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1: 25.19 Tables
  • Abramowitz and Stegun (1964) tabulates: ζ ( n ) , n = 2 , 3 , 4 , , 20D (p. 811); Li 2 ( 1 x ) , x = 0 ( .01 ) 0.5 , 9D (p. 1005); f ( θ ) , θ = 15 ( 1 ) 30 ( 2 ) 90 ( 5 ) 180 , f ( θ ) + θ ln θ , θ = 0 ( 1 ) 15 , 6D (p. 1006). Here f ( θ ) denotes Clausen’s integral, given by the right-hand side of (25.12.9).

  • 2: 32.15 Orthogonal Polynomials
    For this result and applications see Fokas et al. (1991): in this reference, on the right-hand side of Eq. …
    3: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
    4: 16.5 Integral Representations and Integrals
    In the case p = q the left-hand side of (16.5.1) is an entire function, and the right-hand side supplies an integral representation valid when | ph ( z ) | < π / 2 . In the case p = q + 1 the right-hand side of (16.5.1) supplies the analytic continuation of the left-hand side from the open unit disk to the sector | ph ( 1 z ) | < π ; compare §16.2(iii). Lastly, when p > q + 1 the right-hand side of (16.5.1) can be regarded as the definition of the (customarily undefined) left-hand side. In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as z 0 in the sector | ph ( z ) | ( p + 1 q δ ) π / 2 , where δ is an arbitrary small positive constant. …
    5: 24.19 Methods of Computation
    If N ~ 2 n denotes the right-hand side of (24.19.1) but with the second product taken only for p ( π e ) 1 2 n + 1 , then N 2 n = N ~ 2 n for n 2 . …
    6: 19.21 Connection Formulas
    19.21.7 ( x y ) R D ( y , z , x ) + ( z y ) R D ( x , y , z ) = 3 R F ( x , y , z ) 3 y 1 / 2 x 1 / 2 z 1 / 2 ,
    19.21.8 R D ( y , z , x ) + R D ( z , x , y ) + R D ( x , y , z ) = 3 x 1 / 2 y 1 / 2 z 1 / 2 ,
    19.21.10 2 R G ( x , y , z ) = z R F ( x , y , z ) 1 3 ( x z ) ( y z ) R D ( x , y , z ) + x 1 / 2 y 1 / 2 z 1 / 2 , z 0 .
    Because R G is completely symmetric, x , y , z can be permuted on the right-hand side of (19.21.10) so that ( x z ) ( y z ) 0 if the variables are real, thereby avoiding cancellations when R G is calculated from R F and R D (see §19.36(i)). …
    7: 19.25 Relations to Other Functions
    19.25.7 E ( ϕ , k ) = 2 R G ( c 1 , c k 2 , c ) ( c 1 ) R F ( c 1 , c k 2 , c ) c 1 c k 2 / c ,
    All terms on the right-hand sides are nonnegative when k 2 0 , 0 k 2 1 , or 1 k 2 c , respectively. … The transformations in §19.7(ii) result from the symmetry and homogeneity of functions on the right-hand sides of (19.25.5), (19.25.7), and (19.25.14). … The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which ( z ) e j < 0 , for some j . … The sign on the right-hand side of (19.25.40) will change whenever one crosses a curve on which σ j 2 ( z ) < 0 , for some j . …
    8: 16.10 Expansions in Series of F q p Functions
    When | ζ 1 | < 1 the series on the right-hand side converges in the half-plane z < 1 2 . …
    9: 1.6 Vectors and Vector-Valued Functions
    where 𝐧 is the unit vector normal to 𝐚 and 𝐛 whose direction is determined by the right-hand rule; see Figure 1.6.1.
    See accompanying text
    Figure 1.6.1: Vector notation. Right-hand rule for cross products. Magnify
    10: 1.8 Fourier Series
    at every point at which f ( x ) has both a left-hand derivative (that is, (1.4.4) applies when h 0 ) and a right-hand derivative (that is, (1.4.4) applies when h 0 + ). … …