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1: 19.13 Integrals of Elliptic Integrals
§19.13(i) Integration with Respect to the Modulus
§19.13(ii) Integration with Respect to the Amplitude
2: 18.3 Definitions
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
3: 3.11 Approximation Techniques
They enjoy an orthogonal property with respect to integrals: …
4: 36.2 Catastrophes and Canonical Integrals
with the contour passing to the lower right of u = 0 . …with the contour passing to the upper right of u = 0 . … Ψ 1 is related to the Airy function (§9.2): … … Addendum: For further special cases see §36.2(iv)
5: 20.10 Integrals
§20.10(i) Mellin Transforms with respect to the Lattice Parameter
20.10.1 0 x s - 1 θ 2 ( 0 | i x 2 ) d x = 2 s ( 1 - 2 - s ) π - s / 2 Γ ( 1 2 s ) ζ ( s ) ,
20.10.2 0 x s - 1 ( θ 3 ( 0 | i x 2 ) - 1 ) d x = π - s / 2 Γ ( 1 2 s ) ζ ( s ) ,
20.10.3 0 x s - 1 ( 1 - θ 4 ( 0 | i x 2 ) ) d x = ( 1 - 2 1 - s ) π - s / 2 Γ ( 1 2 s ) ζ ( s ) .
§20.10(ii) Laplace Transforms with respect to the Lattice Parameter
6: 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 ,
7.7.2 w ( z ) = 1 π i - e - t 2 d t t - z = 2 z π i 0 e - t 2 d t t 2 - z 2 , z > 0 .
7.7.3 0 e - a t 2 + 2 i z t d t = 1 2 π a e - z 2 / a + i a F ( z a ) , a > 0 .
7.7.9 0 x erf t d t = x erf x + 1 π ( e - x 2 - 1 ) .
In (7.7.13) and (7.7.14) the integration paths are straight lines, ζ = 1 16 π 2 z 4 , and c is a constant such that 0 < c < 1 4 in (7.7.13), and 0 < c < 3 4 in (7.7.14). …
7: 2.8 Differential Equations with a Parameter
dots denoting differentiations with respect to ξ . … The expansions (2.8.11) and (2.8.12) are both uniform and differentiable with respect to ξ . … The expansions (2.8.15) and (2.8.16) are both uniform and differentiable with respect to ξ . … The expansions (2.8.25) and (2.8.26) are both uniform and differentiable with respect to ξ . … The expansions (2.8.29) and (2.8.30) are both uniform and differentiable with respect to ξ . …
8: 2.3 Integrals of a Real Variable
§2.3(i) Integration by Parts
(In other words, differentiation of (2.3.8) with respect to the parameter λ (or μ ) is legitimate.) … derives from the neighborhood of the minimum of p ( t ) in the integration range. … In consequence, the approximation is nonuniform with respect to α and deteriorates severely as α 0 . A uniform approximation can be constructed by quadratic change of integration variable: …
9: 4.37 Inverse Hyperbolic Functions
Elsewhere on the integration paths in (4.37.1) and (4.37.2) the branches are determined by continuity. In (4.37.3) the integration path may not intersect ± 1 . … The principal values (or principal branches) of the inverse sinh , cosh , and tanh are obtained by introducing cuts in the z -plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts. … These functions are analytic in the cut plane depicted in Figure 4.37.1(iv), (v), (vi), respectively. … are respectively given by …
10: 2.4 Contour Integrals
Then by integration by parts the integral … The most successful results are obtained on moving the integration contour as far to the left as possible. …
  • (c)

    Excluding t = a , ( e i θ p ( t ) - e i θ p ( a ) ) is positive when t 𝒫 , and is bounded away from zero uniformly with respect to θ [ θ 1 , θ 2 ] as t b along 𝒫 .

  • The problem of obtaining an asymptotic approximation to I ( α , z ) that is uniform with respect to α in a region containing α ^ is similar to the problem of a coalescing endpoint and saddle point outlined in §2.3(v). The change of integration variable is given by …