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1: 27.9 Quadratic Characters
For an odd prime p , the Legendre symbol ( n | p ) is defined as follows. If p divides n , then the value of ( n | p ) is 0 . …The Legendre symbol ( n | p ) , as a function of n , is a Dirichlet character (mod p ). …
27.9.2 ( 2 | p ) = ( 1 ) ( p 2 1 ) / 8 .
27.9.3 ( p | q ) ( q | p ) = ( 1 ) ( p 1 ) ( q 1 ) / 4 .
2: 27.1 Special Notation
d , k , m , n positive integers (unless otherwise indicated).
( n | p ) Legendre symbol; see §27.9.
3: 19.15 Advantages of Symmetry
These reduction theorems, unknown in the Legendre theory, allow symbolic integration without imposing conditions on the parameters and the limits of integration (see §19.29(ii)). …
4: 34.3 Basic Properties: 3 j Symbol
§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics
34.3.19 P l 1 ( cos θ ) P l 2 ( cos θ ) = l ( 2 l + 1 ) ( l 1 l 2 l 0 0 0 ) 2 P l ( cos θ ) ,
34.3.21 0 π P l 1 ( cos θ ) P l 2 ( cos θ ) P l 3 ( cos θ ) sin θ d θ = 2 ( l 1 l 2 l 3 0 0 0 ) 2 ,
5: 14.6 Integer Order
6: 19.12 Asymptotic Approximations
19.12.1 K ( k ) = m = 0 ( 1 2 ) m ( 1 2 ) m m ! m ! k 2 m ( ln ( 1 k ) + d ( m ) ) , 0 < | k | < 1 ,
19.12.2 E ( k ) = 1 + 1 2 m = 0 ( 1 2 ) m ( 3 2 ) m ( 2 ) m m ! k 2 m + 2 ( ln ( 1 k ) + d ( m ) 1 ( 2 m + 1 ) ( 2 m + 2 ) ) , | k | < 1 ,
7: Errata
  • Chapters 10 Bessel Functions, 18 Orthogonal Polynomials, 34 3j, 6j, 9j Symbols

    The Legendre polynomial P n was mistakenly identified as the associated Legendre function P n in §§10.54, 10.59, 10.60, 18.18, 18.41, 34.3 (and was thus also affected by the bug reported below). These symbols now link correctly to their definitions. Reported by Roy Hughes on 2022-05-23

  • Equation (14.8.3)
    14.8.3 𝖰 ν ( x ) = 1 2 ln ( 2 1 x ) γ ψ ( ν + 1 ) + O ( ( 1 x ) ln ( 1 x ) ) , ν 1 , 2 , 3 ,

    The symbol O ( 1 x ) has been corrected to be O ( ( 1 x ) ln ( 1 x ) ) .

    Reported by Mark Ashbaugh on 2022-02-08

  • Equation (14.8.9)
    14.8.9 𝑸 ν ( x ) = ln ( x 1 ) 2 Γ ( ν + 1 ) + 1 2 ln 2 γ ψ ( ν + 1 ) Γ ( ν + 1 ) + O ( ( x 1 ) ln ( x 1 ) ) , ν 1 , 2 , 3 ,

    The symbol O ( x 1 ) has been corrected to be O ( ( x 1 ) ln ( x 1 ) ) .

    Reported by Mark Ashbaugh on 2022-02-08

  • 8: 10 Bessel Functions
    9: 18 Orthogonal Polynomials
    10: 19.5 Maclaurin and Related Expansions
    19.5.4_1 F ( ϕ , k ) = m = 0 ( 1 2 ) m sin 2 m + 1 ϕ ( 2 m + 1 ) m ! F 1 2 ( m + 1 2 , 1 2 m + 3 2 ; sin 2 ϕ ) k 2 m = sin ϕ F 1 ( 1 2 ; 1 2 , 1 2 ; 3 2 ; sin 2 ϕ , k 2 sin 2 ϕ ) ,
    19.5.4_2 E ( ϕ , k ) = m = 0 ( 1 2 ) m sin 2 m + 1 ϕ ( 2 m + 1 ) m ! F 1 2 ( m + 1 2 , 1 2 m + 3 2 ; sin 2 ϕ ) k 2 m = sin ϕ F 1 ( 1 2 ; 1 2 , 1 2 ; 3 2 ; sin 2 ϕ , k 2 sin 2 ϕ ) ,
    19.5.4_3 Π ( ϕ , α 2 , k ) = m = 0 ( 1 2 ) m sin 2 m + 1 ϕ ( 2 m + 1 ) m ! F 1 ( m + 1 2 ; 1 2 , 1 ; m + 3 2 ; sin 2 ϕ , α 2 sin 2 ϕ ) k 2 m ,