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1: 28.31 Equations of Whittaker–Hill and Ince
§28.31 Equations of Whittaker–Hill and Ince
§28.31(ii) Equation of Ince; Ince Polynomials
The result is the Equation of Ince: …
28.31.5 w o , s ( z ) = = 0 B 2 + s sin ( 2 + s ) z , s = 1 , 2 ,
2: 28.5 Second Solutions fe n , ge n
§28.5(i) Definitions
3: Bibliography I
  • E. L. Ince (1926) Ordinary Differential Equations. Longmans, Green and Co., London.
  • 4: 28.2 Definitions and Basic Properties
    5: 31.14 General Fuchsian Equation
    §31.14 General Fuchsian Equation
    §31.14(i) Definitions
    Heun’s equation (31.2.1) corresponds to N = 3 .
    Normal Form
    An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). …
    6: 1.13 Differential Equations
    (More generally in (1.13.5) for n th-order differential equations, f ( z ) is the coefficient multiplying the ( n 1 ) th-order derivative of the solution divided by the coefficient multiplying the n th-order derivative of the solution, see Ince (1926, §5.2).) …
    7: 32.2 Differential Equations
    §32.2 Differential Equations
    §32.2(i) Introduction
    The six Painlevé equations P I P VI  are as follows: …
    §32.2(ii) Renormalizations
    8: 31.12 Confluent Forms of Heun’s Equation
    Confluent Heun Equation
    Doubly-Confluent Heun Equation
    Biconfluent Heun Equation
    Triconfluent Heun Equation
    9: 29.3 Definitions and Basic Properties
    For each pair of values of ν and k there are four infinite unbounded sets of real eigenvalues h for which equation (29.2.1) has even or odd solutions with periods 2 K or 4 K . … … satisfies the continued-fraction equationThe quantity H = 2 a ν 2 m + 1 ( k 2 ) ν ( ν + 1 ) k 2 satisfies equation (29.3.10) with … The quantity H = 2 b ν 2 m + 1 ( k 2 ) ν ( ν + 1 ) k 2 satisfies equation (29.3.10) with …
    10: Errata
  • Section 1.13

    In Equation (1.13.4), the determinant form of the two-argument Wronskian

    1.13.4 𝒲 { w 1 ( z ) , w 2 ( z ) } = det [ w 1 ( z ) w 2 ( z ) w 1 ( z ) w 2 ( z ) ] = w 1 ( z ) w 2 ( z ) w 2 ( z ) w 1 ( z )

    was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the n -argument Wronskian is given by 𝒲 { w 1 ( z ) , , w n ( z ) } = det [ w k ( j 1 ) ( z ) ] , where 1 j , k n . Immediately below Equation (1.13.4), a sentence was added giving the definition of the n -argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for n th-order differential equations. A reference to Ince (1926, §5.2) was added.