Feynman path integrals
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11: 20.13 Physical Applications
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►In the singular limit , the functions , , become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195).
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12: Bibliography C
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Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams.
J. Comput. Appl. Math. 115 (1-2), pp. 93–99.
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On computing elliptic integrals and functions.
J. Math. and Phys. 44, pp. 36–51.
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Inequalities for a symmetric elliptic integral.
Proc. Amer. Math. Soc. 25 (3), pp. 698–703.
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Remarks on the full asymptotic expansion of Feynman parametrized integrals.
Lett. Nuovo Cimento (2) 13 (8), pp. 310–312.
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Integrals involving complete elliptic integrals.
J. Comput. Appl. Math. 106 (1), pp. 169–175.
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13: 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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14: 31.9 Orthogonality
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►The integration path begins at , encircles once in the positive sense, followed by once in the positive sense, and so on, returning finally to .
The integration path is called a Pochhammer double-loop
contour (compare Figure 5.12.3).
The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning.
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►and the integration paths
, are Pochhammer double-loop contours encircling distinct pairs of singularities , , .
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►For bi-orthogonal relations for path-multiplicative solutions see Schmidt (1979, §2.2).
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15: 9.17 Methods of Computation
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►As described in §3.7(ii), to ensure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows at least as fast as all other solutions of the differential equation.
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§9.17(iii) Integral Representations
►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing in the complex plane, once values of this function can be generated on the positive real axis. … ►In the first method the integration path for the contour integral (9.5.4) is deformed to coincide with paths of steepest descent (§2.4(iv)). …The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). …16: 8.6 Integral Representations
§8.6 Integral Representations
… ►§8.6(ii) Contour Integrals
… ► takes its principal value where the path intersects the positive real axis, and is continuous elsewhere on the path. …where the integration path passes above or below the pole at , according as upper or lower signs are taken. … ►In (8.6.10)–(8.6.12), is a real constant and the path of integration is indented (if necessary) so that in the case of (8.6.10) it separates the poles of the gamma function from the pole at , in the case of (8.6.11) it is to the right of all poles, and in the case of (8.6.12) it separates the poles of the gamma function from the poles at . …17: 1.6 Vectors and Vector-Valued Functions
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§1.6(iv) Path and Line Integrals
… ►A path is defined by , with ranging over an interval and differentiable. …then the length of a path for is …The path integral of a continuous function is …If and , then the reparametrization is orientation-reversing and …18: 15.17 Mathematical Applications
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§15.17(ii) Conformal Mappings
… ►Hypergeometric functions, especially complete elliptic integrals, also play an important role in quasiconformal mapping. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … ►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. …19: 26.2 Basic Definitions
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