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1: 3.3 Interpolation
§3.3(iii) Divided Differences
[ z 0 ] f = f 0 ,
[ z 0 , z 1 ] f = ( [ z 1 ] f [ z 0 ] f ) / ( z 1 z 0 ) ,
[ z 0 , z 1 , z 2 ] f = ( [ z 1 , z 2 ] f [ z 0 , z 1 ] f ) / ( z 2 z 0 ) ,
3.3.39 x ( f ) = [ f 0 ] x + ( f f 0 ) [ f 0 , f 1 ] x + ( f f 0 ) ( f f 1 ) [ f 0 , f 1 , f 2 ] x ;
2: 18.26 Wilson Class: Continued
18.26.14 δ y ( W n ( y 2 ; a , b , c , d ) ) / δ y ( y 2 ) = n ( n + a + b + c + d 1 ) W n 1 ( y 2 ; a + 1 2 , b + 1 2 , c + 1 2 , d + 1 2 ) .
18.26.15 δ y ( S n ( y 2 ; a , b , c ) ) / δ y ( y 2 ) = n S n 1 ( y 2 ; a + 1 2 , b + 1 2 , c + 1 2 ) .
3: 21.7 Riemann Surfaces
21.7.15 4 𝜼 1 ( T ) 𝜼 2 ( T ) = 1 2 ( | T U | g 1 ) ( mod 2 ) ,
4: 20.2 Definitions and Periodic Properties
20.2.6 θ 1 ( z + ( m + n τ ) π | τ ) = ( 1 ) m + n q n 2 e 2 i n z θ 1 ( z | τ ) ,
20.2.7 θ 2 ( z + ( m + n τ ) π | τ ) = ( 1 ) m q n 2 e 2 i n z θ 2 ( z | τ ) ,
20.2.8 θ 3 ( z + ( m + n τ ) π | τ ) = q n 2 e 2 i n z θ 3 ( z | τ ) ,
20.2.12 θ 2 ( z | τ ) = θ 1 ( z + 1 2 π | τ ) = M θ 3 ( z + 1 2 π τ | τ ) = M θ 4 ( z + 1 2 π + 1 2 π τ | τ ) ,
20.2.13 θ 3 ( z | τ ) = θ 4 ( z + 1 2 π | τ ) = M θ 2 ( z + 1 2 π τ | τ ) = M θ 1 ( z + 1 2 π + 1 2 π τ | τ ) ,
5: 18.1 Notation
δ x ( f ( x ) ) = ( f ( x + 1 2 i ) f ( x 1 2 i ) ) / i ,
6: 24.10 Arithmetic Properties
Here and elsewhere two rational numbers are congruent if the modulus divides the numerator of their difference. …
7: 19.36 Methods of Computation
The cases k c 2 / 2 p < and < p < k c 2 / 2 require different treatment for numerical purposes, and again precautions are needed to avoid cancellations. …
8: Errata
  • Equation (3.3.34)

    In the online version, the leading divided difference operators were previously omitted from these formulas, due to programming error.

    Reported by Nico Temme on 2021-06-01

  • Equation (10.22.72)
    10.22.72 0 J μ ( a t ) J ν ( b t ) J ν ( c t ) t 1 μ d t = ( b c ) μ 1 sin ( ( μ ν ) π ) ( sinh χ ) μ 1 2 ( 1 2 π 3 ) 1 2 a μ e ( μ 1 2 ) i π Q ν 1 2 1 2 μ ( cosh χ ) , μ > 1 2 , ν > 1 , a > b + c , cosh χ = ( a 2 b 2 c 2 ) / ( 2 b c )

    Originally, the factor on the right-hand side was written as ( b c ) μ 1 cos ( ν π ) ( sinh χ ) μ 1 2 ( 1 2 π 3 ) 1 2 a μ , which was taken directly from Watson (1944, p. 412, (13.46.5)), who uses a different normalization for the associated Legendre function of the second kind Q ν μ . Watson’s Q ν μ equals sin ( ( ν + μ ) π ) sin ( ν π ) e μ π i Q ν μ in the DLMF.

    Reported by Arun Ravishankar on 2018-10-22

  • 9: 4.24 Inverse Trigonometric Functions: Further Properties
    4.24.2 arccos z = ( 2 ( 1 z ) ) 1 / 2 ( 1 + n = 1 1 3 5 ( 2 n 1 ) 2 2 n ( 2 n + 1 ) n ! ( 1 z ) n ) , | 1 z | 2 .
    10: 7.7 Integral Representations
    7.7.1 erfc z = 2 π e z 2 0 e z 2 t 2 t 2 + 1 d t , | ph z | 1 4 π ,
    7.7.7 x e a 2 t 2 ( b 2 / t 2 ) d t = π 4 a ( e 2 a b erfc ( a x + ( b / x ) ) + e 2 a b erfc ( a x ( b / x ) ) ) , x > 0 , | ph a | < 1 4 π .
    7.7.8 0 e a 2 t 2 ( b 2 / t 2 ) d t = π 2 a e 2 a b , | ph a | < 1 4 π , | ph b | < 1 4 π .
    7.7.10 f ( z ) = 1 π 2 0 e π z 2 t / 2 t ( t 2 + 1 ) d t , | ph z | 1 4 π ,
    7.7.11 g ( z ) = 1 π 2 0 t e π z 2 t / 2 t 2 + 1 d t , | ph z | 1 4 π ,