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L’Hôpital rule

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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: 3.5 Quadrature
§3.5(i) Trapezoidal Rules
The composite trapezoidal rule is …
§3.5(ii) Simpson’s Rule
§3.5(iv) Interpolatory Quadrature Rules
3: 1.4 Calculus of One Variable
Chain Rule
L’Hôpital’s Rule
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: 34.5 Basic Properties: 6 j Symbol
34.5.11 ( 2 j 1 + 1 ) ( ( J 3 + J 2 - J 1 ) ( L 3 + L 2 - J 1 ) - 2 ( J 3 L 3 + J 2 L 2 - J 1 L 1 ) ) { j 1 j 2 j 3 l 1 l 2 l 3 } = j 1 E ( j 1 + 1 ) { j 1 + 1 j 2 j 3 l 1 l 2 l 3 } + ( j 1 + 1 ) E ( j 1 ) { j 1 - 1 j 2 j 3 l 1 l 2 l 3 } ,
L r = l r ( l r + 1 ) ,
Equations (34.5.15) and (34.5.16) are the sum rules. They constitute addition theorems for the 6 j symbol. …
6: 3.2 Linear Algebra
3.2.5 A = L U .
With y = [ y 1 , y 2 , , y n ] T the process of solution can then be regarded as first solving the equation L y = b for y (forward elimination), followed by the solution of U x = y for x (back substitution). … Because of rounding errors, the residual vector r = b - A x is nonzero as a rule. …
3.2.8 L = [ 1 0 0 2 1 0 n - 1 1 0 0 n 1 ] ,
3.2.11 y j = f j - j y j - 1 , j = 2 , , n ,
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: 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). …