normal forms
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21: 33.5 Limiting Forms for Small , Small , or Large
§33.5 Limiting Forms for Small , Small , or Large
►§33.5(i) Small
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33.5.6
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§33.5(iii) Small
… ►§33.5(iv) Large
…22: 28.14 Fourier Series
23: 19.31 Probability Distributions
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and occur as the expectation values, relative to a normal probability distribution in or , of the square root or reciprocal square root of a quadratic form.
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24: 28.12 Definitions and Basic Properties
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►As in §28.7 values of for which (28.2.16) has simple roots are called normal values with respect to .
For real values of and all the are real, and is normal.
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►If is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of and by the normalization
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25: 33.10 Limiting Forms for Large or Large
§33.10 Limiting Forms for Large or Large
►§33.10(i) Large
… ►§33.10(ii) Large Positive
… ►§33.10(iii) Large Negative
…26: 1.6 Vectors and Vector-Valued Functions
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►Suppose is a piecewise smooth surface which forms the complete boundary of a bounded closed point set , and is oriented by its normal being outwards from .
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27: 31.11 Expansions in Series of Hypergeometric Functions
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►Series of Type II (§31.11(iv)) are expansions in orthogonal polynomials, which are useful in calculations of normalization integrals for Heun functions; see Erdélyi (1944) and §31.9(i).
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§31.11(ii) General Form
…28: 30.16 Methods of Computation
29: 7.20 Mathematical Applications
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►The normal distribution function with mean and standard deviation 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).
30: Bibliography C
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Normal elliptic integrals of the first and second kinds.
Duke Math. J. 31 (3), pp. 405–419.
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Intégrandes à deux formes quadratiques.
C. R. Acad. Sci. Paris Sér. A–B 274 (15 May, 1972, Sér. A), pp. 1458–1461 (French).
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Asymptotics and closed form of a generalized incomplete gamma function.
J. Comput. Appl. Math. 67 (2), pp. 371–379.
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Modular Forms and Fermat’s Last Theorem.
Springer-Verlag, New York.
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Algorithm AS 24: From normal integral to deviate.
Appl. Statist. 18 (3), pp. 290–293.
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