Index G

gamma distribution
- incomplete gamma functions §8.23
gamma function Ch.5, see also incomplete gamma functions.
- analytic properties §5.2(i)
- applications
  - mathematical §5.19
  - physical §5.20—§5.20
- approximations
  - Chebyshev series §5.23(ii)
  - complex variables §5.23(iii)
  - rational §5.23(i), §5.23(iii)
- asymptotic expansions §5.11—§5.11(iii)
  - error bounds §5.11(ii)
  - exponentially-improved §5.11(ii)
  - for ratios §5.11(iii)
- Bohr-Mollerup theorem §5.5(iv)
- computation §5.21
- continued fraction §5.10
- definition §5.2(i)
- duplication formula §5.5(iii)
- Euler’s integral §5.2(i)
- extrema
  - asymptotic approximation §5.4(iii)
  - table of §5.4(iii)
- Gauss’s multiplication formula §5.5(iii)
- graphics §5.3—§5.3(ii)
- inequalities §5.6
- infinite products §5.8
- integral representations §5.13—§5.14, §5.9—§5.9(i), §8.21(iii)
  - for derivatives §5.9(ii)
  - multidimensional §5.14—§5.14
- logarithm
  - continued fraction §5.10
  - convexity §5.3(i), §5.5(iv)
  - graphics §5.3(i)
  - integral representations §5.9(i)
  - Taylor series §5.7(i)
- maxima and minima §5.6
- multiplication formulas §5.5(iii)
- multivariate, see multivariate gamma function.
- notation §5.1
- reciprocal
  - analytic properties §5.2(i)
  - graphics §5.3(i), §5.3(ii)
  - Maclaurin series §5.7(i)
  - zeros §5.2(i)
- recurrence relation §5.5(i)
- reflection formula §5.5(ii)
- relations to hypergeometric function §15.4(ii), §15.4(iii)
- scaled §8.18(ii)
- special values §5.4
- tables §5.22(ii), §5.22(iii)
Gaunt coefficient
- $\mathit{3j}$ symbol §34.3(vii)
Gaunt’s integral
- $\mathit{3j}$ symbol §34.3(vii)
Gauss quadrature §3.5(v), §3.5(v)—§3.5(viii)
- Christoffel coefficients (or numbers) §3.5(v)
- comparison with Clenshaw–Curtis formula §3.5(iv)
- eigenvalue/eigenvector characterization §3.5(vi)
- for contour integrals §3.5(viii)
- Gauss–Chebyshev formula §3.5(v)—§3.5(v)
- Gauss–Hermite formula §3.5(v)—§3.5(v)
- Gauss–Jacobi formula §3.5(v)
- Gauss–Laguerre formula §3.5(v)—§3.5(v)
  - generalized §3.5(v)
- Gauss–Legendre formula §3.5(v)—§3.5(v)
- logarithmic weight function §3.5(v)—§3.5(v)
- nodes §3.5(v)
  - tables §3.5(v)—§3.5(viii)
- remainder terms §3.5(v)
- weight functions §3.5(v)
  - tables §3.5(v)—§3.5(viii)
Gauss series
- hypergeometric function §15.2(i)
  - convergence §15.2(i)
Gauss sums
- number theory
  - Dirichlet character §27.10
  - separable §27.10
- theta functions §20.11(i)
Gauss–Christoffel quadrature, see Gauss quadrature.
Gauss’s ${{}_{2}F_{1}}(-1)$ sum
- $q$ -analog §17.7(i)
Gauss’s theorem for vector-valued functions §1.6(v)
Gaussian
- nonperiodic §20.13
Gaussian elimination §3.2(i)—§3.2(ii)
- back substitution §3.2(i)
- forward elimination §3.2(i)
- iterative refinement §3.2(i)
- multipliers §3.2(i), §3.2(i)
- partial pivoting §3.2(i)
- pivot (or pivot element) §3.2(i)
- residual vector §3.2(i)
- triangular decomposition §3.2(i)
- tridiagonal systems §3.2(ii)
Gaussian hypergeometric function, see also hypergeometric function.
- of matrix argument §35.7
  - applications §35.9
  - asymptotic approximations §35.7(iv)
  - basic properties §35.7(ii)
  - case $m=2$ §35.7(ii)
  - computation §35.10
  - confluent form §35.7(ii)
  - definition §35.7(i)
  - Gauss formula §35.7(ii)
  - integral representation §35.7(ii)
  - Jacobi form §35.7(i)
  - notation §35.1
  - partial differential equations §35.7(iii)—§35.7(iii)
  - reflection formula §35.7(ii)
  - transformations of parameters §35.7(ii)
Gaussian noise
- Lambert $W$ -function §4.44
Gaussian polynomials
- definition §26.9(ii)
- tables §26.21
Gaussian probability functions §7.1
Gaussian unitary ensemble
- Painlevé transcendents §32.14
Gegenbauer function
- definition §15.9(iii)
- relation to associated Legendre functions §14.3(iv)
- relation to hypergeometric function §15.9(iii)
Gegenbauer polynomials, see ultraspherical polynomials and also classical orthogonal polynomials.
Gegenbauer’s addition theorem
- Bessel functions §10.23(ii)
- modified Bessel functions §10.44(ii)
general elliptic integrals §19.2(i)
- reduction to basic elliptic integrals §19.29(ii)
- reduction to Legendre’s elliptic integrals §19.14—§19.14(ii)
- reduction to symmetric elliptic integrals §19.29(i)—§19.29(iii)
general orthogonal polynomials Ch.18
- complete §18.2(viii)
- computation Ch.18
- continued fractions §18.2(x)
- difference operators §18.2(ii)
- discriminant §18.2(vi)
- Favard’s theorem §18.2(viii)
- inequality §18.2(xii)
- moments §18.2(ix)
- monic §18.2(iii)
- monotonic weight function §18.2(xii)
- Nevai–Blumenthal class §18.2(xi)
- on countable point sets §18.2(i)
- on general sets §18.2(i)
- on intervals §18.2(i)
- orthonormal §18.2(iii)
- Poisson kernel §18.2(xii)
- quadratic transformations §18.2(vii)
- recurrence relations §18.2(iv)
- sums of products §18.2(v)
- Szegő class §18.2(xi)
- weight functions §18.2(i)
- $x$ -difference operators §18.2(ii)
- zeros §18.2(vi)
generalizations of elliptic integrals §19.35(i)
generalized Airy functions
- from differential equation §9.13—§9.13(i)
  - asymptotic approximations §9.13(i)
  - definitions §9.13(i)
  - relation to Bessel functions §9.13(i)
  - relation to modified Bessel functions §9.13(i)
  - tables §9.18(vii)
- from integral representations §9.13(ii)—§9.13(ii)
  - connection formulas §9.13(ii)
  - definitions §9.13(ii)
  - difference equation §9.13(ii)
  - differential equation §9.13(ii)
generalized exponential integral §8.19
- analytic continuation §8.19(viii)
- applications
  - mathematical §8.22
  - physical §8.24
- approximations §8.27(ii)
- asymptotic expansions
  - exponentially-improved §8.20(i)
  - large parameter §8.20(ii)
  - large variable §8.20(i)
- Chebyshev-series expansions 1st item
- computation Ch.8
- continued fraction §8.19(vii)
- definition §8.19(i)
- derivatives §8.19(v)
- further generalizations §8.19(xi)
- graphics §8.19(ii)—§8.19(ii)
- inequalities §8.19(ix)
- integral representations §8.19(i)
  - Mellin–Barnes type §8.19(i)
- integrals §8.19(x)
- notation §8.1
- of large argument §2.11(ii)
- principal values §8.19(i)
- recurrence relation §8.19(v)
- relations to other functions
  - confluent hypergeometric functions §8.19(vi)
  - error functions §7.11
  - incomplete gamma functions §8.19(i)
- series expansions §8.19(iv)
- special values §8.19(iii)
- tables §8.26(iv)

generalized exponentials §4.12
generalized function
- definition §1.16(i)
generalized functions
- distributions §2.6(iv)
generalized hypergeometric differential equation §16.8(ii)
- confluence of singularities §16.8(iii)
- connection formula §16.8(ii)
- fundamental solutions §16.8(ii)
- singularities §16.8(i)
generalized hypergeometric function ${{}_{0}F_{2}}$
- definition §16.2(i), §16.5
- of large argument §2.10(iii)
generalized hypergeometric functions §16.2(i)
- analytic continuation §16.5
- analytic properties §16.2(ii), §16.2(iii), §16.2(iv), §16.2(v), §16.5
- applications
  - mathematical §16.23
  - physical §16.24
- approximations §16.26
- argument unity §16.4
- as functions of parameters §16.2(v)
- asymptotic expansions
  - formal series §16.11(i)
  - large parameters §16.11(iii)
  - large variable §16.11(ii)
  - small variable §16.5
- balanced §16.4(i)
- bilateral series §16.4(v)
  - Dougall’s bilateral sum §16.4(v)
- computation §16.25
- contiguous balanced series §16.4(iii)
- contiguous functions §16.3(ii)
- contiguous relations §16.4(iii)
- continued fractions §16.4(iv)
- definitions §16.2(i), §16.5
- derivatives §16.3(i)
- differential equation, see generalized hypergeometric differential equation.
- Dixon’s well-poised sum §16.4(ii)
- Dougall’s bilateral sum §16.4(v)
- Dougall’s very well-poised sum §16.4(ii)
- Džrbasjan’s sum §16.4(ii)
- expansions in series of §16.10
- extensions of Kummer’s relations §16.4(iii)
- identities §16.4(iii)—§16.4(iii)
- integral representations §16.5
- integrals §16.5
  - inverse Laplace transform §16.5
  - Laplace transform §16.5
- $k$ -balanced §16.4(i)
- Kummer-type transformations §16.4(iii), §16.6
- monodromy §16.23(i)
- notation §16.2(i)
- of matrix argument §35.8
  - applications §35.9
  - computation §35.10
  - confluence §35.8(iv)
  - convergence properties §35.8(i)
  - definition §35.8(i)
  - Euler integral §35.8(iv)
  - expansion in zonal polynomials §35.8(i)
  - general properties §35.8(iv)
  - invariance §35.8(iv)
  - Kummer transformation §35.8(iii)
  - Laplace transform §35.8(iv)
  - Mellin–Barnes integrals §35.8(v)
  - notation §35.1
  - Pfaff–Saalschütz formula §35.8(iii)
  - relations to other functions §35.8(ii)
  - Thomae transformation §35.8(iii)
  - ${{}_{3}F_{2}}$ case §35.8(iii)
  - value at $\mathbf{T}=\boldsymbol{{0}}$ §35.8(iv)
- Pfaff–Saalschütz balanced sum §16.4(ii)
- polynomial cases §16.2(iv)
- principal branch (value) §16.2(iii)
- products §16.12
- recurrence relations §16.4(iii)
- relations to other functions
  - associated Jacobi polynomials §18.30(i)
  - Bessel functions §10.16
  - classical orthogonal polynomials §18.5(iii)
  - Fresnel integrals §7.11
  - generalized Bessel polynomials §18.34(i)
  - Hahn class orthogonal polynomials §18.20(ii)
  - Jacobi polynomials §18.38(ii)
  - Kummer functions §13.6(vi)
  - Meijer $G$ -function §16.18
  - modified Bessel functions §10.39
  - orthogonal polynomials and other functions §16.7
  - $\mathit{3j},\mathit{6j},\mathit{9j}$ symbols §16.24(iii), §16.4(iii), §16.4(iii)
  - Wilson class orthogonal polynomials §18.26(i)
- Rogers–Dougall very well-poised sum §16.4(ii)
- Saalschützian §16.4(i)
- terminating §16.2(iii), §16.2(iv)
- transformation of variable §16.6
  - cubic §16.6
  - quadratic §16.6
- very well-poised §16.4(i)
- Watson’s sum §16.4(ii)
- well-poised §16.4(i)
- Whipple’s sum §16.4(ii)
- Whipple’s transformation §16.4(iii)
- with two variables Ch.16—§16.16(ii)
- zeros §16.9
generalized hypergeometric series §16.2(i)
generalized integrals
- asymptotic expansions §2.6(i)
generalized logarithms §3.1(iv), §4.12
- applications §4.44
generalized precision §3.1(iv)
generalized sine and cosine integrals §8.21
- analytic properties §8.21(i)
- asymptotic expansions for large variable §8.21(viii)
- auxiliary functions §8.21(vii)
  - asymptotic expansions for large variable §8.21(viii)
  - integral representations §8.21(vii)
- computation Ch.8
- definitions
  - general values §8.21(i)
  - principal values §8.21(ii)
- expansions in series of spherical Bessel functions §8.21(vi)
- integral representations §8.21(iii)
- interrelations §8.21(iv)
- notation §8.1
- power-series expansions §8.21(vi)
- relation to sine and cosine integrals §8.21(v)
- special values §8.21(v)
generating function
- definition §1.10(xi)
Genocchi numbers §24.15(i)
- table Table 24.15.1
genus
- Riemann surface §21.7(i)
geometric mean §1.2(iv), §1.7(iii)
geometric progression (or series) §1.2(ii)
geophysics
- spherical harmonics §14.30(iv)
Gibbs phenomenon
- sine integral §6.16(i)
Glaisher’s constant §2.10(i), §5.17
Glaisher’s notation
- Jacobian elliptic functions §22.1
Goldbach conjecture
- number theory §27.13(ii)
Goodwin–Staton integral
- asymptotic expansion §7.12(iii)
- computation §7.22(ii)
- definition §7.2(v)
- relations to Dawson’s integral and exponential integral §7.5
Graf’s addition theorem
- Bessel functions §10.23(ii)
- modified Bessel functions §10.44(ii)
Gram–Schmidt procedure
- for least squares approximation §3.11(v)
graph theory
- combinatorics §26.19
gravitational radiation
- Coulomb functions 6th item
Green’s theorem for vector-valued functions
- three dimensions §1.6(v)
- two dimensions §1.6(iv)
group representations
- orthogonal polynomials §18.38(iii)
group theory
- hypergeometric function §15.17(v)
Gudermannian function §4.23(viii)
- inverse §4.23(viii)
- relation to amplitude ( $\operatorname{am}$ ) function §22.16(i)
- relation to $R_{C}$ -function §19.10(ii)
- tables §4.46

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Index G

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