Index H

♦A♦B♦C♦D♦E♦F♦G♦H♦I♦J♦K♦L♦M♦N♦O♦P♦Q♦R♦S♦T♦U♦V♦W♦Z♦

Haar measure §35.4(i)
Hadamard’s inequality for determinants ¶ ‣ §1.3(i)
Hahn class orthogonal polynomials §18.19, ¶ ‣ §18.24
- asymptotic approximations §18.24, ¶ ‣ §18.24
- computation Ch.18
- definitions §18.19, ¶ ‣ §18.19
- difference equations on variable §18.22(ii)
- differences §18.22(iii)
- dualities §18.21(i)
- generating functions §18.23
- interrelations with other orthogonal polynomials §18.21, Figure 18.21.1, Figure 18.21.1
- leading coefficients Table 18.19.2
- limit relations §18.21(ii)
- normalizations Table 18.19.1
- notation ¶ ‣ §18.1(ii)
- orthogonality properties ¶ ‣ §18.19, ¶ ‣ §18.19, Table 18.19.1
- recurrence relations §18.22(i)
- relations to confluent hypergeometric functions and generalized hypergeometric functions ¶ ‣ §13.6(v), §18.20(ii)
- relations to hypergeometric function ¶ ‣ §15.9(i), §18.20(ii)
- Rodrigues formula §18.20(i)
- special cases §18.21(ii)
- weight functions Table 18.19.1
Hahn polynomials, see Hahn class orthogonal polynomials.
Hamiltonian systems
- chaotic
  - Lamé functions §29.19(i)
handle
- Riemann surface §21.7(i)
Hankel functions §10.1
- addition theorems ¶ ‣ §10.23(ii), §10.23(ii)
- analytic continuation §10.11
- approximations §10.76(ii)
- asymptotic expansions for large argument §10.17(i), §10.17(v)
  - error bounds §10.17(iii), §10.17(v)
  - exponentially-improved §10.17(v)
- asymptotic expansions for large order §10.19, §10.20(iii)
  - asymptotic forms §10.19(i)
  - double asymptotic properties §10.20(iii), §10.41(v)
  - transition region §10.19(iii)
  - uniform §10.20, §10.20(iii)
- branch conventions ¶ ‣ §10.2(ii)
- computation Ch.10, §10.74(v)
- connection formulas §10.4
- cross-product §10.5
- definitions ¶ ‣ §10.2(ii)
- derivatives §10.6(ii), §10.6(ii)
  - asymptotic expansions for large argument §10.17(ii)
  - asymptotic expansions for large order §10.19(i), §10.19(iii)
  - uniform asymptotic expansions for large order §10.20(i)
  - zeros ¶ ‣ §10.21(ix)
- differential equations §10.2(i), see also Bessel’s equation.
- graphics §10.3(ii)
- incomplete §10.46
- integral representations
  - along real line ¶ ‣ §10.9(i)
  - compendia §10.9(iv)
  - contour integrals ¶ ‣ §10.9(ii)
- integrals, see integrals of Bessel and Hankel functions.
- limiting forms ¶ ‣ §10.2(ii), §10.7
- multiplication theorem §10.23(i)
- notation §10.1
- power series §10.8
- principal branches (or values) ¶ ‣ §10.2(ii)
- recurrence relations §10.6(i)
- relations to other functions
  - Airy functions §9.6(i), §9.6(ii)
  - confluent hypergeometric functions ¶ ‣ §10.16
  - elementary functions ¶ ‣ §10.16
- Wronskians §10.5
- zeros ¶ ‣ §10.21(ix), §10.21(ix)
  - computation §10.74(vi)
  - tables ¶ ‣ §10.75(iii)
  - with respect to order ( $\nu$ -zeros) §10.21(xiv)
Hankel’s expansions
- for Bessel and Hankel functions §10.17(i), §10.17(i)
- for modified Bessel functions §10.40(i)
Hankel’s integrals
- Bessel functions and Hankel functions §10.9(iv)
Hankel’s inversion theorem
- Bessel functions §10.22(v)
Hankel’s loop integral
- gamma function ¶ ‣ §5.9(i)
Hankel transform §10.22(v)
- computation ¶ ‣ §10.74(vii)
harmonic analysis
- hypergeometric function §15.17(iii)
harmonic functions ¶ ‣ §1.9(ii)
- maximum modulus ¶ ‣ §1.10(v)
- mean value property ¶ ‣ §1.9(iii)
- Poisson integral ¶ ‣ §1.9(iii)
harmonic mean §1.2(iv), §1.7(iii)
harmonic oscillators
- Hermite polynomials §18.39(i)
- $q$ -hypergeometric function §17.17
harmonic trapping potentials
- parabolic cylinder functions §12.17
heat conduction in liquids
- Rayleigh function §10.73(v)
heat theory
- conical functions §14.31(ii)
Heaviside function §1.16(iv), §2.6(iii)
- derivative §1.16(iv)
Heine’s formula
- associated Legendre functions §14.28(ii)
Heine’s integral
- Legendre functions ¶ ‣ §14.12(ii)
Helmholtz equation
- $3j,6j,9j$ symbols §34.12
- associated Legendre functions §14.31(iii)
- Bessel functions and modified Bessel functions §10.73(i), §10.73(i)
- parabolic cylinder functions §12.17
- paraboloidal coordinates §28.32(ii)
Hermite–Darboux method
- Heun functions §31.8
Hermite polynomials §18.3, see also classical orthogonal polynomials.
- addition theorem ¶ ‣ §18.18(ii)
- applications
  - integrable systems ¶ ‣ §18.38(ii)
  - random matrix theory ¶ ‣ §18.38(ii)
  - Schrödinger equation §18.39(i)
- asymptotic approximations §18.15(v)
- computation Ch.18
- continued fractions ¶ ‣ §18.13
- definitions Table 18.3.1
- derivatives ¶ ‣ §18.9(iii)
- differential equations Table 18.8.1
- Dirac delta ¶ ‣ §1.17(iii)
- expansions in series of ¶ ‣ §18.18(iii), ¶ ‣ §18.18(iv), ¶ ‣ §18.18(v), ¶ ‣ §18.18(vii), ¶ ‣ §18.18(viii), ¶ ‣ §18.18(i), ¶ ‣ §18.18(ii)
- explicit representations §18.5, ¶ ‣ §18.5(iv)
- Fourier transforms ¶ ‣ §18.17(v)
- generating functions ¶ ‣ §18.12
- graphs Figure 18.4.7, Figure 18.4.7
- inequalities ¶ ‣ §18.14(iii), ¶ ‣ §18.14(i), ¶ ‣ §18.14(ii)
  - Turan-type ¶ ‣ §18.14(ii)
- integral representations ¶ ‣ §18.10(ii), ¶ ‣ §18.10(iv), Table 18.10.1
- integrals ¶ ‣ §18.17(vi), ¶ ‣ §18.17(viii), ¶ ‣ §18.17(i), ¶ ‣ §18.17(iii), §18.17(ix)
  - indefinite ¶ ‣ §18.17(i)
  - Nicholson-type ¶ ‣ §18.17(iii)
- interrelations with other orthogonal polynomials Figure 18.21.1, Figure 18.21.1, ¶ ‣ §18.21(ii), §18.7, ¶ ‣ §18.7(iii)
- Laplace transform ¶ ‣ §18.17(vi)
- leading coefficients Table 18.3.1
- limiting forms as trigonometric functions ¶ ‣ §18.11(ii)
- linearization formulas ¶ ‣ §18.18(v)
- local maxima and minima ¶ ‣ §18.14(iii)
- Mellin transform ¶ ‣ §18.17(vii)
- monic Figure 18.4.7, Figure 18.4.7, ¶ ‣ §3.5(v)
- multiplication theorem ¶ ‣ §18.18(iii)
- normalizations Table 18.3.1
- notation ¶ ‣ §18.1(ii)
- orthogonality properties Table 18.3.1
- Poisson kernels ¶ ‣ §18.18(vii)
- recurrence relations Table 18.9.1
- relations to other functions
  - confluent hypergeometric functions ¶ ‣ §13.18(v), ¶ ‣ §13.6(v), ¶ ‣ §18.11(i)
  - derivatives of the error function §7.10
  - generalized hypergeometric functions ¶ ‣ §18.5(iii)
  - parabolic cylinder functions §12.7(i)
  - repeated integrals of the complementary error function ¶ ‣ §7.18(iv)
- Rodrigues formula Table 18.5.1
- special values Table 18.6.1
- symmetry Table 18.6.1
- tables §18.41(i)
  - of coefficients §18.3
  - of zeros Table 3.5.10, Table 3.5.11, Table 3.5.12, Table 3.5.13
- upper bounds ¶ ‣ §18.14(i)
- zeros §18.16(v), §18.2(vi)
  - asymptotic behavior §18.16(v)
  - tables ¶ ‣ §3.5(v)
Hermitian matrices
- Gaussian unitary ensemble
  - limiting distribution of eigenvalues §32.14
Heun equation, see Heun’s equation.
Heun functions §31.1
- applications
  - mathematical §31.16, §31.16(ii)
  - physical §31.17
- asymptotic approximations §31.13
- computation §31.18
- definition §31.4
- differential equation, see Heun’s equation.
- expansions in series of
  - hypergeometric functions §31.11, §31.11(v)
  - orthogonal polynomials §31.11(iv)
- integral equations and representations §31.10, ¶ ‣ §31.10(ii)
- notation §31.1
- orthogonality
  - double §31.9(ii)
  - single §31.9(i)
- relations to hypergeometric function §31.7(i)
- relations to Lamé functions §31.7(ii)
Heun polynomials §31.5
- applications §31.16(ii)
- definitions §31.5
- integral equations and representations ¶ ‣ §31.10(i)
- notation §31.5
- orthogonality §31.9(i), §31.9(ii)
- products §31.16(ii)
Heun’s equation §31.2(i)
- accessory parameter §31.2(i)
  - asymptotic approximations §31.13
- applications ¶ ‣ §31.12
  - mathematical §31.16, §31.16(ii)
  - physical §31.17
- asymptotic approximations
  - eigenvalues of accessory parameters §31.13
  - solutions near irregular singularities §31.13
  - solutions of confluent forms §31.13
  - solutions with coalescing singularities §31.13
- automorphisms §31.2(v), §31.3(iii), §31.8
  - composite ¶ ‣ §31.2(v)
  - $F$ -homotopic transformations ¶ ‣ §31.2(v)
  - homographic transformations ¶ ‣ §31.2(v)
- basic solutions
  - equivalent expressions §31.3(iii)
  - Fuchs–Frobenius §31.3(i), §31.3(ii)
- biconfluent ¶ ‣ §31.12
- classification of parameters §31.2(i)
- computation of solutions §31.18
- confluent forms §31.12, ¶ ‣ §31.12
  - asymptotic approximations §31.13
  - integral equations ¶ ‣ §31.10(ii)
  - special cases ¶ ‣ §31.12
- doubly-confluent ¶ ‣ §31.12
- doubly-periodic forms
  - Jacobi’s elliptic ¶ ‣ §31.2(iv)
  - Weierstrass’s ¶ ‣ §31.2(iv)
- eigenvalues of accessory parameter §31.4
- expansions of solutions in series of
  - hypergeometric functions §31.11, §31.11(v)
  - orthogonal polynomials §31.11(iv)
- exponent parameters §31.2(i)
- integral equations §31.10, ¶ ‣ §31.10(ii)
- integral representation of solutions §31.10, ¶ ‣ §31.10(ii)
  - kernel functions ¶ ‣ §31.10(i), ¶ ‣ §31.10(ii)
  - separation constant ¶ ‣ §31.10(i)
- inversion problem §31.16(i)
- Jacobi’s elliptic form ¶ ‣ §31.2(iv)
- Liouvillean solutions §31.8
- monodromy group §31.16(i)
- normal form §31.2(ii)
- parameters
  - classification §31.2(i)
- path-multiplicative solutions §31.6
  - biorthogonality §31.9(ii)
  - expansions in series of hypergeometric functions §31.11(v)
- relation to Fuchsian equation ¶ ‣ §31.14(i)
- relation to Lamé’s equation §29.2(ii)
- separation of variables §31.17(i)
- singularities §31.2(i)
- singularity parameter §31.2(i)
- solutions analytic at three singularities, see Heun polynomials.
- solutions analytic at two singularities, see Heun functions.
- solutions via quadratures §31.8
- triconfluent ¶ ‣ §31.12
- trigonometric form §31.2(iii)
- uniformization problem §31.16(i)
- Weierstrass’s form ¶ ‣ §31.2(iv)

Heun’s operator §31.10(i)
hexadecimal number system §3.1(i)
higher-order $3nj$ symbols §34.11
high-frequency scattering
- parabolic cylinder functions §12.17
highway design
- Cornu’s spiral §7.21
Hilbert space
- interrelation between bases
  - Heun polynomial products §31.16(ii), §31.16(ii)
- $L^{2}_{{\rho}}(Q)$ orthonormal basis §31.15(iii)
Hilbert transform
- computation ¶ ‣ §3.5(ix)
- definition §1.14(v)
- Fourier transform of ¶ ‣ §1.14(v)
- inequalities ¶ ‣ §1.14(v)
- inversion ¶ ‣ §1.14(v)
Hill’s equation Ch.28, see also Whittaker–Hill equation.
- antiperiodic solutions §28.29(ii)
- basic solutions §28.29(ii)
- characteristic equation §28.29(ii)
- characteristic exponents §28.29(ii)
- definition §28.29(i)
- discriminant §28.29(iii)
- eigenfunctions §28.30(i)
- eigenvalues §28.29(iii)
- equation of Ince §28.31(ii)
  - Fourier-series solutions §28.31(ii)
  - polynomial solutions, see Ince polynomials.
- expansions in series of eigenfunctions §28.30
- Floquet solutions §28.29(ii)
- Floquet’s theorem §28.29(ii)
- periodic solutions §28.29(ii)
- pseudoperiodic solutions §28.29(ii)
- real case §28.29(iii)
- separation constants §28.32(i), §28.32(ii)
- symmetric case §28.29(ii)
Hölder’s inequalities for sums and integrals ¶ ‣ §1.7(i), ¶ ‣ §1.7(ii)
holomorphic function, see analytic function.
homogeneous harmonic polynomials §14.30(iv)
homographic transformation, see bilinear transformation.
Horner’s scheme for polynomials ¶ ‣ §1.11(i)
- extended ¶ ‣ §1.11(i)
Hurwitz criterion for stable polynomials ¶ ‣ §1.11(v)
Hurwitz system
- Riemann surface §21.10(ii)
Hurwitz zeta function §25.11
- analytic properties §25.11
- asymptotic expansions for large parameter §25.11(xii)
- computation §25.18(i)
- definition §25.11(i)
- derivatives §25.11(vi)
  - asymptotic expansions for large parameter §25.11(xii)
- graphics §25.11(ii)
- integral representations §25.11(vii)
- integrals §25.11(ix)
- relations to other functions
  - Lerch’s transcendent §25.14(i)
  - periodic zeta function §25.13
  - polylogarithms ¶ ‣ §25.12(ii)
  - Riemann zeta function §25.11(i)
- representations by Euler–Maclaurin formula §25.11(iii)
- series representations §25.11(x), §25.11(iv)
- special values §25.11(v)
- sums §25.11(xi)
- tables §25.19
hydrodynamics
- Jacobian elliptic functions §22.19(v)
hyperasymptotic expansions §2.11(v)
hyperbola
- elliptic integrals §19.30(ii)
hyperbolic cosecant function, see hyperbolic functions.
hyperbolic cosine function, see hyperbolic functions.
hyperbolic cotangent function, see hyperbolic functions.
hyperbolic functions Ch.4
- addition formulas §4.35(i)
- analytic properties ¶ ‣ §4.28
- approximations §4.47
- computation ¶ ‣ §4.45(i)
- conformal maps §4.29(ii)
- continued fractions §4.39
- definitions §4.28
- derivatives §4.34
- differential equations §4.34
- elementary properties §4.30
- graphics
  - complex argument §4.29(ii)
  - real argument §4.29(i)
- identities §4.35
- inequalities §4.32
- infinite products §4.36
- integrals
  - definite §4.40(iii)
  - indefinite §4.40(ii)
- inverse, see inverse hyperbolic functions.
- Laurent series §4.33
- limits §4.31
- Maclaurin series §4.33
- moduli §4.35(iv)
- multiples of argument §4.35(iii)
- notation §4.1
- partial fractions §4.36
- periodicity ¶ ‣ §4.28
- poles ¶ ‣ §4.28
- real and imaginary parts §4.35(iv)
- relations to trigonometric functions ¶ ‣ §4.28
- special values §4.31
- squares and products §4.35(ii)
- sums §4.41
- tables §4.46
- zeros ¶ ‣ §4.28
hyperbolic secant function, see hyperbolic functions.
hyperbolic sine function, see hyperbolic functions.
hyperbolic tangent function, see hyperbolic functions.
hyperbolic trigonometric functions, see hyperbolic functions.
hyperbolic umbilic bifurcation set
- formula ¶ ‣ §36.4(i)
- picture §36.4(ii)
hyperbolic umbilic canonical integral ¶ ‣ §36.2(i)
- asymptotic approximations §36.11, §36.12(iii)
- convergent series §36.8
- differential equations §36.10(iii)
- formulas for Stokes set §36.5(iii)
- integral identity §36.9
- picture of Stokes set Figure 36.5.9, Figure 36.5.9
- pictures of modulus Figure 36.3.10, Figure 36.3.10, Figure 36.3.11, Figure 36.3.11, Figure 36.3.12, Figure 36.3.12, Figure 36.3.9, Figure 36.3.9
- pictures of phase Figure 36.3.18, Figure 36.3.18, Figure 36.3.19, Figure 36.3.19, Figure 36.3.20, Figure 36.3.20, Figure 36.3.21, Figure 36.3.21
- scaling laws §36.6
- zeros §36.7(iv)
hyperbolic umbilic catastrophe ¶ ‣ §36.2(i), Figure 36.5.5, Figure 36.5.5, §36.5(iv)
hyperelliptic functions §22.19(iv)
hyperelliptic integrals §19.16(i)
hypergeometric differential equation §15.10(i)
- equivalent equation for contiguous functions §15.5(ii)
- fundamental solutions ¶ ‣ §15.10(i), §15.10(i)
- Kummer’s solutions §15.10(ii)
- singularities ¶ ‣ §15.10(i), ¶ ‣ §15.10(i), ¶ ‣ §15.10(i)
hypergeometric equation, see hypergeometric differential equation.
hypergeometric function §15.2(i), see also Gaussian hypergeometric function.
- analytic properties §15.2(ii)
- applications
  - mathematical §15.17
  - physical §15.18
- asymptotic approximations
  - large $a$ and $b$ §15.12(iii), §15.12(iii), §15.12(iii)
  - large $a$ , $b$ , and $c$ §15.12(iii)
  - large $a$ or $b$ §15.12(iii)
  - large $a$ (or $b$ ) and $c$ §15.12(iii), §15.12(iii)
  - large $c$ §15.12(ii), §15.12(iii), §15.12(iii)
  - large variable §15.12(i)
- branch points §15.2(ii)
- computation §15.19
- contiguous §15.5(ii)
- continued fractions §15.7
- definition §15.2(i)
- derivatives §15.5(i), §15.5(i)
- Fourier transforms §15.14
- graphics §15.3(i), §15.3(ii)
- Hankel transforms §15.14
- integral representations §15.6
  - Mellin–Barnes type §15.6
- integrals §13.16(i), §13.4(i), §13.4(ii), §15.14, §15.14
  - compendia §15.14, §15.14
- Laplace transforms §15.14
- Maclaurin series §15.2(i)
- Mellin transform §15.14
- multivariate §19.16(ii)
- notation §15.1
- Olver’s §15.1
- polynomial cases §15.2(ii)
- principal value (or branch) §15.2(i)
- products
  - series expansions §15.16
- recurrence relations §15.5(ii)
- relations to other functions
  - associated Legendre functions §14.3, §14.3(iii), §15.9(iv)
  - classical orthogonal polynomials §18.5(iii)
  - elementary functions §15.4(i)
  - Ferrers functions §14.3, §14.3(iii), §15.9(iv)
  - gamma function §15.4(ii), §15.4(iii)
  - Gegenbauer function §15.9(iii)
  - Hahn class orthogonal polynomials §18.20(ii)
  - Heun functions §31.7(i)
  - incomplete beta functions §8.17(ii)
  - Jacobi function §15.9(ii)
  - orthogonal polynomials ¶ ‣ §15.9(i), §15.9(i)
  - Painlevé transcendents §15.17(i)
  - Pollaczek polynomials §18.35(i)
  - psi function §15.4(iii)
  - symmetric elliptic integrals §19.25(vii)
  - Szegö–Askey polynomials ¶ ‣ §18.33(iv)
  - Wilson class orthogonal polynomials §18.26(iv)
- singularities §15.2(ii)
- special cases
  - argument a fraction §15.4(iii)
  - argument $\pm 1$ §15.4(ii), §15.4(iii)
  - arguments $e^{{\pm i\pi/3}}$ §15.19(i), §15.4(iii)
  - elementary functions §15.4(i), §15.4(i)
- sums §15.15
  - compendia §15.15
- transformation of variable
  - cubic ¶ ‣ §15.8(v), §15.8(v)
  - linear §15.19(iv), §15.8(i), §15.8(ii)
  - quadratic ¶ ‣ §15.8(iii), §15.8(iii)
- with two variables, see Appell functions.
- Wronskians ¶ ‣ §15.10(i), ¶ ‣ §15.10(i), ¶ ‣ §15.10(i)
- zeros §15.13
hypergeometric functions of matrix argument, see confluent hypergeometric functions of matrix argument, Gaussian hypergeometric functions of matrix argument, and generalized hypergeometric functions of matrix argument.
hypergeometric $R$ -function §19.16(ii)
- derivative §19.18(i)
- differential equation §19.18(ii)
- elliptic cases §19.16(iii)
- integral representations §19.16(ii)