Generalized Bessel Function of Order Greater than One?, Quart. J. Math., (2), 11, 36-48. · Zbl 0023.14101 · doi:10.1093/qmath/os-11.1.36
[324] Wright, E. M. [1941], ?The Asymptotic Expansion of Integral Functions Defined by Taylor Series?, Phil. Trans. Roy. Soc. London Ser. A, 239, 217-232. · Zbl 0061.14506 · doi:10.1098/rsta.1941.0002
[325] Zwaan, A. [1929], ?Intensitäten im Ca Funkenspectrum?, Thesis, Utrecht.
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