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1: 28.2 Definitions and Basic Properties
§28.2(i) Mathieu’s Equation
§28.2(iii) Floquet’s Theorem and the Characteristic Exponents
This is the characteristic equation of Mathieu’s equation (28.2.1). …
§28.2(iv) Floquet Solutions
2: 28.20 Definitions and Basic Properties
§28.20(i) Modified Mathieu’s Equation
When z is replaced by ± i z , (28.2.1) becomes the modified Mathieu’s equation:
28.20.1 w ′′ - ( a - 2 q cosh ( 2 z ) ) w = 0 ,
28.20.2 ( ζ 2 - 1 ) w ′′ + ζ w + ( 4 q ζ 2 - 2 q - a ) w = 0 , ζ = cosh z .
28.20.6 Fe n ( z , q ) = i fe n ( ± i z , q ) , n = 0 , 1 , ,
3: 28.17 Stability as x ±
§28.17 Stability as x ±
See accompanying text
Figure 28.17.1: Stability chart for eigenvalues of Mathieu’s equation (28.2.1). Magnify
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). …
28.7.1 n = 0 ( a 2 n ( q ) - ( 2 n ) 2 ) = 0 ,
28.7.4 n = 0 ( b 2 n + 2 ( q ) - ( 2 n + 2 ) 2 ) = 0 .
5: 28.6 Expansions for Small q
§28.6(i) Eigenvalues
28.6.14 a m ( q ) b m ( q ) } = m 2 + 1 2 ( m 2 - 1 ) q 2 + 5 m 2 + 7 32 ( m 2 - 1 ) 3 ( m 2 - 4 ) q 4 + 9 m 4 + 58 m 2 + 29 64 ( m 2 - 1 ) 5 ( m 2 - 4 ) ( m 2 - 9 ) q 6 + .
28.6.15 a m ( q ) - b m ( q ) = 2 q m ( 2 m - 1 ( m - 1 ) ! ) 2 ( 1 + O ( q 2 ) ) .
Higher coefficients in the foregoing series can be found by equating coefficients in the following continued-fraction equations: …
Table 28.6.1: Radii of convergence for power-series expansions of eigenvalues of Mathieu’s equation.
n ρ n ( 1 ) ρ n ( 2 ) ρ n ( 3 )
6: 28.13 Graphics
§28.13(i) Eigenvalues λ ν ( q ) for General ν
7: 28.5 Second Solutions fe n , ge n
§28.5(i) Definitions
§28.5(ii) Graphics: Line Graphs of Second Solutions of Mathieu’s Equation
8: 28.12 Definitions and Basic Properties
§28.12(i) Eigenvalues λ ν + 2 n ( q )
For given ν (or cos ( ν π ) ) and q , equation (28.2.16) determines an infinite discrete set of values of a , denoted by λ ν + 2 n ( q ) , n = 0 , ± 1 , ± 2 , . … …
28.12.3 λ m ( q ) = { a m ( q ) , m = 0 , 1 , , b - m ( q ) , m = - 1 , - 2 , .
They have the following pseudoperiodic and orthogonality properties: …
9: 28.16 Asymptotic Expansions for Large q
§28.16 Asymptotic Expansions for Large q
28.16.1 λ ν ( h 2 ) - 2 h 2 + 2 s h - 1 8 ( s 2 + 1 ) - 1 2 7 h ( s 3 + 3 s ) - 1 2 12 h 2 ( 5 s 4 + 34 s 2 + 9 ) - 1 2 17 h 3 ( 33 s 5 + 410 s 3 + 405 s ) - 1 2 20 h 4 ( 63 s 6 + 1260 s 4 + 2943 s 2 + 486 ) - 1 2 25 h 5 ( 527 s 7 + 15617 s 5 + 69001 s 3 + 41607 s ) + .
10: 28 Mathieu Functions and Hill’s Equation
Chapter 28 Mathieu Functions and Hill’s Equation