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1: 31.13 Asymptotic Approximations
§31.13 Asymptotic Approximations
For asymptotic approximations for the accessory parameter eigenvalues q m , see Fedoryuk (1991) and Slavyanov (1996). …
2: 31.3 Basic Solutions
31.3.2 a γ c 1 - q c 0 = 0 ,
3: 29.11 Lamé Wave Equation
29.11.1 d 2 w d z 2 + ( h - ν ( ν + 1 ) k 2 sn 2 ( z , k ) + k 2 ω 2 sn 4 ( z , k ) ) w = 0 ,
in which ω is another parameter. … For properties of the solutions of (29.11.1) see Arscott (1956, 1959), Arscott (1964b, Chapter X), Erdélyi et al. (1955, §16.14), Fedoryuk (1989), and Müller (1966a, b, c).
4: 31.14 General Fuchsian Equation
31.14.1 d 2 w d z 2 + ( j = 1 N γ j z - a j ) d w d z + ( j = 1 N q j z - a j ) w = 0 , j = 1 N q j = 0 .
α β = j = 1 N a j q j .
The three sets of parameters comprise the singularity parameters a j , the exponent parameters α , β , γ j , and the N - 2 free accessory parameters q j . With a 1 = 0 and a 2 = 1 the total number of free parameters is 3 N - 3 . …
31.14.3 w ( z ) = ( j = 1 N ( z - a j ) - γ j / 2 ) W ( z ) ,
5: 28.17 Stability as x ±
§28.17 Stability as x ±
If all solutions of (28.2.1) are bounded when x ± along the real axis, then the corresponding pair of parameters ( a , q ) is called stable. …
See accompanying text
Figure 28.17.1: Stability chart for eigenvalues of Mathieu’s equation (28.2.1). Magnify
6: 31.1 Special Notation
x , y real variables.
a complex parameter, | a | 1 , a 1 .
q , α , β , γ , δ , ϵ , ν complex parameters.
Sometimes the parameters are suppressed.
7: 34.8 Approximations for Large Parameters
§34.8 Approximations for Large Parameters
For large values of the parameters in the 3 j , 6 j , and 9 j symbols, different asymptotic forms are obtained depending on which parameters are large. … and the symbol o ( 1 ) denotes a quantity that tends to zero as the parameters tend to infinity, as in §2.1(i). …
8: 28.14 Fourier Series
28.14.1 me ν ( z , q ) = m = - c 2 m ν ( q ) e i ( ν + 2 m ) z ,
28.14.4 q c 2 m + 2 - ( a - ( ν + 2 m ) 2 ) c 2 m + q c 2 m - 2 = 0 , a = λ ν ( q ) , c 2 m = c 2 m ν ( q ) ,
28.14.5 m = - ( c 2 m ν ( q ) ) 2 = 1 ;
28.14.7 c - 2 m - ν ( q ) = c 2 m ν ( q ) ,
28.14.8 c 2 m ν ( - q ) = ( - 1 ) m c 2 m ν ( q ) .
9: 15.7 Continued Fractions
15.7.1 F ( a , b ; c ; z ) F ( a , b + 1 ; c + 1 ; z ) = t 0 - u 1 z t 1 - u 2 z t 2 - u 3 z t 3 - ,
10: 31.6 Path-Multiplicative Solutions