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1: 28.34 Methods of Computation
§28.34(i) Characteristic Exponents
§28.34(ii) Eigenvalues
  • (f)

    Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

  • 2: 2.9 Difference Equations
    where ρ 1 , ρ 2 are the roots of the characteristic equation
    §2.9(ii) Coincident Characteristic Values
    3: 28.29 Definitions and Basic Properties
    §28.29(ii) Floquet’s Theorem and the Characteristic Exponent
    This is the characteristic equation of (28.29.1), and cos ( π ν ) is an entire function of λ . … For a given ν , the characteristic equation ( λ ) - 2 cos ( π ν ) = 0 has infinitely many roots λ . …
    4: 28.7 Analytic Continuation of Eigenvalues
    §28.7 Analytic Continuation of Eigenvalues
    The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). …
    28.7.4 n = 0 ( b 2 n + 2 ( q ) - ( 2 n + 2 ) 2 ) = 0 .
    5: 2.7 Differential Equations
    where λ 1 , λ 2 are the roots of the characteristic equationSee §2.11(v) for other examples. … The transformed differential equation either has a regular singularity at t = , or its characteristic equation has unequal roots. …
    6: 28.2 Definitions and Basic Properties
    28.2.16 cos ( π ν ) = w I ( π ; a , q ) = w I ( π ; a , - q ) .
    This is the characteristic equation of Mathieu’s equation (28.2.1). …If ν ^ = 0 or 1 , or equivalently, ν = n , then ν is a double root of the characteristic equation, otherwise it is a simple root. …
    Distribution
    Change of Sign of q
    7: 28.15 Expansions for Small q
    §28.15(i) Eigenvalues λ ν ( q )
    28.15.2 a - ν 2 - q 2 a - ( ν + 2 ) 2 - q 2 a - ( ν + 4 ) 2 - = q 2 a - ( ν - 2 ) 2 - q 2 a - ( ν - 4 ) 2 - .
    8: 28.16 Asymptotic Expansions for Large q
    §28.16 Asymptotic Expansions for Large q
    9: 28.17 Stability as x ±
    10: 28.12 Definitions and Basic Properties
    §28.12(i) Eigenvalues λ ν + 2 n ( q )
    28.12.2 λ ν ( - q ) = λ ν ( q ) = λ - ν ( q ) .