Bibliography
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.
Airy, G. B. [1838], “On the Intensity of Light in the Neighborhood of a Caustic”, Trans. Camb. Philos. Soc., 6, 379–402.
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.
Bailey, V. A. [1954], “Reflection of Waves by an Inhomogeneous Medium”, Sci. Rep. 67, Ionospheric Res. Lab., Penn. State Univ.
Bailey, V. A. [1954], “Reflections of Waves by an Inhomogeneous Medium”, Phys. Rev. 96, 865–868.
Bell, R. P. [1944], “Eigenvalues and Eigenfunctions for the Operator D xx - ∣x∣”, Phil. Mag., 35, 582–588.
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.
Bellman, R., & Kalaba, R. [1958], “Invariant Imbedding, Wave Propagation, and the WKB Approximation”, Proc. Nat. Acad. Sci., 44, 317–319.
Bellman, R., & Kalaba, R. [1959], “Functional Equations, Wave Propagation, and Invariant Imbedding”, J. Math. Mech., 8, 683–704.
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.
Birkhoff, G. D. [1909], “Singular Points of Linear Ordinary Differential Equations”, Trans. Am. Math. Soc. 10, 436–470.
Birkhoff, G. D. [1933], “Quantum Mechanics and Asymptotic Series”, Bull. Am. Math. Soc., 39, 681–700.
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.
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.
Booker, H. G., & Walkinshaw, W. [1946], “Meteorological Factors in Radio-Wave Propagation”, Joint Conf. Phys. Soc. and Roy. Meteor. Soc., 80–127.
Boris, J. P., & Greene, J. M. [1969], “Determination of Subdominant Solutions Using a Partial Wronskian”, J. Comp. Phys., 4, 30–42.
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.
Brekhovskikh, L. M. [1960], Waves in Layered Media, Academic Press, New York.
Bremmer, H. [1949], Terrestrial Radio Waves, Elsevier Pub. Co., New York.
Bremmer, H. [1951], “The WKB Approximation as the First Term of a Geometric-Optical Series”, Comm. Pure and Appl. Math., 4, 105–115.
Brillouin, L. [1926], “Rémarques sur la méchaniques ondulatoire”, J. Phys. Radium, 7, 353–368.
Budden, K. G. [1961], Radio Waves in the Ionosphere, Cambridge Univ. Press.
Carrier, G. F., Krook, M., & Pearson, C. E. [1966], Functions of a Complex Variable, McGraw-Hill, New York.
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.
Cherry, T. M. [1949], “Uniform Asymptotic Expansions”, J. London Math. Soc., 24, 121–130.
Cherry, T. M. [1950], “Asyptotic Expansions for the Hypergeometric Functions Occurring in Gas Flow Theory”, Proc. Roy. Soc. London Ser. A, 202, 507–522.
Cherry, T. M. [1950], “Uniform Asymptotic Formulas for Functions with Transition Points”, Trans. Am. Math. Soc., 68, 224–257.
Clark, R. A. [1963], “Asymptotic Solution of a Nonhomogeneous Differential Equation with a Turning Point”, Arch. Rational Mech. Anal., 12, 34–51.
Copson, E. [1965], Asyptotic Expansions, Cambridge Univ. Press.
van der Corput, J. A. [1956], “Asymptotic Developments I: Fundamental Theorems of Asymptotics”, J. Anal. Mat., 4, 341–418.
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.
Doetsch, G. [1943], Theorie und Anwendung der Laplace-Transformation, Dover Pub., New York.
Dunham, J. L. [1932], “The WKB Method of Solving the Wave Equation”, Phys. Rev., 41, 713–720.
Dorr, F. W. [1969], “The Asymptotic Behaviour and Numerical Solution of Singular Perturbation Problems with Turning Points”, Thesis, Univ. Wis., Madison, Wisconsin.
Dorr, F. W. [1970], “Some Examples of Singular Perturbation Problems with Turning Points”, SIAM J. Math. Anal., 1, 141–146.
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.
Dorr, F. W., & Parter, S. V. [1970], “Singular Perturbations of Nonlinear Boundary Value Problems with Turning Points”, J. Math. Anal. Appl., 29, 273–293.
van Dyke, M. [1964], Perturbation Methods in Fluid Dynamics, Academic Press, New York.
Eagles, P. M. [1969], “Composite Series in the Orr-Sommerfeld Problem for Symmetric Channel Flow”, Q. J. Mech. Appl. Math., 22, 129–182.
van Engen, H. [1939], “Concerning Gamma Function Expansions”, Tôhoku Math. J., 45, 124–129.
Erdélyi, A. [1954], “Asymptotic Solutions of Differential Equations with Transition Points”, Proc. Intern. Congr. of Math. Amsterdam 1954, 3, 92–101.
Erdélyi, A. [1955], “Differential Equations with Transition Points I: The First Approximation”, Tech. Rep. 6, Dept. of Math., Calif. Inst. Tech., Pasadena, Calif.
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.
Erdélyi, A. [1960] “Asymptotic Solutions of Differential Equations with Transition Points or Singularities”, J. Math. Phys., 1, 16–26.
Erdélyi, A. [1961], “An Expansion Procedure for Singular Perturbations”, Atti della Accad. Sci. Torino I, Classe Sci. Fis., Mat., e Nat., 95, 651–672.
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.
Erdélyi, A., Kennedy, M., & McGregor, J. [1954], “Parabolic Cylinder Functions of Large Order”, J. Rational Mech. Anal., 3, 459–485.
Erdélyi, A., Magnus, W., Oberhettinger, F., & Tricomi, F. [1953], Higher Transcendental Functions I, II, III, McGraw-Hill, New York.
Evgrafov, M. A., & Fedorjuk, M. V. [1966], “Asymptotic Behaviour of Solutions of the Equation w - p (z, λ) w = 0 as λ → ∞ in the Complex z-Plane”, Usp. Mat. Nauk., 21, 3–50.
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.
Evans, R. L. [1953], “Solution of Linear Ordinary Differential Equations Containing a Parameter”, Proc. Am. Math. Soc., 4, 92–94.
Evans, R. L. [1953], “Solution of Linear Ordinary Differential Equations Containing a Large Parameter”, OOR Rep., Contract DA-11-022-ORD-489, Univ. Minn.
Fabry, E. [1885], “Sur les intégrales des équations différentielles linéaires à coéfficients rationnels”, Thèse, Paris.
Fedorjuk, M. [1965a], “Asymptotics of the Discrete Spectrum of the Operator w″ - λ 2 p (x) w = 0”, Mat. Sb. (Russian), 68, 81–110.
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.
Fedorjuk, M. V. [1965c], “Asymptotic Behaviour in a One-Dimensional Scattering Problem”, Dokl. Akad. Nauk. SSSR (Russian), 162, 287–289.
Fedorjuk, M. V. [1966], “Asymptotic Methods in the Theory of One-Dimensional Singular Differential Operators”, Tr. Mosk. Matem. Obsc. (Russian), 15, 296–345.
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.
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.
Ford, W. [1936], “The Asymptotic Developments of Functions Defined by MacLaurin Series”, Univ. of Mich. Science Series, No. 11.
Fowler, R. H. [1920] “The Aerodynamics of a Spinning Shell”, Phil. Trans. Roy. Soc. London Ser. A, 221, 295–387.
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.
Friedrichs, K. O. [1953], Special Topics in Analysis, Part B (Lecture Notes), New York University.
Friedrichs, K. O. [1955], “Asymptotic Phenomena in Mathematical Physics”, Bull. Am. Math. Soc. 61, 485–504.
Fröman, N. [1966], “Detailed Analysis of Some Properties of the JWKB Approximation”, Ark. Fys., 31, 381–408.
Fröman, N, [1966], “A Method for Handling Approximate Solutions of Ordinary Linear Differential Equations”, Ark. Fys., 31, 445–451.
Fröman, N. [1966], “Outline of a General Theory for Higher Order Approximations of the JWKB-Type”, Ark. Fys., 32, 541–548.
Fröman, N. [1970], “Connection Formulas for Certain Higher Order Phase-Integral Approximations”, Ann. Phys., 61, 451–464.
Fröman, N., & Fröman, P. O. [1965], JWKB Approximation: Contributions to the Theory, John Wiley, New York.
Fry, C. G., & Hughes, H. K. [1942], “Asymptotic Developments of Certain Integral Functions”, Duke Math. J., 9, 791–802.
Furry, W. H. [1947], “Two Notes on Phase Integral Methods”, Phys. Rev., 71, 360–371.
Gans, R. C. [1915], “Fortpflanzung des Lichts durch ein inhomogenes Medium”, Ann. Phys.. (Leipzig), 47, 709–736.
Gantacher, F. R. [1959], The Theory of Matrices, Vol. 2, trans. (from Russian) by K. A. Hirsch, Chelsea Pub. Co., New York.
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.
Goetschel, R. H. [1966], “Simplification of Certain Turning Point Problems for Systems of Order Four”, Thesis, University of Wisconsin.
Gol'dman, I., I. [1964], Problems in Quantum Mechanics, trans., ed., and rev. by D. ter Haar, 2nd ed., Academic Press, New York.
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.
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.
Green, G. [1837], “On the Motion of Waves in a Variable Canal of Small Depth and Width”, Camb. Phil. Trans., 6, 457–462.
Hanson, R. [1966], “Reduction Theorem for Systems of Ordinary Differential Equations with Turning Points”, J. Math. Anal. and Appl., 16, 280–301.
Hanson, R. [1967], “Analytic Linear Systems of Differential Equations in Implicit Form”, Funkcial. Ekvac., 10, 123–131.
Hanson, R. [1968], “Simplification of Second Order Systems of Ordinary Differential Equations with Turning Points”, SIAM J. Appl. Math., 16, 1059–1080.
Hanson, R., & Russel, D. [1967], “Classification and Reduction of Second Order Systems at a Turning Point”, J. Math. and Phys., 46, 74–92.
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.
Harris, W. [1960], “Singular Perturbation Problems”, Bol. Soc. Mat. Mex. (2), 5, 245–254.
Harris, W., & Turrittin, H. L. [1957], “Simplification of Systems of Linear Differential Equations Involving a Turning Point”, Rep. 2, Inst. Tech., Univ. Minn.
Heading, J. [1957], “The Stokes Phenomenon and Certain n th Order Differential Equations I, II”, Proc. Camb. Phil. Soc. 53, 399–418, 419–441.
Heading, J. [1960], “The Stokes Phenomenon and Certain n th Order Differential Equations”, Proc. Camb. Phil. Soc., 56, 329–341.
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.
Heading, J. [1962], An Introduction to Phase Integral Methods, Methuen and Co. Ltd., London.
Heading, J. [1962], “The Stokes Phenomenon of the Whittaker Function”, J. Lond. Math. Soc., 37, 195–208.
Heading, J. [1962], “Phase Integral Methods I”, Quart. J. Mech. Appl. Math., 15, 215–244.
Heading, J. [1963], “Uniformly Approximate Solutions of Certain n th Order Differential Equations I”, Proc. Camb. Phil. Soc., 59, 95–110.
Heading, J. [1964], “Transition Point Values”, J. London Math. Soc., 39, 466–480.
Hines, C. O. [1953], “Reflections of Waves from Varying Media”, Quart. Appl. Math., 11, 9–31.
Hinton, D. B., [1968], “Asymptotic Behaviour of Solutions of (ry (m))(k)± qy = 0”, J. Diff. Eqns., 4, 590–596.
Hochstadt, H. [1964], Differential Equations: A Modern Approach, Holt, Rhinehart, & Winston, New York.
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.
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.
Horn, J. [1899a], „Über eine lineare Differentialgleichung Zweiter Ordnung mit einem willkürlichen Parameter“, Math. Ann., 52, 271–292.
Horn, J. [1899b], „Über lineare Differentialgleichungen Zweiter mit einem Veränderlichen Parameter”, Math. Ann., 52, 340–362.
Horn, J. [1912], „Fakultätenreihen in der Theorie der linearen Differentialgleichungen“, Math. Ann., 71, 510–532.
Horn, J. [1915], „Integration linearer Differentialgleichungen durch Laplacesche Integrale und Fakultätenreihen“, Jahresker, Deut. Math. Ver., 24, 309–325; 25, 74–83.
Horn, J. [1944], „Integration von linearer Differentialgleichungen durch Laplacesche Integrale I, II“, Mat. Z., 49, 339–350, 684–701.
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.
Hsieh, P. [1968], “On an Analytic Simplification of a System of Linear Differential Equations Containing a Parameter”, Proc. Am. Math. Soc., 19, 1201–1206.
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.
Hsieh, P., & Sibuya, Y. [1966], “Regular Perturbations of Linear Differential Equations at an Irregular Singular Point”, Funkcial. Ekvac., 8, 99–108.
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.
Hughes, H. [1943], “On a Theorem of Newsom”, Bull. Am. Math. Soc. 49, 288–292.
Hughes, H. [1945], “The Asymptotic Developments of a Class of Entire Functions”, Bull. Am. Math. Soc., 51, 456–461.
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.
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.
Hukuhara, M., & Iwano, M. [1959], “Étude de la convergence des solutions formelles d'un systeme différentiel ordinaire linéaire”, Funkcial. Ekvac., 2, 1–18.
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.
Imai, I. [1948], “On a Refinement of the WKB Method”, in a letter to Phys. Rev., 74, 113.
Imai, I. [1956], “A Refinement of the WKB Method ...”, IRE trans. A. P., 4, 233–239.
Imai, I. [1958], “On the Heat Transfer to Constant-Property Laminar Boundary Layer”, Quart. Appl. Math., 16, 33–45.
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.
Iwano, M. [1964], “On the Behaviour of the Solutions of a n th Order Ordinary Differential Equation”, Japan. J. Math., 34, 1–53.
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.
Iwano, M., & Sibuya, Y. [1963], “Reduction of the Order of a Linear Ordinary Differential Equation containing a Parameter”, Kōdai Math. Sem. Rep., 15, 1–28.
Jeffreys, B. [1956], “The Use of the Airy Functions in a Potential Barrier Problem”, Proc. Camb. Phil. Soc., 52, 273–279.
Jeffreys, H. [1924], “On Certain Approximate Solutions of Linear Differential Equations of the Second Order”, Proc. London Math. Soc., (2), 23, 428–436.
Jeffreys, H. [1942], “Asymptotic Solutions of Linear Differential Equations”, Philos. Mag., 33, 451–456.
Jeffreys, H. [1953], “On Approximate Solutions of Linear Differential Equations”, Proc. Camb. Phil. Soc., 49, 601–611.
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.
Jeffreys, H. [1962], Asymptotic Approximation, Oxford University Press.
Jeffreys, H. & Jeffreys, B. [1956], Methods of Mathematical Physics, Cambridge University Press.
Jenssen, O. [1960], “Asymptotic Integration of the Differential Equation for a Special Case of Symmetrically Loaded Toroidal Shells”, J. Math. Phys., 39, 1–17.
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.
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.
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.
Kazarinoff, N. [1955], “Asymptotic Expansions for the Whittaker Functions of Large Complex Order”, Trans. Am. Math. Soc., 78, 305–328.
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.
Kazarinoff, N. [1958a], “Asymptotic Forms for the Whittaker Functions with Both Parameters Large”, J. Math. Mech., 6, 341–360.
Kazarinoff, N. [1958b], “Asymptotic Theory of Second Order Differential Equations with Two Simple Turning Points”, Arch. Rat. Mech. Anal., 2, 129–150.
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.
Kazarinoff, H., & McKelvey, R. [1956], “Asymptotic Solutions of Differential Equations in a Domain Containing a Regular Singular Point”, Can. J. Math., 8, 97–104.
Keller, H. B., & Keller, J. B. [1951], “On Systems of Linear Differential Equations”, Rep. EM-33, New York University.
Kemble, E. C. [1935], The Fundamental Principles of Quantum Mechanics, McGraw-Hill, New York.
Kemble, E. C. [1935], “A Contribution to the Theory of the WKB Method”, Phys. Rev., 48, 549–561.
Kemp, H., & Levinson, N. [1959], “On y xx +(1+λg(x))y=0”, Proc. Am. Math. Soc., 10, 82–86.
Kiyek, K. [1967], “Über eine spezielle Klasse linearer Differentialgleichungen mit einem kleinem Parameter”, Arch. Rational Mech. Anal., 25, 135–147.
Kramers, H. A. [1926], “Wellenmechanik und Halbzahlige Quantisierung”, Z. Physik., 39, 828–840.
Kuhn, T. S. [1950], “An Application of the WKB Method to the Cohesive Energy of Monovalent Metals”, Phys. Rev., 79, 515–519.
Kurss, H. [1957], “The Solution of Some Turning Point Problems”, Rep. IMM240, New York University.
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.
Lakin, W. D., & Reid, W. H. [1970], “Stokes Multipliers for the Orr-Sommerfeld Equation”, Phil. Trans. Roy. Soc. London Ser. A, 268, 325–349.
Lakin, W. D., & Sanchez, D. A. [1970], Topics in Ordinary Differential Equations: A Potpourri, Prindle, Weber & Schmidt Inc., Boston, Massachusetts.
Landau, L., & Lifshitz, E. [1958], Quantum Mechanics, Pergamon Press, London.
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.
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.
Langer, R. [1934a], “The Asymptotic Solutions of Certain Linear Differential Equations of the Second Order”, Trans. Am. Math. Soc., 36, 90–106.
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.
Langer, R. [1934c], “The Asymptotic Solutions of Linear Ordinary Differential Equations with Reference to the Stokes Phenomenon”, Bull. Am. Math. Soc., 40, 545–582.
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.
Langer, R. [1937], “On the Connection Formulas and the Solution of the Wave Equation”, Phys. Rev., 51, 669–676.
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.
Langer, R. [1950], “Asymptotic Solutions of a Differential Equation in the Theory of Microwave Propagation”, Comm. Pure Appl. Math., 3, 427–438.
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.
Langer, R. [1955], “The Solutions of the Differential Equation: y‴λ2 z y′+3 μλ2 y=0” Duke. Math. J., 22, 525–542.
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.
Langer, R. [1956b], “On the Construction of a Related Differential Equation”, Trans. Am. Math. Soc., 81, 394–410.
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.
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.
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.
Langer, R. [1959c], “Asymptotic Theories for Linear Ordinary Differential Equations Depending Upon a Parameter”, SIAM J. Appl. Math., 7, 298–305.
Langer, R. [1960], “Turning Points in Linear Asymptotic Theory”, Bol. Soc. Mat. Mex. (2), 5, 1–12.
Leavitt, W. G. [1948], “A Normal Form for Matrices whose Elements are Holomorphic Functions”, Duke Math. J., 15, 463–472.
Leavitt, W. G. [1951], “On Systems of Linear Differential Equations”, Am. J. Math., 73, 690–696.
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.
Lee, R. Y. [1968], “Turning Point Problems of Almost Diagonal Systems”, J. Math. Anal. Appl., 24, 509–526.
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.
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.
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.
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.
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.
Lin, C. C. [1966], The Theory of Hydrodynamic Stability, Cambridge University Press.
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.
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.
Linstone, H. A. [1954], “Singular Peturbations of Linear Differential Equations in the Complex Domain”, Thesis, Univ. Calif., Los Angeles.
Liouville, J. [1837], “Sur les développement des fonctions ...”, J. Math. Pures Appl. (1), 2, 16–35.
Lynn, R. [1968], “Uniform Asymptotic Expansions of Second Order Differential Equations with Turning Points”, Thesis, New York University.
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.
Malmquist, J. [1941], “Sur l'étude analytique ..., I, II, III”, Acta Math., 73, 87–129; 74, 1–64, 109–128.
Malmquist, J. [1943], “Sur les points singuliers des équations différentielles”, Ark. Math. Astr. Fys., 29A, No. 18.
McGuinness, D. L. [1965], “Differential Equations with Second Order Turning Points”, Thesis, Case Inst. Tech.
McGuiness, D. L. [1966], “Nonhomogeneous Differential Equation with a Second Order Turning Point”, J. Math. Phys., 7, 1030–1037.
McHugh, J. A. M. [1970], “New Results in Turning Point Theory”, Thesis, New York University.
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.
McKelvey, R. [1957], “Solution About a Singular Point of a Linear Differential Equation Involving a Large Parameter”, Trans. Am. Math. Soc., 91, 410–424.
McKelvey, & Kazarinoff, N. [1956], “Asymptotic Solutions of Differential Equations in a Domain Containing a Regular Singular Point”, Can. J. Math., 8, 97–104.
McLeod, J. B. [1961], “The Determination of the Transmission Coefficient”, Quart, J. Math. (2), 12, 153–158.
Meksyn, D. [1947], “Asymptotic Integration of a Fourth Order Differential Equation Containing a Large Parameter”, Proc. London Math. Soc. (2), 49, 436–457.
Miller, J. C. P. [1946], “The Airy Integral”, British Assoc. of Math. Tables, Cambridge Univ. Press.
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).
Miller, S. G., & Good, R. H. [1953], “A WKB Type Approximation to the Schroedinger Equation”, Phys. Rev., 91, 174–179.
Millington, G. [1969], “Stokes Phenomenon”, Radio Sci., 4, 95–115.
Moriguchi, H. [1959a], “Connection Formulas for WKB Solutions with Two Turning Points”, J. Phys. Soc. Japan., 14, 968.
Moriguchi, H. [1959b], “An Improvement of the WBK Method...”, J. Phys. Soc. Japan. 14, 1771–1796.
Morse, P. M., & Feshback, H. [1953], Methods of Theoretical Physics, Part 2, McGraw-Hill Book Co., Inc., New York.
Mullin, F. E. [1968], “On the Regular Perturbation of the Subdominant Solution to Second Order Linear Ordinary Differential Equations”, Funkcial. Ekvac., 11, 1–38.
Murphy, E. L., & Good, R. H. [1964], “WKB Connection Formulas”, J. Math. and Phys., 43, 251–254.
Newsom, C. V., & Franck, A. [1940], “On the Asymptotic Representation of Functions of the Bessel Type”, Bol. Mat., 13, 11–14.
Newsom, C. V. [1943], “The Asymptotic Behaviour of a Class of Entire Functions”, Am. J. Math., 65, 450–454.
Nishimoto, T. [1965], “On Matching Problems for a Linear Ordinary Differential Equation Containing a Parameter, I, II, III”, Kōdai Math. Sem. Rep., 17, 198–221, 307–328; 18, 61–86.
Nishimoto, T. [1967], “On Matching Methods for a Linear Ordinary Differential Equation Containing a Parameter”, Kōdai Math. Sem. Rep., 19, 80–94.
Nishimoto, T. [1968], “A Turning Point Problem of an n th Order Differential Equation of Hydrodynamic Type”, Kōdai Math. Sem. Rep., 20, 218–256.
Nishimoto, T. [1969], “A Remark on a Turning Point Problem”, Kōdai Math. Sem. Rep., 21, 58–63.
Nishimoto, T., & Okubo, K. [1966], “A Connection Problem for a Nonhomogeneous System of Linear Ordinary Differential Equations”, Funkcial. Ekvac., 9, 291–298.
Noaillon, P. [1912], “Dévelopments asymptotiques dans les équations linéaires à paramètre variable”, Mem. Soc. Roy. Sci. Liège IIIe Ser., 9.
Nörlund, N. [1926], “Leçons sur les Séries d'Interpolation”, Gautiers-Villars, Paris.
Okubo, K. [1961], “On Certain Reduction Theorems for Systems of Differential Equation which Contain a Turning Point”, Proc. Japan Acad., 37, 544–549.
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.
Okubo, K. [1965], “A Connection Problem Involving a Logarithmic Function”, Publ. Res. Inst. Math. Sci. Univ. Kyoto, Ser. A., 1, 99–128.
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.
Olver, F. W. J. [1954b], “The Asymptotic Expansion of Bessel Functions of Large Order”, Phil. Trans. Roy. Soc. London Ser. A 247, 328–368.
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.
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.
Olver, F. W. J. [1959a], “Linear Differential Equations of Second Order with a Large Parameter”, SIAM J. Appl. Math., 7, 306–310.
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.
Olver, F. W. J. [1961], “Error Bounds for the Liouville-Green (WKB) Approximations”, Proc. Camb. Phil. Soc., 57, 790–810.
Olver, F. W. J. [1963], “Error Bounds for First Approximations in Turning Point Problems”, SIAM J. Appl. Math., 11, 748–772.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Pitts, C. G. [1966] “y xx +(λ-q(x))y=0 on x≧0”, Quart. J. Math., (2), 17, 307–320.
Poincaré, H. [1886], “Sur les intégrales irrégulières des équations linéaires”, Acta. Math., 8, 295–344.
Ráb, M. [1966], “Note sur les formules asymptotiques pour les solutions d'un système d'équations différentielles linéaires”, Czechoslovak Math. J., 16, 127–129.
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.
Rabenstein, A. L. [1958], “Asymptotic Solutions of u (4)+λ2(zu″+αu′+βu)=0 for Large |λ|”, Arch. Rational Mech. Anal., 1, 418–435.
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.
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.
Russell, D. L. [1967], “Analytic Simplification of Second Order Systems with Combined Transition Point—Regular Singular Point”, Funkcial. Ekvac., 10, 15–34.
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.
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.
Saito, T. [1962], “On a Singular Point of a Second Order Linear Differential Equation Containing a Parameter”, Funkcial. Ekvac., 5, 1–29.
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.
Scheffé, H. [1942], “Linear Differential Equations with Two Term Recurrence Formulas”, J. Math. Phys., 21, 240–249.
Schlesinger, L. [1907], “Über asymptotische Darstellungen der Lösungen linearer Differentialsysteme als Funktionen eines Parameters”, Math. Ann., 63, 277–300.
Schwid, N. [1935], “The Asymptotic Forms of the Hermite and Weber Functions”, Trans. Am. Math. Soc., 37, 339–362.
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.
Sibuya, Y. [1958a], “Second Order Linear Ordinary Differential Equations Containing a Large Parameter”, Proc. Japan. Acad., 34, 229–234.
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.
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.
Sibuya, Y. [1960a], “On Perturbations of Discontinuous Solutions of Ordinary Differential Equations”, Nat. Sci. Rep., Ochaniomizu Univ., 11, 1–18.
Sibuya, Y. [1960b], “On Nonlinear Ordinary Differential Equations Containing a Parameter”, J. Math. Mech., 9, 369–398.
Sibuya, Y. [1962a], “Simplification of a System of Linear Ordinary Differential Equations about a Singular Point”, Funkcial. Ekvac., 4, 29–56.
Sibuya, Y. [1962b], “Asymptotic Solutions of a System of Linear Ordinary Differential Equations Containing a Parameter”, Funkcial. Ekvac., 4, 83–113.
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.
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.
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.
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.
Sibuya, Y. [1967], “Subdominant Solutions of the Differential Equation y″=λ2 (x-a 1) ... (x-a m )y″, Acta. Math., 119, 235–272.
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.
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.
Slepian, D. [1965], “Some Asymptotic Expansions for Prolate Spheroidal Wave Functions”, J. Math. and Phys., 44, 99–140.
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.
Stengle, G. [1969], “Uniform Asymptotic Solution of Second Order Linear Differential Equation without Turning Varieties”, Math. Comp., 23, 1–22.
Stokes, G. G. [1857], “On the Discontinuity of Arbitrary Constants which Appear in Divergent Developments”, Trans. Camb. Phil. Soc., 10, 105–128.
Stokes, G. G. [1868], “Supplement to a Paper on the Discontinuity of Arbitrary Constants, etc.”, Trans. Camb. Phil. Soc., 11, 412–425.
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.
Streifer, W. [1968], “Uniform Asymptotic Expansions for Prolate Spheriodal Wave Functions”, J. Math. and Phys., 47, 407–415.
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.
Swann, D. W. [1970b], “Improved Solutions to an Underwater Sound Propagation Problem”, Internal Memorandum, Bell Telephone Laboratories.
Swanson, C. A. [1956], “Differential Equations with Singular Points”, Techn. Rep. 16, Contract Nonr. 220(11), Dept. of Math., Cal. Inst. Techn., Pasadena, Calif.
Swanson, C. A., & Headley, V. B. [1967], “An Extension of Airy's Equation”, SIAM J. Appl. Math., 15, 1400–1412.
Synge, J. [1938], “Hydrodynamical Stability”, Semi-Centenial Publications of Am. Math. Soc., 2, 227–269.
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.
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.
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.
Thorne, R. C. [1957a], “The Asymptotic Solution of Differential Equations with a Turning Point and Singularities”, Proc. Cam. Phil. Soc., 53, 382–398.
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.
Titchmarsh, E. C. [1946], Eigenfunctions Expansions, Vol. 1, Oxford.
Tollmien, W. [1929], “Über die Entstehung der Turbulenz”, Nachr. Ges. Wiss. Göttingen, Math. Phys. Klasse, 21–44.
Tollmien, W. [1947], “Asymptotische Integration der Störungsdifferentialgleichung ebener laminarer Strömungen bei hohen Reynoldschen Zahlen”, Z. Angew. Math. Mech., 25/27, 33–50, 70–83.
Tollmien, W. [1948], “Laminare Grenzschichten”, appearing in Fiat Rev. German Sci. 1939–40, Hydro and Aero-Dynamics, Wiesbaden.
Trjitzinsky, W. J. [1934], “Analytic Theory of Linear Differential Equations”, Acta. Math., 62, 167–226.
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.
Trjitzinsky, W. J. [1936], “Theory of Linear Differential Equations Containing a Parameter”, Acta. Math., 67, 1–50.
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.
Turrittin, H. L. [1950], “Stokes Multipliers for Asymptotic Solutions of a Certain Differential Equation”, Trans. Am. Math. Soc., 68, 304–329.
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.
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.
Turrittin, H. L. [1963], “Reducing the Rank of Ordinary Differential Equations”, Duke Math. J., 30, 271–274.
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.
Turrittin, H. L. [1966], “Stokes Multipliers for the Differential Equation x y (n)=y”, Funkcial. Ekvac., 9, 261–272.
Turrittin, H. L., & Harris, W. [1957], “Simplification of Systems of Linear Differential Equations Involving a Turning Point”, Tech. Rep. 2, Inst. Tech., Univ. Minn.
Wasow, W. [1944], “On the Asymptotic Solution of Boundary Value Problems for Ordinary Differential Equations Containing a Parameter”, J. Math. Phys., 32, 173–183.
Wasow, W. [1948], “The Complex Asymptotic Theory of a Fourth Order Differential Equation of Hydrodynamics”, Ann. Math., 49, 852–871.
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.
Wasow, W. [1950b], “A Study of the Solutions of the Differential Equation y (4)+λ2(x y″+y)=0 for Large Values of λ”, Ann. Math., (2), 52, 350–361.
Wasow, W. [1953], “Asymptotic Solution of the Differential Equation of Hydrodynamic Stability in a Domain Containing a Transition Point”, Ann. Math., 58, 222–252.
Wasow, W. [1956], “Singular Perturbations of Boundary Value Problems for Nonlinear Differential Equations of the Second Order”, Comm. Pure Appl. Math., 9, 93–113.
Wasow, W. [1959], “Solution of Nonlinear Differential Equations with a Parameter by Asymptotic Series”, Ann. Math., 69, 486–509.
Wasow, W. [1960], “A Turning Point Problem for a System of Two Linear Differential Equations”, J. Math. Phys., 38, 257–278.
Wasow, W. [1961], “Turning Point Problems for Systems of Linear Equations, I. — The Formal Theory”, Comm. Pure Appl. Math., 14, 657–673.
Wasow, W. [1962a], “Turning Point Problems for Systems of Linear Differential Equations, II. — The Analytic Theory”, Comm. Pure Appl. Math., 15, 173–187.
Wasow, W. [1962b], “On Holomorphically Similar Matrices”, J. Math. Anal. Appl., 4, 202–206.
Wasow, W. [1963a], “Simplification of Turning Point Problems for System of Linear Differential Equations”, Trans. Am. Math. Soc., 106, 100–114.
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.
Wasow, W. [1965], Asymptotic Expansions for Ordinary Differential Equations, Interscience, New York.
Wasow, W. [1966a], “Almost Diagonal Systems”, Funkcial. Ekvac., 8, 143–171.
Wasow, W. [1966b], “On Analytic Validity of Formal Simplifications of Linear Differential Equations, I”, Funkcial. Ekvac., 9, 83–92.
Wasow, W. [1967], “On the Analytic Validity of Formal Simplifications of Linear Differential Equations, II”, Funkcial. Ekvac., 10, 107–122.
Wasow, W. [1968], “Connection Problems for Asymptotic Series”, Bull. Am. Math. Soc., 74, 831–853.
Wasow, W. [1970a], “Simple Turning Point Problems in Unbounded Domains”, SIAM J. Math. Anal., 1, 153–170.
Wasow, W. [1970b], “The Central Connection Problem at Turning Points of Linear Differential Equations”, to be published.
Watson, G. N. [1948], A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge Univ. Press.
Wentzel, G. [1926], “Eine Verallgemeinerung der Quantenbedingung für die Zwecke der Wellenmechanik”, Z. Physik., 38, 518–529.
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.
Wright, E. M. [1935], “The Asymptotic Expansion of the Generalized Bessel Function”, Proc. London Math. Soc. (2), 38, 257–270.
Wright, E. M. [1935], “The Asymptotic Expansion of the Generalized Hypergeometric Function”, J. London Math. Soc., 10, 286–293.
Wright, E. M. [1940a], “The Asymptotic Expansion of Integral Functions Defined by Taylor Series”, Phil. Trans. Roy. Soc. London Ser. A, 238, 423–451.
Wright, E. M. [1940b], “The Asymptotic Expansion of the Generalized Hypergeometric Function”, Proc. London Math. Soc., (2), 46, 389–408.
Wright, E. M. [1940c], “The Generalized Bessel Function of Order Greater than One”, Quart. J. Math., (2), 11, 36–48.
Wright, E. M. [1941], “The Asymptotic Expansion of Integral Functions Defined by Taylor Series”, Phil. Trans. Roy. Soc. London Ser. A, 239, 217–232.
Zwaan, A. [1929], “Intensitäten im Ca Funkenspectrum”, Thesis, Utrecht.
Further Bibliography
Braaksma, B. L. J. [1971], “Asymptotic Analysis of a Differential Equation of Turritin”, SIAM J. Math. Anal., 2, 1–16.
Butuzov, V. F., Vasil'eva, A. B., & Fedoryuk, M. V. [1970], “Asymptotic Methods in the Theory of Ordinary Differential Equations”, appearing in Progress in Mathematics, vol. 8: Mathematical Analysis, R. V. Gamkrelidze, ed., pp. 1–82, Plenum Press, New York-London.
Dorodynitsyn, A. A. [1952], “Asymptotic Distribution Laws for the Eigenvalues of Some Special Forms of Second Order Differential Equations”, Usp. Matem. Nauk., 7, 3–96.
Kiyek, K. [1963], “Zur Theorie der linearen Differentialgleichungssysteme mit einem großen Parameter”, Inaugurald-Dissertation zur Erlangung der Doktorwürde der Hohen Naturwissenschaftlichen Fakultät der Julius-Maximilians-Universität Würzburg, Würzburg.
Kohno, M. [1966], “A Two-Point Connection Problem Involving Logarithmic Polynomials”, Publications Res. Inst. Math. Sci., A2, 269–305.
Narayan, J., & Stengle, G. [1971], “Uniform Asymptotic Splitting of Linear Differential Equations”, appearing in Analytic Theory of Differential Equations, Lecture Notes in Mathematics, No. 183, 170–177, Springer-Verlag, New York.
Nishimoto, T. [1970], “On the Central Connection Problem at a Turning Point”, Kōdai Math. Sem. Rep., 22, 30–44.
Saito, T. [1959], “The W.K.B. Method for the Differential Equations of the Fourth Order”, J. Phys. Soc. Japan, 14, 1816–1819.
Thorne, R. C. [1960], “Asymptotic Formulae for Solutions of Linear Second Order Differential Equations with a Large Parameter”, J. Austral. Math. Soc., 1, 439–464.
Turrittin, H. L. [1971], “Stokes Multipliers for the Equations y‴−y/x 2=0″, appearing in Analytic Theory of Differential Equations, Lecture Notes in Mathematics, No. 183, 145–157, Springer-Verlag, New York.
Wasow, W. [1971], “The Central Connection Problem at Turning Points of Linear Differential Equations”, appearing in Analytic Theory of Differential Equations, No. 183, 158–164, Springer-Verlag, New York.
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McHugh, J.A.M. An historical survey of ordinary linear differential equations with a large parameter and turning points. Arch. Rational Mech. 7, 277–324 (1971). https://doi.org/10.1007/BF00328046
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DOI: https://doi.org/10.1007/BF00328046