# limit forms as ℑτ→0+

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##### 1: 20.13 Physical Applications
In the singular limit $\Im\tau\rightarrow 0+$, the functions $\theta_{j}\left(z\middle|\tau\right)$, $j=1,2,3,4$, 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 $\mathbb{R}$ which vanishes on $(-\infty,0]$, 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 equationsWrite $\tau_{j}=z_{j+1}-z_{j}$, $j=0,1,\dots,P$, expand $w(z)$ and $w^{\prime}(z)$ in Taylor series (§1.10(i)) centered at $z=z_{j}$, and apply (3.7.2). …where $\mathbf{A}(\tau,z)$ is the matrix …and $\mathbf{b}(\tau,z)$ 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 $\sigma\in[c,\infty)$. 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 $\omega_{0}=\operatorname{ph}\left(p^{\prime\prime}(t_{0})\right)$ is the one satisfying $|\theta+2\omega+\omega_{0}|\leq\frac{1}{2}\pi$, where $\omega$ is the limiting value of $\operatorname{ph}\left(t-t_{0}\right)$ as $t\to t_{0}$ from $b$. …