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Lamé equation

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1: 29.19 Physical Applications
§29.19(i) Lamé Functions
Simply-periodic Lamé functions ( ν noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. …
§29.19(ii) Lamé Polynomials
2: 29.11 Lamé Wave Equation
§29.11 Lamé Wave Equation
The Lamé (or ellipsoidal) wave equation is given by …In the case ω = 0 , (29.11.1) reduces to Lamé’s equation (29.2.1). …
3: 29.9 Stability
§29.9 Stability
The Lamé equation (29.2.1) with specified values of k , h , ν is called stable if all of its solutions are bounded on ; otherwise the equation is called unstable. …
4: 29.2 Differential Equations
§29.2(i) Lamé’s Equation
§29.2(ii) Other Forms
we have …For the Weierstrass function see §23.2(ii). …
5: 29.3 Definitions and Basic Properties
Table 29.3.1: Eigenvalues of Lamé’s equation.
eigenvalue h parity period
satisfies the continued-fraction equation
Table 29.3.2: Lamé functions.
boundary conditions
eigenvalue
h
eigenfunction
w ( z )
parity of
w ( z )
parity of
w ( z - K )
period of
w ( z )
6: 29.7 Asymptotic Expansions
29.7.1 a ν m ( k 2 ) p κ - τ 0 - τ 1 κ - 1 - τ 2 κ - 2 - ,
29.7.5 b ν m + 1 ( k 2 ) - a ν m ( k 2 ) = O ( ν m + 3 2 ( 1 - k 1 + k ) ν ) , ν .
Weinstein and Keller (1985) give asymptotics for solutions of Hill’s equation28.29(i)) that are applicable to the Lamé equation.
7: 29.4 Graphics
§29.4(i) Eigenvalues of Lamé’s Equation: Line Graphs
§29.4(ii) Eigenvalues of Lamé’s Equation: Surfaces
8: 29.1 Special Notation
( s ν m ( k 2 ) ) 2 = 4 π 0 K ( Es ν m ( x , k 2 ) ) 2 d x .
9: 29.5 Special Cases and Limiting Forms
29.5.4 lim k 1 - a ν m ( k 2 ) = lim k 1 - b ν m + 1 ( k 2 ) = ν ( ν + 1 ) - μ 2 ,
10: 29.17 Other Solutions
§29.17(i) Second Solution