# limit forms as ℑτ→0+

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## 4 matching pages

##### 1: 20.13 Physical Applications

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►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).
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##### 2: 2.6 Distributional Methods

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►This leads to integrals of the form
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►The distribution method outlined here can be extended readily to functions $f(t)$ having an asymptotic expansion of the form
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►To define convolutions of distributions, we first introduce the space ${K}^{+}$ of all distributions of the form
${\mathit{D}}^{n}f$, where $n$ is a nonnegative integer, $f$ is a locally integrable function on $\mathbb{R}$ which vanishes on $(-\mathrm{\infty},0]$, and ${\mathit{D}}^{n}f$ denotes the $n$th derivative of the distribution associated with $f$.
…It is easily seen that ${K}^{+}$
forms a commutative, associative linear algebra.
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►On inserting this identity into (2.6.54), we immediately encounter divergent integrals of the form
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##### 3: 3.7 Ordinary Differential Equations

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►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 $\mathbf{A}(\tau ,z)$ is the matrix … ►The remaining two equations are supplied by boundary conditions of the form … ►with limits taken in (3.7.16) when $a$ or $b$, or both, are infinite. …##### 4: 2.4 Contour Integrals

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►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).
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►If this integral converges uniformly at each limit for all sufficiently large $t$, then by the Riemann–Lebesgue lemma (§1.8(i))
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►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 $b$.
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