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solution by variation of parameters

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1: 1.13 Differential Equations
Variation of Parameters
2: 9.12 Scorer Functions
Solutions of this equation are the Scorer functions and can be found by the method of variation of parameters1.13(iii)). …
3: 10.15 Derivatives with Respect to Order
10.15.2 Y ν ( z ) ν = cot ( ν π ) ( J ν ( z ) ν π Y ν ( z ) ) csc ( ν π ) J ν ( z ) ν π J ν ( z ) .
10.15.3 J ν ( z ) ν | ν = n = π 2 Y n ( z ) + n ! 2 ( 1 2 z ) n k = 0 n 1 ( 1 2 z ) k J k ( z ) k ! ( n k ) .
10.15.4 Y ν ( z ) ν | ν = n = π 2 J n ( z ) + n ! 2 ( 1 2 z ) n k = 0 n 1 ( 1 2 z ) k Y k ( z ) k ! ( n k ) ,
10.15.5 J ν ( z ) ν | ν = 0 = π 2 Y 0 ( z ) , Y ν ( z ) ν | ν = 0 = π 2 J 0 ( z ) .
10.15.6 J ν ( x ) ν | ν = 1 2 = 2 π x ( Ci ( 2 x ) sin x Si ( 2 x ) cos x ) ,
4: 2.8 Differential Equations with a Parameter
§2.8 Differential Equations with a Parameter
§2.8(i) Classification of Cases
Zeros of f ( z ) are also called turning points. …
§2.8(vi) Coalescing Transition Points
5: Bibliography M
  • R. S. Maier (2007) The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.
  • M. Mazzocco (2001a) Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.
  • M. Mazzocco (2001b) Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321 (1), pp. 157–195.
  • R. J. Muirhead (1978) Latent roots and matrix variates: A review of some asymptotic results. Ann. Statist. 6 (1), pp. 5–33.
  • M. E. Muldoon (1981) The variation with respect to order of zeros of Bessel functions. Rend. Sem. Mat. Univ. Politec. Torino 39 (2), pp. 15–25.