# Stokes line

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## 4 matching pages

##### 1: 2.11 Remainder Terms; Stokes Phenomenon
The choice of $\alpha^{\prime}$ and $\beta^{\prime}$ is facilitated by a knowledge of the relevant Stokes lines; see §2.11(iv) below. … Rays (or curves) on which one contribution in a compound asymptotic expansion achieves maximum dominance over another are called Stokes lines ($\theta=\pi$ in the present example). … The relevant Stokes lines are $\operatorname{ph}z=\pm\pi$ for $w_{1}(z)$, and $\operatorname{ph}z=0,2\pi$ for $w_{2}(z)$. … For example, extrapolated values may converge to an accurate value on one side of a Stokes line2.11(iv)), and converge to a quite inaccurate value on the other.
##### 3: 8.22 Mathematical Applications
8.22.1 $F_{p}\left(z\right)=\frac{\Gamma\left(p\right)}{2\pi}z^{1-p}E_{p}\left(z\right% )=\frac{\Gamma\left(p\right)}{2\pi}\Gamma\left(1-p,z\right),$
plays a fundamental role in re-expansions of remainder terms in asymptotic expansions, including exponentially-improved expansions and a smooth interpretation of the Stokes phenomenon. … The function $\Gamma\left(a,z\right)$, with $|\operatorname{ph}a|\leq\tfrac{1}{2}\pi$ and $\operatorname{ph}z=\tfrac{1}{2}\pi$, has an intimate connection with the Riemann zeta function $\zeta\left(s\right)$25.2(i)) on the critical line $\Re s=\tfrac{1}{2}$. …
##### 4: Bibliography D
• T. M. Dunster (1990a) Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.
• T. M. Dunster (1996b) Asymptotics of the generalized exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes discontinuities. Proc. Roy. Soc. London Ser. A 452, pp. 1351–1367.