About the Project
NIST

hypergeometric equation

AdvancedHelp

(0.003 seconds)

1—10 of 114 matching pages

1: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
15.10.1 z ( 1 - z ) d 2 w d z 2 + ( c - ( a + b + 1 ) z ) d w d z - a b w = 0 .
This is the hypergeometric differential equation. …
Singularity z = 0
2: 15.17 Mathematical Applications
The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … …
3: 31.12 Confluent Forms of Heun’s Equation
This is analogous to the derivation of the confluent hypergeometric equation from the hypergeometric equation in §13.2(i). …
4: 15.11 Riemann’s Differential Equation
15.11.3 w = P { α β γ a 1 b 1 c 1 z a 2 b 2 c 2 } .
The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by …
15.11.8 z λ ( 1 - z ) μ P { 0 1 a 1 b 1 c 1 z a 2 b 2 c 2 } = P { 0 1 a 1 + λ b 1 + μ c 1 - λ - μ z a 2 + λ b 2 + μ c 2 - λ - μ } ,
5: 16.8 Differential Equations
§16.8(ii) The Generalized Hypergeometric Differential Equation
When no b j is an integer, and no two b j differ by an integer, a fundamental set of solutions of (16.8.3) is given by … We have the connection formula …
§16.8(iii) Confluence of Singularities
6: 13.28 Physical Applications
§13.28(i) Exact Solutions of the Wave Equation
7: 16.23 Mathematical Applications
These equations are frequently solvable in terms of generalized hypergeometric functions, and the monodromy of generalized hypergeometric functions plays an important role in describing properties of the solutions. … …
8: 19.18 Derivatives and Differential Equations
§19.18(ii) Differential Equations
If n = 2 , then elimination of 2 v between (19.18.11) and (19.18.12), followed by the substitution ( b 1 , b 2 , z 1 , z 2 ) = ( b , c - b , 1 - z , 1 ) , produces the Gauss hypergeometric equation (15.10.1). …
9: Bibliography N
  • J. Negro, L. M. Nieto, and O. Rosas-Ortiz (2000) Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.
  • 10: Bibliography S
  • F. C. Smith (1939a) On the logarithmic solutions of the generalized hypergeometric equation when p = q + 1 . Bull. Amer. Math. Soc. 45 (8), pp. 629–636.
  • F. C. Smith (1939b) Relations among the fundamental solutions of the generalized hypergeometric equation when p = q + 1 . II. Logarithmic cases. Bull. Amer. Math. Soc. 45 (12), pp. 927–935.
  • C. Snow (1952) Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, D.C..