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1: 28.20 Definitions and Basic Properties
§28.20(iv) Radial Mathieu Functions Mc n ( j ) , Ms n ( j )
28.20.15 Mc n ( j ) ( z , h ) = M n ( j ) ( z , h ) , n = 0 , 1 , ,
28.20.16 Ms n ( j ) ( z , h ) = ( - 1 ) n M - n ( j ) ( z , h ) , n = 1 , 2 , .
28.20.17 Ie n ( z , h ) = i - n Mc n ( 1 ) ( z , i h ) ,
§28.20(vii) Shift of Variable
2: 28.21 Graphics
Radial Mathieu Functions: Surfaces
See accompanying text
Figure 28.21.6: Ms 1 ( 2 ) ( x , h ) for 0.2 h 2 , 0 x 2 . Magnify 3D Help
3: 28.22 Connection Formulas
The joining factors in the above formulas are given by …
28.22.9 f e , m ( h ) = - π / 2 g e , m ( h ) Mc m ( 2 ) ( 0 , h ) ,
ge m ( 0 , h 2 ) = 1 2 π S m ( h 2 ) ( g o , m ( h ) ) 2 se m ( 0 , h 2 ) .
4: 28.23 Expansions in Series of Bessel Functions
§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 ) ,
5: 28.28 Integrals, Integral Representations, and Integral Equations
§28.28(i) Equations with Elementary Kernels
28.28.23 2 π 0 π 𝒞 2 + 2 ( j ) ( 2 h R ) sin ( ( 2 + 2 ) ϕ ) se 2 m + 2 ( t , h 2 ) d t = ( - 1 ) + m B 2 + 2 2 m + 2 ( h 2 ) Ms 2 m + 2 ( j ) ( z , h ) .
§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order
28.28.49 α ^ n , m ( c ) = 1 2 π 0 2 π cos t ce n ( t , h 2 ) ce m ( t , h 2 ) d t = ( - 1 ) p + 1 2 i π ce n ( 0 , h 2 ) ce m ( 0 , h 2 ) h Dc 0 ( n , m , 0 ) .
§28.28(v) Compendia
6: 28.1 Special Notation
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 ) .
The radial functions Mc n ( j ) ( z , h ) and Ms n ( j ) ( z , h ) are denoted by Mc n ( j ) ( z , q ) and Ms n ( j ) ( z , q ) , respectively.
7: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
§28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
28.24.2 ε s Mc 2 m ( j ) ( z , h ) = ( - 1 ) m = 0 ( - 1 ) A 2 2 m ( h 2 ) A 2 s 2 m ( h 2 ) ( J - s ( h e - z ) 𝒞 + s ( j ) ( h e z ) + J + s ( h e - z ) 𝒞 - s ( j ) ( h e z ) ) ,
28.24.3 Mc 2 m + 1 ( j ) ( z , h ) = ( - 1 ) m = 0 ( - 1 ) A 2 + 1 2 m + 1 ( h 2 ) A 2 s + 1 2 m + 1 ( h 2 ) ( J - s ( h e - z ) 𝒞 + s + 1 ( j ) ( h e z ) + J + s + 1 ( h e - z ) 𝒞 - s ( j ) ( h e z ) ) ,
28.24.4 Ms 2 m + 1 ( j ) ( z , h ) = ( - 1 ) m = 0 ( - 1 ) B 2 + 1 2 m + 1 ( h 2 ) B 2 s + 1 2 m + 1 ( h 2 ) ( J - s ( h e - z ) 𝒞 + s + 1 ( j ) ( h e z ) - J + s + 1 ( h e - z ) 𝒞 - s ( j ) ( h e z ) ) ,
For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).
8: 28.26 Asymptotic Approximations for Large q
28.26.1 Mc m ( 3 ) ( z , h ) = e i ϕ ( π h cosh z ) 1 / 2 ( Fc m ( z , h ) - i Gc m ( z , h ) ) ,
28.26.2 i Ms m + 1 ( 3 ) ( z , h ) = e i ϕ ( π h cosh z ) 1 / 2 ( Fs m ( z , h ) - i Gs m ( z , h ) ) ,
9: Bibliography B
  • G. Blanch and D. S. Clemm (1962) Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..
  • G. Blanch and D. S. Clemm (1965) Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..
  • 10: Bibliography L
  • T. M. Larsen, D. Erricolo, and P. L. E. Uslenghi (2009) New method to obtain small parameter power series expansions of Mathieu radial and angular functions. Math. Comp. 78 (265), pp. 255–274.