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1: 1.13 Differential Equations
§1.13(vii) Closed-Form Solutions
2: 20.13 Physical Applications
This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.
3: Bibliography O
  • M. K. Ong (1986) A closed form solution of the s -wave Bethe-Goldstone equation with an infinite repulsive core. J. Math. Phys. 27 (4), pp. 1154–1158.
  • 4: 32.2 Differential Equations
    For arbitrary values of the parameters α , β , γ , and δ , the general solutions of P I P VI  are transcendental, that is, they cannot be expressed in closed-form elementary functions. …
    5: Bibliography C
  • J. Camacho, R. Guimerà, and L. A. N. Amaral (2002) Analytical solution of a model for complex food webs. Phys. Rev. E 65 (3), pp. (030901–1)–(030901–4).
  • B. C. Carlson (1972b) Intégrandes à deux formes quadratiques. C. R. Acad. Sci. Paris Sér. A–B 274 (15 May, 1972, Sér. A), pp. 1458–1461 (French).
  • M. A. Chaudhry, N. M. Temme, and E. J. M. Veling (1996) Asymptotics and closed form of a generalized incomplete gamma function. J. Comput. Appl. Math. 67 (2), pp. 371–379.
  • G. Cornell, J. H. Silverman, and G. Stevens (Eds.) (1997) Modular Forms and Fermat’s Last Theorem. Springer-Verlag, New York.
  • Cunningham Project (website)
  • 6: Bibliography K
  • K. Kajiwara and Y. Ohta (1996) Determinant structure of the rational solutions for the Painlevé II equation. J. Math. Phys. 37 (9), pp. 4693–4704.
  • A. I. Kheyfits (2004) Closed-form representations of the Lambert W function. Fract. Calc. Appl. Anal. 7 (2), pp. 177–190.
  • S. Kida (1981) A vortex filament moving without change of form. J. Fluid Mech. 112, pp. 397–409.
  • N. Koblitz (1993) Introduction to Elliptic Curves and Modular Forms. 2nd edition, Graduate Texts in Mathematics, Vol. 97, Springer-Verlag, New York.
  • Y. A. Kravtsov (1964) Asymptotic solution of Maxwell’s equations near caustics. Izv. Vuz. Radiofiz. 7, pp. 1049–1056.