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functions f(ϵ,ℓ;r),h(ϵ,ℓ;r)

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11—20 of 438 matching pages

11: 7.12 Asymptotic Expansions
7.12.2 f ( z ) 1 π z m = 0 ( 1 ) m ( 1 2 ) 2 m ( π z 2 / 2 ) 2 m ,
7.12.4 f ( z ) = 1 π z m = 0 n 1 ( 1 ) m ( 1 2 ) 2 m ( π z 2 / 2 ) 2 m + R n ( f ) ( z ) ,
12: 10 Bessel Functions
Chapter 10 Bessel Functions
13: 15.1 Special Notation
14: 27.5 Inversion Formulas
27.5.1 h ( n ) = d | n f ( d ) g ( n d ) ,
The set of all number-theoretic functions f with f ( 1 ) 0 forms an abelian group under Dirichlet multiplication, with the function 1 / n in (27.2.5) as identity element; see Apostol (1976, p. 129). …
27.5.3 g ( n ) = d | n f ( d ) f ( n ) = d | n g ( d ) μ ( n d ) .
27.5.6 G ( x ) = n x F ( x n ) F ( x ) = n x μ ( n ) G ( x n ) ,
27.5.8 g ( n ) = d | n f ( d ) f ( n ) = d | n ( g ( n d ) ) μ ( d ) .
15: 7.14 Integrals
7.14.1 0 e 2 i a t erfc ( b t ) d t = 1 a π F ( a b ) + i 2 a ( 1 e ( a / b ) 2 ) , a , | ph b | < 1 4 π .
7.14.5 0 e a t C ( t ) d t = 1 a f ( a π ) , a > 0 ,
16: 7.23 Tables
  • Abramowitz and Stegun (1964, Chapter 7) includes erf x , ( 2 / π ) e x 2 , x [ 0 , 2 ] , 10D; ( 2 / π ) e x 2 , x [ 2 , 10 ] , 8S; x e x 2 erfc x , x 2 [ 0 , 0.25 ] , 7D; 2 n Γ ( 1 2 n + 1 ) i n erfc ( x ) , n = 1 ( 1 ) 6 , 10 , 11 , x [ 0 , 5 ] , 6S; F ( x ) , x [ 0 , 2 ] , 10D; x F ( x ) , x 2 [ 0 , 0.25 ] , 9D; C ( x ) , S ( x ) , x [ 0 , 5 ] , 7D; f ( x ) , g ( x ) , x [ 0 , 1 ] , x 1 [ 0 , 1 ] , 15D.

  • 17: 16.1 Special Notation
    The main functions treated in this chapter are the generalized hypergeometric function F q p ( a 1 , , a p b 1 , , b q ; z ) , the Appell (two-variable hypergeometric) functions F 1 ( α ; β , β ; γ ; x , y ) , F 2 ( α ; β , β ; γ , γ ; x , y ) , F 3 ( α , α ; β , β ; γ ; x , y ) , F 4 ( α , β ; γ , γ ; x , y ) , and the Meijer G -function G p , q m , n ( z ; a 1 , , a p b 1 , , b q ) . Alternative notations are F q p ( 𝐚 𝐛 ; z ) , F q p ( a 1 , , a p ; b 1 , , b q ; z ) , and F q p ( 𝐚 ; 𝐛 ; z ) for the generalized hypergeometric function, F 1 ( α , β , β ; γ ; x , y ) , F 2 ( α , β , β ; γ , γ ; x , y ) , F 3 ( α , α , β , β ; γ ; x , y ) , F 4 ( α , β ; γ , γ ; x , y ) , for the Appell functions, and G p , q m , n ( z ; 𝐚 ; 𝐛 ) for the Meijer G -function.
    18: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    Functions f , g L 2 ( X , d α ) for which f g , f g = 0 are identified with each other. The space L 2 ( X , d α ) becomes a separable Hilbert space with inner product
    1.18.12 f , g = a b f ( x ) g ( x ) ¯ d α ( x ) ,
    Spectral expansions of T , and of functions F ( T ) of T , these being expansions of T and F ( T ) in terms of the eigenvalues and eigenfunctions summed over the spectrum, then follow:where f ^ ( λ n ) = 1 π 0 π f ( y ) e 2 i n y d y = f , ϕ exp ( n ) , being that of (1.8.3) and (1.8.4). The eigenfunction expansions of (1.8.1) and (1.8.2) follow from Cases 1, 2, above.
    1.18.47 f ^ ( λ ) = f , ϕ λ .
    19: 33.3 Graphics
    §33.3(i) Line Graphs of the Coulomb Radial Functions F ( η , ρ ) and G ( η , ρ )
    33.3.1 M ( η , ρ ) = ( F 2 ( η , ρ ) + G 2 ( η , ρ ) ) 1 / 2 = | H ± ( η , ρ ) | .
    §33.3(ii) Surfaces of the Coulomb Radial Functions F 0 ( η , ρ ) and G 0 ( η , ρ )
    20: 32.14 Combinatorics
    32.14.1 lim N Prob ( N ( 𝝅 ) 2 N N 1 / 6 s ) = F ( s ) ,
    where the distribution function F ( s ) is defined here by
    32.14.2 F ( s ) = exp ( s ( x s ) w 2 ( x ) d x ) ,
    The distribution function F ( s ) given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of n × n Hermitian matrices; see Tracy and Widom (1994). …