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## 11—20 of 69 matching pages

##### 11: 19.31 Probability Distributions

###### §19.31 Probability Distributions

► ${R}_{G}(x,y,z)$ and ${R}_{F}(x,y,z)$ occur as the expectation values, relative to a normal probability distribution in ${\mathbb{R}}^{2}$ or ${\mathbb{R}}^{3}$, of the square root or reciprocal square root of a quadratic form. …##### 12: 11.12 Physical Applications

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►Applications of Struve functions occur in water-wave and surface-wave problems (Hirata (1975) and Ahmadi and Widnall (1985)), unsteady aerodynamics (Shaw (1985) and Wehausen and Laitone (1960)), distribution of fluid pressure over a vibrating disk (McLachlan (1934)), resistive MHD instability theory (Paris and Sy (1983)), and optical diffraction (Levine and Schwinger (1948)).
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##### 13: 15.18 Physical Applications

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►More varied applications include photon scattering from atoms (Gavrila (1967)), energy distributions of particles in plasmas (Mace and Hellberg (1995)), conformal field theory of critical phenomena (Burkhardt and Xue (1991)), quantum chromo-dynamics (Atkinson and Johnson (1988)), and general parametrization of the effective potentials of interaction between atoms in diatomic molecules (Herrick and O’Connor (1998)).

##### 14: 25.17 Physical Applications

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►Analogies exist between the distribution of the zeros of $\zeta \left(s\right)$ on the critical line and of semiclassical quantum eigenvalues.
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##### 15: 13.22 Zeros

###### §13.22 Zeros

…##### 16: 25.10 Zeros

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###### §25.10(i) Distribution

… ►In the region $$, called the*critical strip*, $\zeta \left(s\right)$ has infinitely many zeros, distributed symmetrically about the real axis and about the*critical line*$\mathrm{\Re}s=\frac{1}{2}$. …##### 17: Foreword

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►That 1046-page tome proved to be an invaluable reference for the many scientists and engineers who use the special functions of applied mathematics in their day-to-day work, so much so that it became the most widely distributed and most highly cited NIST publication in the first 100 years of the institution’s existence.
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##### 18: 7.20 Mathematical Applications

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►The normal distribution function with mean $m$ and standard deviation $\sigma $ is given by
…For applications in statistics and probability theory, also for the role of the normal distribution functions (the error functions and probability integrals) in the asymptotics of arbitrary probability density functions, see Johnson et al. (1994, Chapter 13) and Patel and Read (1982, Chapters 2 and 3).

##### 19: Notices

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►Limited copying and internal distribution of the content of these pages is permitted for research and teaching.
Reproduction, copying, or distribution for any commercial purpose is strictly prohibited.
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##### 20: 10.42 Zeros

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►The distribution of the zeros of ${K}_{n}\left(nz\right)$ in the sector $-\frac{3}{2}\pi \le \mathrm{ph}z\le \frac{1}{2}\pi $ in the cases $n=1,5,10$ is obtained on rotating Figures 10.21.2, 10.21.4, 10.21.6, respectively, through an angle $-\frac{1}{2}\pi $ so that in each case the cut lies along the positive imaginary axis.
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