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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: 36.15 Methods of Computation
§36.15(iii) Integration along Deformed Contour
Direct numerical evaluation can be carried out along a contour that runs along the segment of the real t -axis containing all real critical points of Φ and is deformed outside this range so as to reach infinity along the asymptotic valleys of exp ( i Φ ) . …
§36.15(iv) Integration along Finite Contour
This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of Φ , with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …
4: 5.21 Methods of Computation
Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour. …
5: 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). …
6: 13.4 Integral Representations
§13.4(ii) Contour Integrals
The contour of integration starts and terminates at a point α on the real axis between 0 and 1 . …At the point where the contour crosses the interval ( 1 , ) , t b and the 𝐅 1 2 function assume their principal values; compare §§15.1 and 15.2(i). …The contour cuts the real axis between 1 and 0 . …
7: 5.12 Beta Function
with the contour as shown in Figure 5.12.1. … when b > 0 , a is not an integer and the contour cuts the real axis between 1 and the origin. … where the contour starts from an arbitrary point P in the interval ( 0 , 1 ) , circles 1 and then 0 in the positive sense, circles 1 and then 0 in the negative sense, and returns to P . It can always be deformed into the contour shown in Figure 5.12.3.
See accompanying text
Figure 5.12.3: t -plane. Contour for Pochhammer’s integral. Magnify
8: 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). …
9: 15.6 Integral Representations
In (15.6.2) the point 1 / z lies outside the integration contour, t b 1 and ( t 1 ) c b 1 assume their principal values where the contour cuts the interval ( 1 , ) , and ( 1 z t ) a = 1 at t = 0 . In (15.6.3) the point 1 / ( z 1 ) lies outside the integration contour, the contour cuts the real axis between t = 1 and 0 , at which point ph t = π and ph ( 1 + t ) = 0 . In (15.6.4) the point 1 / z lies outside the integration contour, and at the point where the contour cuts the negative real axis ph t = π and ph ( 1 t ) = 0 . In (15.6.5) the integration contour starts and terminates at a point A on the real axis between 0 and 1 . …However, this reverses the direction of the integration contour, and in consequence (15.6.5) would need to be multiplied by 1 . …
10: 12.5 Integral Representations
§12.5(ii) Contour Integrals
For the particular loop contour, see Figure 5.9.1. … where the contour separates the poles of Γ ( t ) from those of Γ ( 1 2 + a 2 t ) . … where the contour separates the poles of Γ ( t ) from those of Γ ( 1 2 a 2 t ) . …