limit forms as ℑτ→0+

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1: 20.13 Physical Applications

… ►In the singular limit $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, 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 form … ►The distribution method outlined here can be extended readily to functions $f(t)$ having an asymptotic expansion of the form … ►To 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 $ℝ$ which vanishes on $(-\mathrm{\infty },0]$, 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 equations … ►Write ${\tau }_j=z_{j+1}-z_j$, $j=0,1,\mathrm{\dots },P$, expand $w(z)$ and $w^{\prime }(z)$ in Taylor series (§1.10(i)) centered at $z=z_j$, and apply (3.7.2). …where $𝐀(\tau ,z)$ is the matrix …and $𝐛(\tau ,z)$ is the vector … ►with 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 $\sigma \in [c,\mathrm{\infty })$. Furthermore, as $t\to 0+$, $q(t)$ 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 ${\omega }_0=\mathrm{ph}\left(p^{\prime \prime }(t_0)\right)$ is the one satisfying $|\theta +2\omega +{\omega }_0|\le \frac{1}{2}\pi$, where $\omega$ is the limiting value of $\mathrm{ph}\left(t-t_0\right)$ as $t\to t_0$ from $b$. …