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1: 31.18 Methods of Computation
Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of z ; see Laĭ (1994) and Lay et al. (1998). …
2: 25.13 Periodic Zeta Function
25.13.2 F ( x , s ) = Γ ( 1 - s ) ( 2 π ) 1 - s ( e π i ( 1 - s ) / 2 ζ ( 1 - s , x ) + e π i ( s - 1 ) / 2 ζ ( 1 - s , 1 - x ) ) , 0 < x < 1 , s > 1 ,
25.13.3 ζ ( 1 - s , x ) = Γ ( s ) ( 2 π ) s ( e - π i s / 2 F ( x , s ) + e π i s / 2 F ( - x , s ) ) , 0 < x < 1 , s > 0 .
3: 14.21 Definitions and Basic Properties
§14.21(iii) Properties
This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …
4: 10.4 Connection Formulas
§10.4 Connection Formulas
5: 28.22 Connection Formulas
§28.22 Connection Formulas
6: 19.21 Connection Formulas
§19.21 Connection Formulas
The complete cases of R F and R G have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … Connection formulas for R - a ( b ; z ) are given in Carlson (1977b, pp. 99, 101, and 123–124). …
7: 14.9 Connection Formulas
§14.9 Connection Formulas
§14.9(i) Connections Between P ν ± μ ( x ) , P - ν - 1 ± μ ( x ) , Q ν ± μ ( x ) , Q - ν - 1 μ ( x )
§14.9(ii) Connections Between P ν ± μ ( ± x ) , Q ν - μ ( ± x ) , Q ν μ ( x )
§14.9(iii) Connections Between P ν ± μ ( x ) , P - ν - 1 ± μ ( x ) , Q ν ± μ ( x ) , Q - ν - 1 μ ( x )
8: Bibliography O
  • F. W. J. Olver (1977a) Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.
  • F. W. J. Olver (1977b) Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.
  • F. W. J. Olver (1978) General connection formulae for Liouville-Green approximations in the complex plane. Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548.
  • 9: 10.27 Connection Formulas
    §10.27 Connection Formulas
    10: 9.2 Differential Equation
    §9.2(v) Connection Formulas