critical phenomena

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11: Foreword

… ► The success of the original handbook, widely referred to as “Abramowitz and Stegun” (“A&S”), derived not only from the fact that it provided critically useful scientific data in a highly accessible format, but also because it served to standardize definitions and notations for special functions. …

12: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

… ►This reference includes exponentially-improved asymptotic expansions for $E_{a,b}\left(z\right)$ when $|z|\to \mathrm{\infty }$, together with a smooth interpretation of Stokes phenomena. …

13: 36.11 Leading-Order Asymptotics

… ►With real critical points (36.4.1) ordered so that …

14: Preface

… ►The validators played a critical role in the project, one that was absent in its 1964 counterpart: to provide critical, independent reviews during the development of each chapter, with attention to accuracy and appropriateness of subject coverage. …

15: 9.16 Physical Applications

… ►Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. …

16: 2.4 Contour Integrals

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§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method

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§2.4(vi) Other Coalescing Critical Points

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17: 25.15 Dirichlet $L$-functions

… ►There are also infinitely many zeros in the critical strip $0\le \mathrm{\Re }s\le 1$, located symmetrically about the critical line $\mathrm{\Re }s=\frac{1}{2}$, but not necessarily symmetrically about the real axis. …

18: 8.22 Mathematical Applications

… ►The function $\mathrm{\Gamma }(a,z)$, with $|\mathrm{ph}a|\le \frac{1}{2}\pi$ and $\mathrm{ph}z=\frac{1}{2}\pi$, has an intimate connection with the Riemann zeta function $\zeta \left(s\right)$ (§25.2(i)) on the critical line $\mathrm{\Re }s=\frac{1}{2}$. …

19: 10.72 Mathematical Applications

… ►In (10.72.1) assume $f(z)=f(z,\alpha )$ and $g(z)=g(z,\alpha )$ depend continuously on a real parameter $\alpha$, $f(z,\alpha )$ has a simple zero $z=z_0(\alpha )$ and a double pole $z=0$, except for a critical value $\alpha =a$, where $z_0(a)=0$. …

20: Bibliography O

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On higher-order Stokes phenomena of an inhomogeneous linear ordinary differential equation. J. Comput. Appl. Math. 169 (1), pp. 235–246.

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External Links:

ISSN 0377-0427, Document, MathReview (Jorge Mozo Fernández), ZentralBlatt

Referenced by:

§2.11(iv).

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