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1: 20.2 Definitions and Periodic Properties
The four points ( 0 , π , π + τ π , τ π ) are the vertices of the fundamental parallelogram in the z -plane; see Figure 20.2.1. …
Figure 20.2.1: z -plane. Fundamental parallelogram. …
2: Howard S. Cohl
Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and q -series. …
3: Leonard C. Maximon
Maximon published numerous papers on the fundamental processes of quantum electrodynamics and on the special functions of mathematical physics. …
4: 28.29 Definitions and Basic Properties
28.29.4 w I ( z + π , λ ) = w I ( π , λ ) w I ( z , λ ) + w I ( π , λ ) w II ( z , λ ) ,
28.29.5 w II ( z + π , λ ) = w II ( π , λ ) w I ( z , λ ) + w II ( π , λ ) w II ( z , λ ) .
28.29.8 [ w I ( π , λ ) w II ( π , λ ) w I ( π , λ ) w II ( π , λ ) ] .
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. …
5: 23.5 Special Lattices
Then Δ > 0 and the parallelogram with vertices at 0 , 2 ω 1 , 2 ω 1 + 2 ω 3 , 2 ω 3 is a rectangle. … The parallelogram 0 , 2 ω 1 , 2 ω 1 + 2 ω 3 , 2 ω 3 is a square, and … The parallelogram 0 , 2 ω 1 2 ω 3 , 2 ω 1 , 2 ω 3 , is a rhombus: see Figure 23.5.1. …
6: 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
28.2.5 [ w I ( 0 ; a , q ) w II ( 0 ; a , q ) w I ( 0 ; a , q ) w II ( 0 ; a , q ) ] = [ 1 0 0 1 ] .
28.2.6 𝒲 { w I , w II } = 1 ,
28.2.7 w I ( z ± π ; a , q ) = w I ( π ; a , q ) w I ( z ; a , q ) ± w I ( π ; a , q ) w II ( z ; a , q ) ,
28.2.8 w II ( z ± π ; a , q ) = ± w II ( π ; a , q ) w I ( z ; a , q ) + w II ( π ; a , q ) w II ( z ; a , q ) ,
7: 3.12 Mathematical Constants
The fundamental constant …
8: 8.24 Physical Applications
With more general values of p , E p ( x ) supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).
9: 16.21 Differential Equation
A fundamental set of solutions of (16.21.1) is given by …For other fundamental sets see Erdélyi et al. (1953a, §5.4) and Marichev (1984).
10: 15.10 Hypergeometric Differential Equation
§15.10(i) Fundamental Solutions
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. … (a) If c equals n = 1 , 2 , 3 , , and a = 1 , 2 , , n 1 , then fundamental solutions in the neighborhood of z = 0 are given by (15.10.2) with the interpretation (15.2.5) for f 2 ( z ) . … 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. …