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1: 28.4 Fourier Series
§28.4(ii) Recurrence Relations
§28.4(iii) Normalization
§28.4(v) Change of Sign of q
§28.4(vi) Behavior for Small q
§28.4(vii) Asymptotic Forms for Large m
2: 28.11 Expansions in Series of Mathieu Functions
28.11.3 1 = 2 n = 0 A 0 2 n ( q ) ce 2 n ( z , q ) ,
28.11.4 cos 2 m z = n = 0 A 2 m 2 n ( q ) ce 2 n ( z , q ) , m 0 ,
28.11.5 cos ( 2 m + 1 ) z = n = 0 A 2 m + 1 2 n + 1 ( q ) ce 2 n + 1 ( z , q ) ,
28.11.6 sin ( 2 m + 1 ) z = n = 0 B 2 m + 1 2 n + 1 ( q ) se 2 n + 1 ( z , q ) ,
28.11.7 sin ( 2 m + 2 ) z = n = 0 B 2 m + 2 2 n + 2 ( q ) se 2 n + 2 ( z , q ) .
3: 28.35 Tables
§28.35 Tables
  • Ince (1932) includes eigenvalues a n , b n , and Fourier coefficients for n = 0 or 1 ( 1 ) 6 , q = 0 ( 1 ) 10 ( 2 ) 20 ( 4 ) 40 ; 7D. Also ce n ( x , q ) , se n ( x , q ) for q = 0 ( 1 ) 10 , x = 1 ( 1 ) 90 , corresponding to the eigenvalues in the tables; 5D. Notation: a n = be n - 2 q , b n = bo n - 2 q .

  • National Bureau of Standards (1967) includes the eigenvalues a n ( q ) , b n ( q ) for n = 0 ( 1 ) 3 with q = 0 ( .2 ) 20 ( .5 ) 37 ( 1 ) 100 , and n = 4 ( 1 ) 15 with q = 0 ( 2 ) 100 ; Fourier coefficients for ce n ( x , q ) and se n ( x , q ) for n = 0 ( 1 ) 15 , n = 1 ( 1 ) 15 , respectively, and various values of q in the interval [ 0 , 100 ] ; joining factors g e , n ( q ) , f e , n ( q ) for n = 0 ( 1 ) 15 with q = 0 ( .5  to  10 ) 100 (but in a different notation). Also, eigenvalues for large values of q . Precision is generally 8D.

  • Stratton et al. (1941) includes b n , b n , and the corresponding Fourier coefficients for Se n ( c , x ) and So n ( c , x ) for n = 0 or 1 ( 1 ) 4 , c = 0 ( .1 or .2 ) 4.5 . Precision is mostly 5S. Notation: c = 2 q , b n = a n + 2 q , b n = b n + 2 q , and for Se n ( c , x ) , So n ( c , x ) see §28.1.

  • Blanch and Clemm (1969) includes eigenvalues a n ( q ) , b n ( q ) for q = ρ e i ϕ , ρ = 0 ( .5 ) 25 , ϕ = 5 ( 5 ) 90 , n = 0 ( 1 ) 15 ; 4D. Also a n ( q ) and b n ( q ) for q = i ρ , ρ = 0 ( .5 ) 100 , n = 0 ( 2 ) 14 and n = 2 ( 2 ) 16 , respectively; 8D. Double points for n = 0 ( 1 ) 15 ; 8D. Graphs are included.

  • 4: 28.14 Fourier Series
    28.14.4 q c 2 m + 2 - ( a - ( ν + 2 m ) 2 ) c 2 m + q c 2 m - 2 = 0 , a = λ ν ( q ) , c 2 m = c 2 m ν ( q ) ,
    28.14.5 m = - ( c 2 m ν ( q ) ) 2 = 1 ;
    5: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
    28.24.6 ε s Ie 2 m ( z , h ) = ( - 1 ) s = 0 ( - 1 ) A 2 2 m ( h 2 ) A 2 s 2 m ( h 2 ) ( I - s ( h e - z ) I + s ( h e z ) + I + s ( h e - z ) I - s ( h e z ) ) ,
    28.24.7 Io 2 m + 2 ( z , h ) = ( - 1 ) s = 0 ( - 1 ) B 2 + 2 2 m + 2 ( h 2 ) B 2 s + 2 2 m + 2 ( h 2 ) ( I - s ( h e - z ) I + s + 2 ( h e z ) - I + s + 2 ( h e - z ) I - s ( h e z ) ) ,
    28.24.8 Ie 2 m + 1 ( z , h ) = ( - 1 ) s = 0 ( - 1 ) B 2 + 1 2 m + 1 ( h 2 ) B 2 s + 1 2 m + 1 ( h 2 ) ( I - s ( h e - z ) I + s + 1 ( h e z ) + I + s + 1 ( h e - z ) I - s ( h e z ) ) ,
    28.24.9 Io 2 m + 1 ( z , h ) = ( - 1 ) s = 0 ( - 1 ) A 2 + 1 2 m + 1 ( h 2 ) A 2 s + 1 2 m + 1 ( h 2 ) ( I - s ( h e - z ) I + s + 1 ( h e z ) - I + s + 1 ( h e - z ) I - s ( h e z ) ) ,
    28.24.10 ε s Ke 2 m ( z , h ) = = 0 A 2 2 m ( h 2 ) A 2 s 2 m ( h 2 ) ( I - s ( h e - z ) K + s ( h e z ) + I + s ( h e - z ) K - s ( h e z ) ) ,
    6: 28.10 Integral Equations
    With the notation of §28.4 for Fourier coefficients,
    28.10.1 2 π 0 π / 2 cos ( 2 h cos z cos t ) ce 2 n ( t , h 2 ) d t = A 0 2 n ( h 2 ) ce 2 n ( 1 2 π , h 2 ) ce 2 n ( z , h 2 ) ,
    28.10.2 2 π 0 π / 2 cosh ( 2 h sin z sin t ) ce 2 n ( t , h 2 ) d t = A 0 2 n ( h 2 ) ce 2 n ( 0 , h 2 ) ce 2 n ( z , h 2 ) ,
    28.10.3 2 π 0 π / 2 sin ( 2 h cos z cos t ) ce 2 n + 1 ( t , h 2 ) d t = - h A 1 2 n + 1 ( h 2 ) ce 2 n + 1 ( 1 2 π , h 2 ) ce 2 n + 1 ( z , h 2 ) ,
    28.10.4 2 π 0 π / 2 cos z cos t cosh ( 2 h sin z sin t ) ce 2 n + 1 ( t , h 2 ) d t = A 1 2 n + 1 ( h 2 ) 2 ce 2 n + 1 ( 0 , h 2 ) ce 2 n + 1 ( z , h 2 ) ,
    7: 28.15 Expansions for Small q
    §28.15 Expansions for Small q
    8: 29.20 Methods of Computation
    Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. … A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). … The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …
    9: 28.23 Expansions in Series of Bessel Functions
    28.23.6 Mc 2 m ( j ) ( z , h ) = ( - 1 ) m ( ce 2 m ( 0 , h 2 ) ) - 1 = 0 ( - 1 ) A 2 2 m ( h 2 ) 𝒞 2 ( j ) ( 2 h cosh z ) ,
    28.23.7 Mc 2 m ( j ) ( z , h ) = ( - 1 ) m ( ce 2 m ( 1 2 π , h 2 ) ) - 1 = 0 A 2 2 m ( h 2 ) 𝒞 2 ( j ) ( 2 h sinh z ) ,
    28.23.8 Mc 2 m + 1 ( j ) ( z , h ) = ( - 1 ) m ( ce 2 m + 1 ( 0 , h 2 ) ) - 1 = 0 ( - 1 ) A 2 + 1 2 m + 1 ( h 2 ) 𝒞 2 + 1 ( j ) ( 2 h cosh z ) ,
    28.23.9 Mc 2 m + 1 ( j ) ( z , h ) = ( - 1 ) m + 1 ( ce 2 m + 1 ( 1 2 π , h 2 ) ) - 1 coth z = 0 ( 2 + 1 ) A 2 + 1 2 m + 1 ( h 2 ) 𝒞 2 + 1 ( j ) ( 2 h sinh z ) ,
    28.23.10 Ms 2 m + 1 ( j ) ( z , h ) = ( - 1 ) m ( se 2 m + 1 ( 0 , h 2 ) ) - 1 tanh z = 0 ( - 1 ) ( 2 + 1 ) B 2 + 1 2 m + 1 ( h 2 ) 𝒞 2 + 1 ( j ) ( 2 h cosh z ) ,
    10: 27.17 Other Applications
    Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. …