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1: 28.20 Definitions and Basic Properties
§28.20(i) Modified Mathieu’s Equation
28.20.1 w ′′ - ( a - 2 q cosh ( 2 z ) ) w = 0 ,
§28.20(ii) Solutions Ce ν , Se ν , Me ν , Fe n , Ge n
§28.20(iv) Radial Mathieu Functions Mc n ( j ) , Ms n ( j )
§28.20(vi) Wronskians
2: 28.21 Graphics
§28.21 Graphics
See accompanying text
Figure 28.21.1: Mc 0 ( 1 ) ( x , h ) for 0 h 3 , 0 x 2 . Magnify 3D Help
See accompanying text
Figure 28.21.2: Mc 1 ( 1 ) ( x , h ) for 0 h 3 , 0 x 2 . Magnify 3D Help
See accompanying text
Figure 28.21.3: Mc 0 ( 2 ) ( x , h ) for 0.1 h 2 , 0 x 2 . Magnify 3D Help
See accompanying text
Figure 28.21.6: Ms 1 ( 2 ) ( x , h ) for 0.2 h 2 , 0 x 2 . Magnify 3D Help
3: 10.26 Graphics
§10.26(i) Real Order and Variable
See accompanying text
Figure 10.26.7: I ~ 1 / 2 ( x ) , K ~ 1 / 2 ( x ) , 0.01 x 3 . Magnify
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Figure 10.26.8: I ~ 1 ( x ) , K ~ 1 ( x ) , 0.01 x 3 . Magnify
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Figure 10.26.9: I ~ 5 ( x ) , K ~ 5 ( x ) , 0.01 x 3 . Magnify
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Figure 10.26.10: K ~ 5 ( x ) , 0.01 x 3 . Magnify
4: 10.29 Recurrence Relations and Derivatives
§10.29(i) Recurrence Relations
With 𝒵 ν ( z ) defined as in §10.25(ii), …
I 0 ( z ) = I 1 ( z ) ,
For results on modified quotients of the form z 𝒵 ν ± 1 ( z ) / 𝒵 ν ( z ) see Onoe (1955) and Onoe (1956).
§10.29(ii) Derivatives
5: 28.27 Addition Theorems
§28.27 Addition Theorems
They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …
6: 10.28 Wronskians and Cross-Products
§10.28 Wronskians and Cross-Products
10.28.1 𝒲 { I ν ( z ) , I - ν ( z ) } = I ν ( z ) I - ν - 1 ( z ) - I ν + 1 ( z ) I - ν ( z ) = - 2 sin ( ν π ) / ( π z ) ,
10.28.2 𝒲 { K ν ( z ) , I ν ( z ) } = I ν ( z ) K ν + 1 ( z ) + I ν + 1 ( z ) K ν ( z ) = 1 / z .
7: 28.1 Special Notation
and the modified Mathieu functions
Ce ν ( z , q ) , Se ν ( z , q ) , Fe n ( z , q ) , Ge n ( z , q ) ,
Me ν ( z , q ) , M ν ( j ) ( z , h ) , Mc n ( j ) ( z , h ) , Ms n ( j ) ( z , h ) ,
The functions Mc n ( j ) ( z , h ) and Ms n ( j ) ( z , h ) are also known as the radial Mathieu functions. … 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.
8: 10.45 Functions of Imaginary Order
§10.45 Functions of Imaginary Order
and I ~ ν ( x ) , K ~ ν ( x ) are real and linearly independent solutions of (10.45.1): … The corresponding result for K ~ ν ( x ) is given by …
9: 10.35 Generating Function and Associated Series
§10.35 Generating Function and Associated Series
10.35.4 1 = I 0 ( z ) - 2 I 2 ( z ) + 2 I 4 ( z ) - 2 I 6 ( z ) + ,
10.35.5 e ± z = I 0 ( z ) ± 2 I 1 ( z ) + 2 I 2 ( z ) ± 2 I 3 ( z ) + ,
cosh z = I 0 ( z ) + 2 I 2 ( z ) + 2 I 4 ( z ) + 2 I 6 ( z ) + ,
sinh z = 2 I 1 ( z ) + 2 I 3 ( z ) + 2 I 5 ( z ) + .
10: 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 ) .
Here me ν ( 0 , h 2 ) ( 0 ) is given by (28.14.1) with z = 0 , and M ν ( 1 ) ( 0 , h ) is given by (28.24.1) with j = 1 , z = 0 , and n chosen so that | c 2 n ν ( h 2 ) | = max ( | c 2 ν ( h 2 ) | ) , where the maximum is taken over all integers . …