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Lamé functions

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11: 29.4 Graphics
§29.4(i) Eigenvalues of Lamé’s Equation: Line Graphs
See accompanying text
Figure 29.4.12: b ν 2 ( k 2 ) as a function of ν and k 2 . Magnify 3D Help
§29.4(iii) Lamé Functions: Line Graphs
§29.4(iv) Lamé Functions: Surfaces
See accompanying text
Figure 29.4.32: 𝐸𝑠 2.5 2 ( x , k 2 ) as a function of x and k 2 . Magnify 3D Help
12: 29.1 Special Notation
All derivatives are denoted by differentials, not by primes. The main functions treated in this chapter are the eigenvalues a ν 2 m ( k 2 ) , a ν 2 m + 1 ( k 2 ) , b ν 2 m + 1 ( k 2 ) , b ν 2 m + 2 ( k 2 ) , the Lamé functions 𝐸𝑐 ν 2 m ( z , k 2 ) , 𝐸𝑐 ν 2 m + 1 ( z , k 2 ) , 𝐸𝑠 ν 2 m + 1 ( z , k 2 ) , 𝐸𝑠 ν 2 m + 2 ( z , k 2 ) , and the Lamé polynomials 𝑢𝐸 2 n m ( z , k 2 ) , 𝑠𝐸 2 n + 1 m ( z , k 2 ) , 𝑐𝐸 2 n + 1 m ( z , k 2 ) , 𝑑𝐸 2 n + 1 m ( z , k 2 ) , 𝑠𝑐𝐸 2 n + 2 m ( z , k 2 ) , 𝑠𝑑𝐸 2 n + 2 m ( z , k 2 ) , 𝑐𝑑𝐸 2 n + 2 m ( z , k 2 ) , 𝑠𝑐𝑑𝐸 2 n + 3 m ( z , k 2 ) . … The relation to the Lamé functions L c ν ( m ) , L s ν ( m ) of Jansen (1977) is given by …The relation to the Lamé functions Ec ν m , Es ν m of Ince (1940b) is given by …
( s ν m ( k 2 ) ) 2 = 4 π 0 K ( 𝐸𝑠 ν m ( x , k 2 ) ) 2 d x .
13: 14 Legendre and Related Functions
Chapter 14 Legendre and Related Functions
14: 10 Bessel Functions
Chapter 10 Bessel Functions
15: 18 Orthogonal Polynomials
16: 1 Algebraic and Analytic Methods
17: Hans Volkmer
18: 29.12 Definitions
§29.12(i) Elliptic-Function Form
The Lamé functions 𝐸𝑐 ν m ( z , k 2 ) , m = 0 , 1 , , ν , and 𝐸𝑠 ν m ( z , k 2 ) , m = 1 , 2 , , ν , are called the Lamé polynomials. There are eight types of Lamé polynomials, defined as follows:
29.12.1 𝑢𝐸 2 n m ( z , k 2 ) = 𝐸𝑐 2 n 2 m ( z , k 2 ) ,
In consequence they are doubly-periodic meromorphic functions of z . …
19: 29.6 Fourier Series
§29.6 Fourier Series
§29.6(i) Function 𝐸𝑐 ν 2 m ( z , k 2 )
In addition, if H satisfies (29.6.2), then (29.6.3) applies. …
§29.6(ii) Function 𝐸𝑐 ν 2 m + 1 ( z , k 2 )
§29.6(iii) Function 𝐸𝑠 ν 2 m + 1 ( z , k 2 )
20: 31.7 Relations to Other Functions
§31.7(ii) Relations to Lamé Functions