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1: 1.13 Differential Equations
Fundamental Pair
Two solutions w 1 ( z ) and w 2 ( z ) are called a fundamental pair if any other solution w ( z ) is expressible as …A fundamental pair can be obtained, for example, by taking any z 0 D and requiring that … The following three statements are equivalent: w 1 ( z ) and w 2 ( z ) comprise a fundamental pair in D ; 𝒲 { w 1 ( z ) , w 2 ( z ) } does not vanish in D ; w 1 ( z ) and w 2 ( z ) are linearly independent, that is, the only constants A and B such that … If w 0 ( z ) is any one solution, and w 1 ( z ) , w 2 ( z ) are a fundamental pair of solutions of the corresponding homogeneous equation (1.13.1), then every solution of (1.13.8) can be expressed as …
2: 13.2 Definitions and Basic Properties
Fundamental pairs of solutions of (13.2.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are … A fundamental pair of solutions that is numerically satisfactory near the origin is … When b = n + 1 = 1 , 2 , 3 , , a fundamental pair that is numerically satisfactory near the origin is M ( a , n + 1 , z ) and … When b = n = 0 , 1 , 2 , , a fundamental pair that is numerically satisfactory near the origin is z n + 1 M ( a + n + 1 , n + 2 , z ) and …
3: 28.29 Definitions and Basic Properties
If ν ( 0 , 1 ) is a solution of (28.29.9), then F ν ( z ) , F ν ( z ) comprise a fundamental pair of solutions of Hill’s equation. …
4: 13.14 Definitions and Basic Properties
Fundamental pairs of solutions of (13.14.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are … A fundamental pair of solutions that is numerically satisfactory in the sector | ph z | π near the origin is …
5: 28.2 Definitions and Basic Properties
(28.2.1) possesses a fundamental pair of solutions w I ( z ; a , q ) , w II ( z ; a , q ) called basic solutions with …
6: 15.10 Hypergeometric Differential Equation
When none of the exponent pairs differ by an integer, that is, when none of c , c a b , a b is an integer, we have the following pairs f 1 ( z ) , f 2 ( z ) of fundamental solutions. … The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions. …
7: 18.38 Mathematical Applications
Classical OP’s play a fundamental role in Gaussian quadrature. …
Integrable Systems
It has elegant structures, including N -soliton solutions, Lax pairs, and Bäcklund transformations. …
8: 3.11 Approximation Techniques
The pair of vectors { 𝐟 , 𝐚 } …is called a discrete Fourier transform pair. … is of fundamental importance in the FFT algorithm. …