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1: 5.21 Methods of Computation
An effective way of computing Γ ( z ) in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). …
2: About the Project
Refer to caption
Figure 1: The Editors and 9 of the 10 Associate Editors of the DLMF Project (photo taken at 3rd Editors Meeting, April, 2001). The front row, from left to right: Ronald F. …The back row, from left to right: William P. …
3: 4.42 Solution of Triangles
§4.42(i) Planar Right Triangles
See accompanying text
Figure 4.42.1: Planar right triangle. Magnify
4: 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).

  • 5: 32.15 Orthogonal Polynomials
    For this result and applications see Fokas et al. (1991): in this reference, on the right-hand side of Eq. …
    6: 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. …
    7: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
    8: 2.3 Integrals of a Real Variable
    assume a and b are finite, and q ( t ) is infinitely differentiable on [ a , b ] . … When p ( t ) is real and x is a large positive parameter, the main contribution to the integral
    2.3.13 I ( x ) = a b e x p ( t ) q ( t ) d t
    When the parameter x is large the contributions from the real and imaginary parts of the integrand in
    2.3.19 I ( x ) = a b e i x p ( t ) q ( t ) d t
    9: 2.4 Contour Integrals
    Let 𝒫 denote the path for the contour integral
    2.4.10 I ( z ) = a b e z p ( t ) q ( t ) d t ,
    2.4.14 I ( z ) = t 0 b e z p ( t ) q ( t ) d t t 0 a e z p ( t ) q ( t ) d t ,
    and apply the result of §2.4(iii) to each integral on the right-hand side, the role of the series (2.4.11) being played by the Taylor series of p ( t ) and q ( t ) at t = t 0 . …Thus the right-hand side of (2.4.14) reduces to the error terms. …
    10: 19.32 Conformal Map onto a Rectangle
    As p proceeds along the entire real axis with the upper half-plane on the right, z describes the rectangle in the clockwise direction; hence z ( x 3 ) is negative imaginary. …