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Functions

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Field Theory Handbook

Abstract

The purpose of this section is to provide a summary of the mathematical functions obtained as solutions of the differential equations of field theory. The summary is necessarily incomplete, since many of the functions have never been thoroughly investigated. In particular, almost nothing is known about the properties of the various wave functions Recent tabulation of spheroidal wave functions is a beginning in this direction, but much remains to be done.

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Bibliography

  1. Frobenius, G.: Über die Integration der linearen Differentialgleichungen durch Reihen. J. reine angew. Math. 76, 214 (1873).

    MATH  Google Scholar 

  2. Forsyth, A. R.: Theory of differential equations. Cambridge: Cambridge Univ. Press 1902; New York: Dover Publications 1959, Vol. IV.

    Google Scholar 

  3. Ince, E. L.: Ordinary differential equations. London: Longmans, Green & Co. 1927; New York: Dover Publications 1959, p. 396.

    Google Scholar 

  4. Moon, P., and D. E. SPENCER: Series solutions of Bâcher equations. (To be published.)

    Google Scholar 

  5. Nielsen, N.: Handbuch der Theorie der Cylinderfunktionen. Leipzig: B. G. Teubner 1904.

    Google Scholar 

  6. Watson, G. N.: Theory of Bessel functions, p. 64. Cambridge: Cambridge Univ. Press 1944.

    Google Scholar 

  7. Sturm, C., and J. Liouville: Extrait d’un mémoire sur le développement des fonctions en séries. J. de Math. 2, 220 (1837).

    Google Scholar 

  8. Liouville, J.: Premier mémoire sur la théorie des équations différentielles linéaires et sur le développement des fonctions en séries. J. de Math. 3, 561 (1838).

    Google Scholar 

  9. Churchill, R. V.: Fourier series and boundary value problems, p. 46. New York: McGraw-Hill Book Co. 1941.

    Google Scholar 

  10. Stratton, J. A.: Electromagnetic theory. New York: McGraw-Hill Book Co. 1941

    MATH  Google Scholar 

  11. Ramo, S., and J. R. Whinnery: Fields and waves in modern radio. New York: John Wiley & Sons 1953.

    Google Scholar 

  12. Smythe, W. R.: Static and dynamic electricity. New York: McGraw-Hill Book Co. 1950.

    Google Scholar 

  13. Southworth, C.: Principles and applications of waveguide transmission. Princeton, N. J.: D. Van Nostrand Co. 1950.

    Google Scholar 

  14. Marcuvitz, N.: Waveguide handbook. New York: McGraw-Hill Book Co. 1951.

    Google Scholar 

  15. Mclachlan, N. W.: Bessel functions for engineers. London: Oxford Univ. Press 1934.

    Google Scholar 

  16. Mclachlan, N. W.: Theory and applications of Mathieu functions. London: Oxford

    Google Scholar 

  17. Aharoni, J.: Antennae. London: Oxford Univ. Press 1946.

    Google Scholar 

  18. Kraus, J. D.: Antennas. New York: McGraw-Hill Book Co. 1950.

    Google Scholar 

  19. Silver, S.: Microwave antenna theory and design. New York: McGraw Hill Book Co. 1949.

    Google Scholar 

  20. Ollendorff, F.: Potentialfelder der Elektrotechnik. Berlin: Springer 1932.

    Google Scholar 

  21. Slater, J. C.: Microwave electronics. Princeton, N. J.: D. Van Nostrand Co. 1950.

    Google Scholar 

  22. Fano, R. M., L. J. Chu and R. B. Adler: Electromagnetic fields, energy, and forces. New York: John Wiley & Sons 1960.

    Google Scholar 

  23. Adler, R. B., L. J. Chu and R. M. Fano: Electromagnetic energy transmission and radiation. New York: John Wiley & Sons 1960.

    Google Scholar 

  24. Wait, J. R.: Electromagnetic radiation from cylindrical structures. New York: Pergamon Press 1959.

    MATH  Google Scholar 

  25. Morse, P. M., and H. Feshbach: Methods of theoretical physics. New York: McGraw-Hill Book Co. 1953.

    MATH  Google Scholar 

  26. Carslaw, H. S., and J. C. J AEGE R: Conduction of heat in solids. London: Oxford Univ. Press 1947.

    Google Scholar 

  27. Ingersoll, L. R., O. J. ZOBEL and A. C. INGERSOLL: Heat conduction, with engineering and geological applications. New York: McGraw-Hill Book Co. 1945.

    Google Scholar 

  28. Muskat, M.: The flow of homogeneous fluids through porous media. New York: McGraw-Hill Book Co. 1937.

    MATH  Google Scholar 

  29. Born, M., and E. Wolf: Principles of optics. New York: Pergamon Press 1959.

    MATH  Google Scholar 

  30. Weber, H.: Über die Integration der partiellen Differentialgleichung: i22L/Óx2+ ö210 y2 T k2u = O. Math. Ann. 1, 1 (1869).

    MathSciNet  MATH  Google Scholar 

  31. Hermite, C.: Sur un nouveau développement en série des functions. C. R. Acad. Sci., Paris 58, 93 (1864).

    Google Scholar 

  32. Whittaker, E. T., and G. N. WATSON: Modern analysis, p. 347. Cambridge: Cambridge Univ. Press 1946.

    Google Scholar 

  33. Ince, E. L.: Ordinary differential equations, p. 159. London: Longmans, Green & Co. 1927.

    Google Scholar 

  34. Whittaker, E. T.: On the functions associated with the parabolic cylinder in harmonic analysis. Proc. London Math. Soc. 35, 417 (1903).

    MATH  Google Scholar 

  35. Watson, G. N.: The harmonic functions associated with the parabolic cylinder. Proc. London Math. Soc. 8, 393 (1910).

    MATH  Google Scholar 

  36. Milne, A.: On the equation of the parabolic cylinder functions. Proc. Edinburgh Math. Soc. 32, 2 (1914).

    MATH  Google Scholar 

  37. Hille, E.: On the zeros of the functions of the parabolic cylinder.:Ark. Math., Astron. Fys. 18, 26 (1924).

    Google Scholar 

  38. Wells, C. P., and R. D. SPENCE: The parabolic cylinder functions. J. Math. Phys. 24, 51 (1945).

    MathSciNet  MATH  Google Scholar 

  39. Darwin, C. G.: On Weber’s function. Quart. J. Mech. Appl. Math. 2, 311 (1949).

    MathSciNet  MATH  Google Scholar 

  40. Miller, J. C. P.: On the choice of standard solutions for a homogeneous linear differential equation of the second order. Quar. J. Mech. Appl. Math. 3, 225 (1950).

    MATH  Google Scholar 

  41. Miller, J. C. P.: A method for the determination of converging factors, applied to the asymptotic expansions for the parabolic cylinder functions. Proc. Cambridge Phil. Soc. 48, 243 (1952).

    MATH  Google Scholar 

  42. Magnus, W., and F. Oberhettinger: Formeln und Sätze fiir die speziellen Funktionen der mathematischen Physik. Berlin: Springer 1943; New York: Chelsea Publ. Co. 1949.

    Google Scholar 

  43. ErdÉlyi, A., W. Magnus, F. Oberhettinger, and F. G. Tricomi: Higher transcendental functions, Vol. II, p. 115. New York: McGraw-Hill Book Co. 1953.

    Google Scholar 

  44. Emde, F.: Tafeln höherer Funktionen, p. 27. Leipzig: B. G. Teubner 1948. Tables of the confluent hypergeometric function Fan, 2; x) and related functions. Washington, D.C.: Nat. Bu. Stds. 1949.

    Google Scholar 

  45. Emde, F.: Tables of Weber parabolic cylinder functions. London: H. M. Stationery Office 1955.

    Google Scholar 

  46. Bessel, F. W.: Untersuchung des Teils der planetarischen Störung, welcher aus Bewegung der Sonne entsteht. Berl. Abh. 1 (1824).

    Google Scholar 

  47. Watson, G. N.: Theory of Bessel functions. Cambridge: Cambridge Univ. Press 1944.

    MATH  Google Scholar 

  48. Whittaker, E. T., and G. N. Watson: Modern analysis, Chap. 17. Cambridge: Cambridge Univ. Press 1946.

    Google Scholar 

  49. Gray, A., G. B. Mathewsand T. M. Macrobert: A treatise on Bessel functions. London: Macmillan Co. 1931.

    Google Scholar 

  50. Mclachlan, N. W.: Bessel functions for engineers. London: Oxford Univ. Press 1941.

    Google Scholar 

  51. Neumann, F.: Beiträge zur Theorie der Kugelfunktionen. Leipzig: B. G. Teubner 1878.

    Google Scholar 

  52. Heine, H. E.: Handbuch der Kugelfunktionen. Berlin: G. Reimer 1878.

    Google Scholar 

  53. Nielsen, N.: Handbuch der Theorie der Cylinderfunktionen. Leipzig: B. G. Teubner 1904.

    Google Scholar 

  54. Prasad, G.: A treatise on spherical harmonics and the functions of Bessel and Lamé. 2 vols. Benares: S. C. Chatterji 1932.

    Google Scholar 

  55. Lense, J.: Kugelfunktionen. Leipzig: Akad. Verlagsges. 1950.

    MATH  Google Scholar 

  56. Byerly, W. E.: Fourier’s series and spherical, cylindrical, and ellipsoidal harmonics. Boston: Ginn & Co. 1893.

    Google Scholar 

  57. Pockels, F.: Über die partielle Differentialgleichung A u K2u = 0 und deren Auftreten in der mathematischen Physik. Leipzig: B. G. Teubner 1891.

    Google Scholar 

  58. Bateman, H., and R. C. ARCHIBALD: A guide to tables of Bessel functions. Math. Tables and Other Aids to Comp. 1, 205 (1944).

    MATH  Google Scholar 

  59. Fletcher, A., J. C. P. MILLER and L. ROSENHEAD: Index of mathematical tables. London: Sci. Computing Service 1946.

    MATH  Google Scholar 

  60. Jahnke, E., and F. Emde: Tables of functions, p. 126. New York: Dover Publications 1943.

    Google Scholar 

  61. Watson, G. N.: Theory of Bessel functions, p. 666. Cambridge: Cambridge Univ. Press 1944.

    Google Scholar 

  62. Gray, A., G. B. Mathews and T. M. Macrobert: A treatise on Bessel functions. London: Macmillan Co. 1931.

    Google Scholar 

  63. Mclachlan, N. W.: Bessel functions for engineers. London: Oxford Univ. Press 1941.

    Google Scholar 

  64. Mclachlan, N. W.: British Assn. for Adv. Sci.: Bessel functions, Vol. 6 and 10. Cambridge: Cambridge Univ. Press 1937 and 1952.

    Google Scholar 

  65. Mclachlan, N. W.: Harvard Computation Lab.: Tables of the Bessel functions of the first kind, Vols. 3 to 14. Cambridge, Mass.: Harvard Univ. Press 1947 to 1951.

    Google Scholar 

  66. Mclachlan, N. W.: Tables of the modified Hankel functions of order one-third and their derivatives. Cambridge, Mass.: Harvard Univ. Press 1945.

    Google Scholar 

  67. Cambi, E.: Tables of Bessel functions of the first kind, to all significant orders. New York: Dover Publications 1948.

    MATH  Google Scholar 

  68. Cambi, E.: Nat. Bu. Stds.: Table of J0(z) and J1(z) for complex arguments. New York: Columbia Univ. Press 1947.

    Google Scholar 

  69. Cambi, E.: Tables of spherical Bessel functions. New York: Columbia Univ. Press 1947.

    Google Scholar 

  70. Cambi, E.: Tables of Bessel functions Y (z) and Y (z) for complex arguments. New York: Columbia Univ. Press 1950.

    Google Scholar 

  71. Cambi, E.: Tables of Bessel functions of fractional order, 2 vols. New York: Columbia Univ. Press 1948.

    Google Scholar 

  72. Onoe, M.: Tables of modified quotients of Bessel functions of the first kind and real and imaginary arguments. New York: Columbia Univ. Press 1958.

    Google Scholar 

  73. Onoe, M.: Tables of the Bessel functions Yo (x), Y (x), Kp (x), K, (x), 0 < x < 1. Washington, D. C.: Nat. Bu. Stds. 1948.

    Google Scholar 

  74. Baer, C.: Die Funktion des parabolischen Cylinders. Gymnasialprogramm, Cüstrin, 1883.

    Google Scholar 

  75. Baer, C.: Parabolische Coordinaten. Frankfurt 1888.

    Google Scholar 

  76. Mathieu, E.: Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique. J. de Math. 13, 137 (1868).

    Google Scholar 

  77. Curtis, M. F.: The existence of the functions of the elliptic cylinder. Ann. of Math. 20, 23 (1918).

    MathSciNet  MATH  Google Scholar 

  78. Humbert, P.: Fonctions de Lamé et fonctions de Mathieu. ( Mém. des sci. math.) Paris: Gauthier-Villars 1926.

    MATH  Google Scholar 

  79. Dhar, S. C.: Elliptic cylinder functions of second kind. Bull. Calcutta Math. Soc. 18, 11 (1927).

    Google Scholar 

  80. Ince, E. L.: The elliptic cylinder functions of the second kind. Proc. Edinburgh Math. Soc. 33, 2 (1914).

    Google Scholar 

  81. Ince, E. L.: Researches into the characteristic numbers of the Mathieu equation. Proc. Roy. Soc. Edinburgh 46, 20, 316 (1925); 47, 294 (1927).

    MATH  Google Scholar 

  82. Ince, E. L.: Ordinary differential equations, p. 175. London: Longmans, Green & Co. 1927.

    Google Scholar 

  83. Ince, E. L.: Tables of elliptic-cylinder functions. Proc. Roy. Soc. Edinburgh 52, 355 (1932).

    Google Scholar 

  84. Goldstein, S.: Mathieu functions. Trans. Cambridge Phil. Soc. 23, 303 (1927).

    Google Scholar 

  85. Goldstein, S.: The second solution of Mathieu’s differential equation. Proc. Cambridge Phil. Soc. 24, 223 (1928).

    MATH  Google Scholar 

  86. Varma, R. S.: On Mathieu functions. J. Indian Math. Soc. 19, 49 (1931).

    Google Scholar 

  87. Barrow, W. L.: Untersuchen über den Heulsummer. ann. cl. Phys. 11, 147 (1931).

    Google Scholar 

  88. Strutt, M. J. O.: Lamésche, Mathieusche und verwandte Funktionen in Physik und Technik. Berlin: Springer 1932.

    Google Scholar 

  89. Whittaker, E. T., and G. N. WATSON: Modern analysis, Chap. 19. Cambridge: Cambridge Univ. Press. 1946.

    Google Scholar 

  90. Blanch, G.: Computation of Mathieu functions. J. Math. Phys. 25, 1 (1946).

    MathSciNet  MATH  Google Scholar 

  91. Mclachlan, N. W.: Mathieu functions and their classification. J. Math. Phys. 25, 209 (1946).

    MathSciNet  MATH  Google Scholar 

  92. Mclachlan, N. W.: Theory and application of Mathieu functions, p. 19. London: Oxford Univ. Press 1947.

    Google Scholar 

  93. Morse, P. M., and H. FESHBACH: Methods of mathematical physics. New York: McGraw-Hill Book Co. 1953.

    Google Scholar 

  94. MEIxNER, J., u. F. W. ScHARKE: Mathieusche Funktionen und Sphäroidfunktionen. Berlin: Springer 1954.

    Google Scholar 

  95. Campbell, R.: Théorie générale de l’équation de Mathieu. Paris: Masson & Cie. 1955.

    Google Scholar 

  96. Fletcher, A. F., J. C. P. MILLER and L. ROSENHEAD: An index of mathematical tables, p. 328. London: Sci. Comp. Service 1946.

    Google Scholar 

  97. Stratton, J. A., P. M. MORSE, L. J. CHU and R. A. HUTNER: Elliptic cylinder and spheroidal wave functions. New York: John Wiley & Sons 1941. Tables relating to Mathieu functions. New York: Columbia Univ. Press 1951.

    Google Scholar 

  98. Emde, F.: Tafeln höherer Funktionen. Leipzig: B. G. Teubner 1948.

    MATH  Google Scholar 

  99. Ince, E. L.: Tables of elliptic-cylinder functions. Proc. Roy. Soc. Edinburgh 52

    Google Scholar 

  100. Goldstein, S.: Mathieu functions. Trans. Cambridge Phil. Soc. 23, 303 (1927).

    Google Scholar 

  101. Legendre, A. M.: Sur l’attraction des sphéroïdes. Mém. Math. Phys. 10 (1785).

    Google Scholar 

  102. Hobson, E. W.: The theory of spherical and ellipsoidal harmonics. Cambridge: Cambridge Univ. Press 1931.

    Google Scholar 

  103. Heine, E.: Kugelfunktionen. Berlin: G. Reimer 1878.

    Google Scholar 

  104. Byerlv, W. E.: Fourier’s series and spherical, cylindrical, and ellipsoidal liarmonics. Boston: Ginn & Co. 1893.

    Google Scholar 

  105. Prasad, G.: Spherical harmonics and the functions of Bessel and Lamé, 2 vols. Henares: S. C. Chatterji 1930.

    Google Scholar 

  106. Whittaker, E. T., and G. N. WATSON: Modern analysis, Chap. 15. Cambridge: Cambridge Univ. Press 1946.

    Google Scholar 

  107. Macrobert, T. M.: Spherical harmonics. New York: Dover Publications 1948.

    MATH  Google Scholar 

  108. Robin, L.: Fonctions sphériques de Legendre et fonctions sphéroidales, 3 vols. Paris: Gauthier-Villars 1959.

    Google Scholar 

  109. Davis, H. T.: A bibliography and index of mathematical tables. Evanston, Ill. 1949.

    Google Scholar 

  110. Jahnke, E., and F. Emde: Tables of functions, p. 119. Leipzig: B. G. Teubner 1938.

    Google Scholar 

  111. Fowle, F. E.: Smithsonian physical tables, p. 66. Washington, D. C. 1933.

    Google Scholar 

  112. Nat. Bu. Stds.: Table of associated Legendre functions. New York: Columbia Univ. Press 1945.

    Google Scholar 

  113. Mursi, Z.: Table of Legendre associated functions. Cairo.: Fouad I Univ. 1941.

    Google Scholar 

  114. Tallquist, H.: Tafeln der Kugelfunktionen. Acta Soc. Sci. Fenn. 32, No. 6 (1906); 33, Nos. 4 and 9 (1908); 6, Nos. 3 and 10 (1932); 2, No. 4 (1937); 2, No. 11 (1938).

    Google Scholar 

  115. Stratton, J. A., P. M. MORSE, L. J. CHU, J. D. C. LITTLE, and F. J.CORBÂTO: Spheroidal wave functions. New York: John Wiley & Sons 1956.

    Google Scholar 

  116. Flammer, C.: Spheroidal wave functions. Stanford, Cal.: Stanford Univ. Press 1957.

    Google Scholar 

  117. LamÉ, G.: Leçons sur les coordonnées curvilignes et leurs diverses applications. Paris: Mallet-Bachelier 1859.

    Google Scholar 

  118. Todhunter, I.: An elementary treatise on Laplace’s functions, Lamé’s functions, and Bessel’s functions. London: Macmillan Co. 1875.

    Google Scholar 

  119. Heine, H. E.: Handbuch der Kugelfunktionen. Berlin: G. Reimer 1878.

    Google Scholar 

  120. Klein, F.: Über Lamésche Funktionen. Math. Ann. 18, 237 (1881).

    MathSciNet  MATH  Google Scholar 

  121. Byerly, W. E.: Fourier’s series and spherical, cylindrical, and ellipsoidal harmonics. Boston: Ginn & Co. 1893.

    Google Scholar 

  122. Niven, W. D.: On ellipsoidal harmonics. Phil. Trans. Roy. Soc. Lond. 182, 231 (1892).

    ADS  Google Scholar 

  123. Maclauren, R.: On the solutions of the equation (V2 + K2) i = 0 in elliptic coordinates and their physical applications. Trans. Cambridge Phil. Soc. 17, 41 (1898).

    Google Scholar 

  124. Darwin, G. H.: Ellipsoidal harmonic analysis. Phil. Trans. Roy. Soc. Lond. 197, 461 (1901).

    ADS  MATH  Google Scholar 

  125. Humbert, P.: Fonctions de Lamé et fonctions de Mathieu. Paris: Gauthier-Villars 1926.

    MATH  Google Scholar 

  126. Hobson, E. W.: The theory of spherical and ellipsoidal harmonics, Chap. 11. Cambridge: Cambridge Univ. Press 1931.

    Google Scholar 

  127. Prasad, G.: A treatise on spherical harmonics and the functions of Bessel and Lamé, 2 vols. Benares: S. C. Chatterji 1932.

    Google Scholar 

  128. Strutt, M. J. O.: Lamésche, Mathieusche und verwandte Funktionen in Physik und Technik. Berlin: Springer 1932.

    Google Scholar 

  129. Wangerin, A.: Über die Reduktion der Gleichung a2 V/a x2 + a2 V/ô y2 + a2 V/ôz2 = o auf gewöhnliche Differentialgleichungen, Mber. Akad. Wiss. Berlin 152 (1878).

    Google Scholar 

  130. Heine, H. E.: Handbuch der Kugelfunktionen. Berlin: G. Reimer 1878.

    Google Scholar 

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  1. Massachusetts Institute of Technology, Cambridge, Mass., USA

    Parry Moon

  2. University of Connecticut, Storrs, Conn., USA

    Domina Eberle Spencer

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Moon, P., Spencer, D.E. (1961). Functions. In: Field Theory Handbook. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-53060-9_7

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