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1: 19.8 Quadratic Transformations
§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)
As n , a n and g n converge to a common limit M ( a 0 , g 0 ) called the AGM (Arithmetic-Geometric Mean) of a 0 and g 0 . …showing that the convergence of c n to 0 and of a n and g n to M ( a 0 , g 0 ) is quadratic in each case. … Again, p n and ε n converge quadratically to M ( a 0 , g 0 ) and 0, respectively, and Q n converges to 0 faster than quadratically. …
2: 28.1 Special Notation
ce ν ( z , q ) , se ν ( z , q ) , fe n ( z , q ) , ge n ( z , q ) , me ν ( z , q ) ,
Ce ν ( z , q ) , Se ν ( z , q ) , Fe n ( z , q ) , Ge n ( z , q ) ,
Abramowitz and Stegun (1964, Chapter 20)
Se n ( s , z ) = ce n ( z , q ) ce n ( 0 , q ) ,
Se n ( c , z ) = ce n ( z , q ) ce n ( 0 , q ) ,
3: 18.42 Software
For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C3). …
4: 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 .

  • Kirkpatrick (1960) contains tables of the modified functions Ce n ( x , q ) , Se n + 1 ( x , q ) for n = 0 ( 1 ) 5 , q = 1 ( 1 ) 20 , x = 0.1 ( .1 ) 1 ; 4D or 5D.

  • 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.

  • Zhang and Jin (1996, pp. 521–532) includes the eigenvalues a n ( q ) , b n + 1 ( q ) for n = 0 ( 1 ) 4 , q = 0 ( 1 ) 50 ; n = 0 ( 1 ) 20 ( a ’s) or 19 ( b ’s), q = 1 , 3 , 5 , 10 , 15 , 25 , 50 ( 50 ) 200 . Fourier coefficients for ce n ( x , 10 ) , se n + 1 ( x , 10 ) , n = 0 ( 1 ) 7 . Mathieu functions ce n ( x , 10 ) , se n + 1 ( x , 10 ) , and their first x -derivatives for n = 0 ( 1 ) 4 , x = 0 ( 5 ) 90 . Modified Mathieu functions Mc n ( j ) ( x , 10 ) , Ms n + 1 ( j ) ( x , 10 ) , and their first x -derivatives for n = 0 ( 1 ) 4 , j = 1 , 2 , x = 0 ( .2 ) 4 . Precision is mostly 9S.

  • Ince (1932) includes the first zero for ce n , se n for n = 2 ( 1 ) 5 or 6 , q = 0 ( 1 ) 10 ( 2 ) 40 ; 4D. This reference also gives zeros of the first derivatives, together with expansions for small q .

  • 5: 28.2 Definitions and Basic Properties
    Period π means that the eigenfunction has the property w ( z + π ) = w ( z ) , whereas antiperiod π means that w ( z + π ) = w ( z ) . Even parity means w ( z ) = w ( z ) , and odd parity means w ( z ) = w ( z ) . …
    ce 0 ( z , 0 ) = 1 / 2 ,
    ce n ( z , 0 ) = cos ( n z ) ,
    6: 28.11 Expansions in Series of Mathieu Functions
    28.11.1 f ( z ) = α 0 ce 0 ( z , q ) + n = 1 ( α n ce n ( z , q ) + β n se n ( z , q ) ) ,
    α n = 1 π 0 2 π f ( x ) ce n ( x , q ) d x ,
    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 ) ,
    7: 28.3 Graphics
    See accompanying text
    Figure 28.3.1: ce 2 n ( x , 1 ) for 0 x π / 2 , n = 0 , 1 , 2 , 3 . Magnify
    See accompanying text
    Figure 28.3.2: ce 2 n ( x , 10 ) for 0 x π / 2 , n = 0 , 1 , 2 , 3 . Magnify
    See accompanying text
    Figure 28.3.3: ce 2 n + 1 ( x , 1 ) for 0 x π / 2 , n = 0 , 1 , 2 , 3 . Magnify
    See accompanying text
    Figure 28.3.4: ce 2 n + 1 ( x , 10 ) for 0 x π / 2 , n = 0 , 1 , 2 , 3 . Magnify
    See accompanying text
    Figure 28.3.9: ce 0 ( x , q ) for 0 x 2 π , 0 q 10 . Magnify 3D Help
    8: 28.9 Zeros
    For real q each of the functions ce 2 n ( z , q ) , se 2 n + 1 ( z , q ) , ce 2 n + 1 ( z , q ) , and se 2 n + 2 ( z , q ) has exactly n zeros in 0 < z < 1 2 π . …For q the zeros of ce 2 n ( z , q ) and se 2 n + 1 ( z , q ) approach asymptotically the zeros of 𝐻𝑒 2 n ( q 1 / 4 ( π 2 z ) ) , and the zeros of ce 2 n + 1 ( z , q ) and se 2 n + 2 ( z , q ) approach asymptotically the zeros of 𝐻𝑒 2 n + 1 ( q 1 / 4 ( π 2 z ) ) . …Furthermore, for q > 0 ce m ( z , q ) and se m ( z , q ) also have purely imaginary zeros that correspond uniquely to the purely imaginary z -zeros of J m ( 2 q cos z ) 10.21(i)), and they are asymptotically equal as q 0 and | z | . …
    9: 28.10 Integral Equations
    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 ) ,
    10: 28.5 Second Solutions fe n , ge n
    28.5.8 𝒲 { ce n , fe n } = ce n ( 0 , q ) fe n ( 0 , q ) ,
    For further information on C n ( q ) , S n ( q ) , and expansions of f n ( z , q ) , g n ( z , q ) in Fourier series or in series of ce n , se n functions, see McLachlan (1947, Chapter VII) or Meixner and Schäfke (1954, §2.72). …
    See accompanying text
    Figure 28.5.1: fe 0 ( x , 0.5 ) for 0 x 2 π and (for comparison) ce 0 ( x , 0.5 ) . Magnify
    See accompanying text
    Figure 28.5.2: fe 0 ( x , 1 ) for 0 x 2 π and (for comparison) ce 0 ( x , 1 ) . Magnify
    See accompanying text
    Figure 28.5.3: fe 1 ( x , 0.5 ) for 0 x 2 π and (for comparison) ce 1 ( x , 0.5 ) . Magnify