hypergeometric functions

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11: 16.26 Approximations

§16.26 Approximations

►For discussions of the approximation of generalized hypergeometric functions and the Meijer $G$-function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).

12: 17.17 Physical Applications

§17.17 Physical Applications

… ►See Kassel (1995). …

13: 17.15 Generalizations

§17.15 Generalizations

…

14: 17.18 Methods of Computation

§17.18 Methods of Computation

… ►The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. …

15: 16.24 Physical Applications

§16.24 Physical Applications

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§16.24(i) Random Walks

►Generalized hypergeometric functions and Appell functions appear in the evaluation of the so-called Watson integrals which characterize the simplest possible lattice walks. … ►

§16.24(iii) $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ Symbols

… ►They can be expressed as ${}_3{}^{}F_2^{}$ functions with unit argument. …

16: 16.25 Methods of Computation

§16.25 Methods of Computation

… ►They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19. …

17: 35.10 Methods of Computation

§35.10 Methods of Computation

… ►See Yan (1992) for the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $𝐎(m)$ applied to a generalization of the integral (35.5.8). …

18: 15 Hypergeometric Function

Chapter 15 Hypergeometric Function

…

19: Adri B. Olde Daalhuis

… ►

13 Confluent Hypergeometric Functions

►

15 Hypergeometric Function

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16 Generalized Hypergeometric Functions & Meijer G-Function

20: 6.11 Relations to Other Functions

… ►

Confluent Hypergeometric Function

►

6.11.2 $$E_1\left(z\right)={\mathrm{e}}^{-z}U(1,1,z),$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $z$: complex variable

Referenced by:

§6.11, 5th item, §6.6

Permalink:

http://dlmf.nist.gov/6.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

►

6.11.3 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)=U(1,1,-\mathrm{i}z).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.11, 5th item

Permalink:

http://dlmf.nist.gov/6.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6