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1: 9.14 Incomplete Airy Functions
Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …
2: 9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
3: 12.16 Mathematical Applications
PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). …
4: 21.1 Special Notation
g , h positive integers.
a ω line integral of the differential ω over the cycle a .
5: 27.18 Methods of Computation: Primes
An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). …
6: 21.7 Riemann Surfaces
21.7.5 a k ω j = δ j , k , j , k = 1 , 2 , , g .
21.7.6 Ω j k = b k ω j , j , k = 1 , 2 , , g ,
7: 12.5 Integral Representations
§12.5(ii) Contour Integrals
8: 8.6 Integral Representations
§8.6(ii) Contour Integrals
9: 11.5 Integral Representations
§11.5(ii) Contour Integrals
10: 2.4 Contour Integrals
§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 assigning an appropriate value to c to modify the contour, the approximating integral is reducible to an Airy function or a Scorer function (§§9.2, 9.12). …