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limit forms as ℑτ→0+

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1: 20.13 Physical Applications
In the singular limit τ 0 + , the functions θ j ( z | τ ) , j = 1 , 2 , 3 , 4 , become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …
2: 2.6 Distributional Methods
This leads to integrals of the formThe distribution method outlined here can be extended readily to functions f ( t ) having an asymptotic expansion of the formTo define convolutions of distributions, we first introduce the space K + of all distributions of the form D n f , where n is a nonnegative integer, f is a locally integrable function on which vanishes on ( - , 0 ] , and D n f denotes the n th derivative of the distribution associated with f . …It is easily seen that K + forms a commutative, associative linear algebra. … On inserting this identity into (2.6.54), we immediately encounter divergent integrals of the form
3: 3.7 Ordinary Differential Equations
Consideration will be limited to ordinary linear second-order differential equationsWrite τ j = z j + 1 - z j , j = 0 , 1 , , P , expand w ( z ) and w ( z ) in Taylor series (§1.10(i)) centered at z = z j , and apply (3.7.2). …where A ( τ , z ) is the matrix … The remaining two equations are supplied by boundary conditions of the formwith limits taken in (3.7.16) when a or b , or both, are infinite. …
4: 2.4 Contour Integrals
is seen to converge absolutely at each limit, and be independent of σ [ c , ) . Furthermore, as t 0 + , q ( t ) has the expansion (2.3.7). … If this integral converges uniformly at each limit for all sufficiently large t , then by the Riemann–Lebesgue lemma (§1.8(i)) … The final expansion then has the form …The branch of ω 0 = ph ( p ′′ ( t 0 ) ) is the one satisfying | θ + 2 ω + ω 0 | 1 2 π , where ω is the limiting value of ph ( t - t 0 ) as t t 0 from b . …