Index H

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, Figure 18.21.1
- leading coefficients Table 18.19.2
- limit relations §18.21(ii)
- 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)
- relations to hypergeometric function §15.9(i)
- relations to other functions
  - generalized hypergeometric functions §18.20(ii)
  - hypergeometric function §18.20(ii)
  - $3j$ symbols §18.38(iii)
- Rodrigues formula §18.20(i)
- special cases §18.21(ii)
- standardizations Table 18.19.1
- weight functions Table 18.19.1
- which are also Sheffer polynomials §18.2(xii)
Hahn polynomials, see Hahn class orthogonal polynomials.
Hamiltonian systems
- chaotic
  - Lamé functions §29.19(i)
handle
- Riemann surface §21.7(i)
Hankel determinant §18.2(ix)
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)—§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 2nd item
  - with respect to order ( $\nu$ -zeros) §10.21(xiv)
Hankel transform §10.22(v)
- Bessel eigenfunction expansion §1.18(vi)
- computation §10.74(vii)
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)
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 oscillator §18.39(i)
- Hermite polynomials §18.39(i)
harmonic oscillators
- $q$ -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
- $\mathit{3j},\mathit{6j},\mathit{9j}$ 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)
Herglotz generating function §14.30(ii)
Hermite differential operator
- eigenfunction expansion §1.18(v)
Hermite EOP’s §18.36(vi)
- solution of a rational Schrödinger equation §18.39(i)
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)
- as Sheffer polynomials §18.2(xii)
- 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(ii), §18.18(iii), §18.18(iv), §18.18(v), §18.18(vii), §18.18(viii), §18.18(i)
- 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, Figure 18.4.7
- harmonic oscillator §18.39(i)
- inequalities §18.14(iii), §18.14(i), §18.14(ii)
  - Turán-type §18.14(ii)
- integral representations §18.10(ii), §18.10(iv), Table 18.10.1
- integrals §18.17(ix), §18.17(vi), §18.17(viii), §18.17(i), §18.17(iii)
  - indefinite §18.17(i)
  - Nicholson-type §18.17(iii)
- interrelations with other orthogonal polynomials Figure 18.21.1, 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)
- Mehler formula §18.18(vii)
- Mellin transform §18.17(vii)
- monic Figure 18.4.7, Figure 18.4.7, Figure 18.4.7, §3.5(v)
- multiplication theorem §18.18(iii)
- notation §18.1(ii)
- orthogonality properties Table 18.3.1
- Poisson kernels §18.18(vii)
- recurrence relations Table 18.9.1, Table 18.9.2
- 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
- standardizations Table 18.3.1
- 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)
Hermite–Darboux method
- Heun functions §31.8
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)
high-frequency scattering
- parabolic cylinder functions §12.17
higher-order $3nj$ symbols §34.11
highway design
- Cornu’s spiral §7.21
Hilbert space §1.18(i)
- expansion of arbitrary function §1.18(ii)
- expansion of arbitrary vector §1.18(i)
- formally self-adjoint differential operators §1.18(iii)
- formally self-adjoint operators §1.18(iii)
- interrelation between bases
  - Heun polynomial products §31.16(ii)—§31.16(ii)
- $L^{2}_{\rho}(Q)$ orthonormal basis §31.15(iii)
- $\ell^{2}$ space §1.18(i)
- linear operators §1.18(iii)
- self-adjoint differential operators §1.18(iii)
- Stieltjes measure §1.18(ii)
- symmetric operators §1.18(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)
Hille–Hardy formula §18.18(vii)
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 1st item
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)—§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(iv), §25.11(x)
- special values §25.11(v)
- sums §25.11(xi)
- tables 3rd item
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)—§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, Figure 36.5.9
- pictures of modulus Figure 36.3.10, Figure 36.3.10, Figure 36.3.10, Figure 36.3.11, Figure 36.3.11, Figure 36.3.11, Figure 36.3.12, Figure 36.3.12, Figure 36.3.12, Figure 36.3.9, Figure 36.3.9, Figure 36.3.9, Erratum (V1.0.14) for Figures 36.3.9, 36.3.10, 36.3.11, 36.3.12
- 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), §36.5(iv), §36.5(iv)—§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 $a$ (or $b$ ) and $c$ §15.12(iii), §15.12(iii)
  - large $a$ and $b$ §15.12(iii), §15.12(iii)
  - large $a$ or $b$ §15.12(iii)
  - large $a$ , $b$ , and $c$ §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—§15.6
  - Mellin–Barnes type §15.6—§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)
  - psi function §15.4(iii)
  - symmetric elliptic integrals §19.25(vii)
  - Szeő–Askey polynomials §18.33(iv)
  - type 2 Pollaczek polynomials §18.35(i)
  - Wilson class orthogonal polynomials §18.26(iv)
- singularities §15.2(ii)
- special cases
  - argument $\pm 1$ §15.4(ii), §15.4(iii)
  - argument a fraction §15.4(iii)
  - arguments ${\mathrm{e}}^{\pm\mathrm{i}\pi/3}$ §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)—§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)

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