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modified Mathieu functions

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1: 28.20 Definitions and Basic Properties
§28.20(ii) Solutions Ce ν , Se ν , Me ν , Fe n , Ge n
For other values of z , h , and ν the functions M ν ( j ) ( z , h ) , j = 1 , 2 , 3 , 4 , are determined by analytic continuation. …
§28.20(iv) Radial Mathieu Functions Mc n ( j ) , Ms n ( j )
§28.20(vi) Wronskians
§28.20(vii) Shift of Variable
2: 28.27 Addition Theorems
§28.27 Addition Theorems
3: 28.22 Connection Formulas
§28.22 Connection Formulas
The joining factors in the above formulas are given by …
28.22.13 M ν ( 1 ) ( z , h ) = M ν ( 1 ) ( 0 , h ) me ν ( 0 , h 2 ) Me ν ( z , h 2 ) .
4: 28.23 Expansions in Series of Bessel Functions
§28.23 Expansions in Series of Bessel Functions
28.23.2 me ν ( 0 , h 2 ) M ν ( j ) ( z , h ) = n = - ( - 1 ) n c 2 n ν ( h 2 ) 𝒞 ν + 2 n ( j ) ( 2 h cosh z ) ,
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 ) ,
28.23.8 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 ) ,
5: 28.1 Special Notation
m , n

integers.

ν

order of the Mathieu function or modified Mathieu function. (When ν is an integer it is often replaced by n .)

and the modified Mathieu functions
Ce ν ( z , q ) , Se ν ( z , q ) , Fe n ( z , q ) , Ge n ( z , q ) ,
The functions Mc n ( j ) ( z , h ) and Ms n ( j ) ( z , h ) are also known as the radial Mathieu functions. …
f o , n ( h ) .
6: 28.35 Tables
§28.35 Tables
  • 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.

  • 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.

  • 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.

  • §28.35(iii) Zeros
    7: 28.34 Methods of Computation
    §28.34(iv) Modified Mathieu Functions
    8: 28.33 Physical Applications
    §28.33 Physical Applications
    28.33.2 V n ( ξ , η ) = ( c n M n ( 1 ) ( ξ , q ) + d n M n ( 2 ) ( ξ , q ) ) me n ( η , q ) ,
    28.33.3 M n ( 1 ) ( ξ 0 , q ) M n ( 2 ) ( ξ 1 , q ) - M n ( 1 ) ( ξ 1 , q ) M n ( 2 ) ( ξ 0 , q ) = 0 .
  • Torres-Vega et al. (1998) for Mathieu functions in phase space.

  • 9: 28.28 Integrals, Integral Representations, and Integral Equations
    §28.28(i) Equations with Elementary Kernels
    28.28.15 0 cos ( 2 h cos y cosh t ) Ce 2 n ( t , h 2 ) d t = ( - 1 ) n + 1 1 2 π Mc 2 n ( 2 ) ( 0 , h ) ce 2 n ( y , h 2 ) ,
    §28.28(iv) Integrals of Products of Mathieu Functions of Integer Order
    §28.28(v) Compendia
    10: 28.21 Graphics
    §28.21 Graphics
    See accompanying text
    Figure 28.21.6: Ms 1 ( 2 ) ( x , h ) for 0.2 h 2 , 0 x 2 . Magnify 3D Help