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1: 25.15 Dirichlet L -functions
§25.15 Dirichlet L -functions
§25.15(i) Definitions and Basic Properties
The notation L ( s , χ ) was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series … …
§25.15(ii) Zeros
2: 23.21 Physical Applications
In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form ( 1 - x 2 ) ( 1 - k 2 x 2 ) . …
23.21.1 x 2 ρ - e 1 + y 2 ρ - e 2 + z 2 ρ - e 3 = 1 ,
23.21.3 f ( ρ ) = 2 ( ( ρ - e 1 ) ( ρ - e 2 ) ( ρ - e 3 ) ) 1 / 2 .
23.21.5 ( ( v ) - ( w ) ) ( ( w ) - ( u ) ) ( ( u ) - ( v ) ) 2 = ( ( w ) - ( v ) ) 2 u 2 + ( ( u ) - ( w ) ) 2 v 2 + ( ( v ) - ( u ) ) 2 w 2 .
3: 31.15 Stieltjes Polynomials
31.15.12 ρ ( z ) = ( j = 1 N - 1 k = 1 N | z j - a k | γ k - 1 ) ( j < k N - 1 ( z k - z j ) ) .
The normalized system of products (31.15.8) forms an orthonormal basis in the Hilbert space L ρ 2 ( Q ) . For further details and for the expansions of analytic functions in this basis see Volkmer (1999).
4: 18.4 Graphics
See accompanying text
Figure 18.4.5: Laguerre polynomials L ( n ) ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
See accompanying text
Figure 18.4.6: Laguerre polynomials L 3 ( α ) ( x ) , α = 0 , 1 , 2 , 3 , 4 . Magnify
See accompanying text
Figure 18.4.8: Laguerre polynomials L 3 ( α ) ( x ) , 0 α 3 , 0 x 10 . Magnify 3D Help
See accompanying text
Figure 18.4.9: Laguerre polynomials L 4 ( α ) ( x ) , 0 α 3 , 0 x 10 . Magnify 3D Help
5: 18.41 Tables
Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates T n ( x ) , U n ( x ) , L ( n ) ( x ) , and H n ( x ) for n = 0 ( 1 ) 12 . The ranges of x are 0.2 ( .2 ) 1 for T n ( x ) and U n ( x ) , and 0.5 , 1 , 3 , 5 , 10 for L ( n ) ( x ) and H n ( x ) . … For P n ( x ) , L ( n ) ( x ) , and H n ( x ) see §3.5(v). …
6: 30.15 Signal Analysis
The sequence ϕ n , n = 0 , 1 , 2 , forms an orthonormal basis in the space of σ -bandlimited functions, and, after normalization, an orthonormal basis in L 2 ( - τ , τ ) . … taken over all f L 2 ( - , ) subject to …
7: 23.10 Addition Theorems and Other Identities
23.10.17 ( c z | c 𝕃 ) = c - 2 ( z | 𝕃 ) ,
23.10.18 ζ ( c z | c 𝕃 ) = c - 1 ζ ( z | 𝕃 ) ,
23.10.19 σ ( c z | c 𝕃 ) = c σ ( z | 𝕃 ) .
Also, when 𝕃 is replaced by c 𝕃 the lattice invariants g 2 and g 3 are divided by c 4 and c 6 , respectively. …
8: 23.14 Integrals
23.14.2 2 ( z ) d z = 1 6 ( z ) + 1 12 g 2 z ,
9: 23.2 Definitions and Periodic Properties
The generators of a given lattice 𝕃 are not unique. …where a , b , c , d are integers, then 2 χ 1 , 2 χ 3 are generators of 𝕃 iff … When z 𝕃 the functions are related by … When it is important to display the lattice with the functions they are denoted by ( z | 𝕃 ) , ζ ( z | 𝕃 ) , and σ ( z | 𝕃 ) , respectively. … If 2 ω 1 , 2 ω 3 is any pair of generators of 𝕃 , and ω 2 is defined by (23.2.1), then …
10: 19.33 Triaxial Ellipsoids
The external field and the induced magnetization together produce a uniform field inside the ellipsoid with strength H / ( 1 + L c χ ) , where L c is the demagnetizing factor, given in cgs units by
19.33.7 L c = 2 π a b c 0 d λ ( a 2 + λ ) ( b 2 + λ ) ( c 2 + λ ) 3 = V R D ( a 2 , b 2 , c 2 ) .
19.33.8 L a + L b + L c = 4 π ,
where L a and L b are obtained from L c by permutation of a , b , and c . …