contour
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1: 36.15 Methods of Computation
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§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 -axis containing all real critical points of and is deformed outside this range so as to reach infinity along the asymptotic valleys of . … ►§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 , with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …2: 9.14 Incomplete Airy Functions
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►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.
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3: 9.15 Mathematical Applications
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►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.
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4: 5.21 Methods of Computation
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►Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour.
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5: 12.16 Mathematical Applications
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►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).
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6: 5.12 Beta Function
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►with the contour as shown in Figure 5.12.1.
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►when , is not an integer and the contour cuts the real axis between and the origin.
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►where the contour starts from an arbitrary point in the interval , circles and then in the positive sense, circles and then in the negative sense, and returns to .
It can always be deformed into the contour shown in Figure 5.12.3.
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7: 13.4 Integral Representations
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§13.4(ii) Contour Integrals
… ►The contour of integration starts and terminates at a point on the real axis between and . …The contour cuts the real axis between and . … ►Again, and the function assume their principal values where the contour intersects the positive real axis. … ►8: 27.18 Methods of Computation: Primes
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►An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000).
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9: 15.6 Integral Representations
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►In (15.6.2) the point lies outside the integration contour, and assume their principal values where the contour cuts the interval , and at .
►In (15.6.3) the point lies outside the integration contour, the contour cuts the real axis between and , at which point and .
►In (15.6.4) the point lies outside the integration contour, and at the point where the contour cuts the negative real axis and .
►In (15.6.5) the integration contour starts and terminates at a point on the real axis between and .
…However, this reverses the direction of the integration contour, and in consequence (15.6.5) would need to be multiplied by .
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10: 5.19 Mathematical Applications
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►Many special functions can be represented as a Mellin–Barnes
integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of , the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends.
…By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of for large , or small , can be obtained complete with an integral representation of the error term.
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