Index D

Darboux’s method
- asymptotic approximations of sums and sequences §2.10(iv)—§2.10(iv)
Dawson’s integral §7.2(ii)
- applications §7.21
- approximations 3rd item, §7.24(ii)
- computation §7.22(i)
- definition §7.2(ii)
- generalized §7.16
- graphics Figure 7.3.2, Figure 7.3.2, Figure 7.3.2
- integral representation §7.7(i)
- notation §7.1
- relation to error functions §7.5
- relation to parabolic cylinder functions §12.7(ii)
- tables §7.23(ii)
de Branges–Wilson beta integral §5.13
De Moivre’s theorem
- trigonometric functions §4.21(iii)
Debye functions §8.22(ii)
Dedekind modular function §27.14(iv)
- functional equation §27.14(iv)
Dedekind sums
- number theory §27.14(iii)
Dedekind’s eta function, see modular functions.
Dedekind’s modular function, see modular functions.
deficiency indices
- self-adjoint extensions of a symmetric operator §1.18(ix)
del operator §1.6(iii)
Delannoy numbers
- definition §26.6(i)
- generating functions §26.6(ii)
- recurrence relation §26.6(iii)
- relation to lattice paths §26.6(i)
- table Table 26.6.1
delta sequence §1.17(i)
delta wing equation
- Lamé polynomials §29.19(ii)
denominator polynomials §18.2(x), §18.30(vi)
- Markov’s theorem §18.30(vi)
derivatives
- chain rule §1.4(iii), §1.5(i)
- definition §1.4(iii), §1.5(i)
- distributional §1.16(ii)
- Faà di Bruno’s formula §1.4(iii)
- Jacobian §1.5(vi)
- L’Hôpital’s rule §1.4(iii)
- left-hand §1.8(ii)
- Leibniz’s formula §1.4(iii)
- mean value theorem §1.4(iii)
- notation §1.4(iii), §1.5(i)
- of distribution §1.16(ii)
- partial §1.5(i)
- right-hand §1.8(ii)
Descartes’ rule of signs (for polynomials) §1.11(ii)
determinants
- alternants §1.3(ii)
- Cauchy §1.3(ii)
- circulant §1.3(ii)
- cofactor §1.3(i)
- definition §1.3(i)
- Hadamard’s inequality §1.3(i)
- Hankel §24.14(ii), §3.9(iv)
- inequalities §1.3(i)
- infinite
  - convergence §1.3(iii)
  - Hill’s type §1.3(iii)
- Krattenthaler’s formula §1.3(ii)
- minor §1.3(i)
- notation §1.3(i)
- persymmetric §24.14(ii)
- properties §1.3(i)
- Vandermonde §1.3(ii)
diatomic molecules
- hypergeometric function §15.18
difference equations
- asymptotic solutions, see asymptotic solutions of difference equations.
- distinguished solutions §3.6(ii)
- minimal solutions §3.6(ii)
- numerical solution §3.6—§3.6(vii)
  - backward recursion method §3.6(ii), §3.6(iv)
  - boundary-value methods §3.6(iv), §3.6(vii)
  - homogeneous equations §3.6(ii)—§3.6(iii)
  - inhomogeneous equations §3.6(iv)
  - normalizing factor §3.6(iii)
  - stability §3.6(ii)
- recessive solutions §3.6(ii)
difference operators §18.1(i)
- backward §18.1(i)
- central in imaginary direction §18.1(i)
- forward §18.1(i)
differentiable functions §1.4(iii), §1.9(ii)
differential equations
- asymptotic solutions, see asymptotic solutions of differential equations.
- change of variables
  - elimination of first derivative §1.13(iv)—§1.13(iv)
  - Liouville transformation §1.13(iv)
  - point at infinity §1.13(iv)
- classification of singularities §16.8(i), §2.7(i)
- closed-form solutions §1.13(vii)
- dominant solutions §2.7(iii)
- Fuchs–Frobenius theory §2.7(i)
- homogeneous §1.13(iii), §3.7(i)
- indices differing by an integer §2.7(i)
- indicial equation §2.7(i)
- inhomogeneous §1.13(iii), §3.7(i)
  - solution by variation of parameters §1.13(iii)
- irregular singularity §2.7(ii)
- nonhomogeneous, see inhomogeneous.
- numerical solution
  - boundary-value problems §3.7(iii)
  - eigenfunctions §3.7(iv)
  - eigenvalues §3.7(iv)
  - initial-value problems §3.7(ii)
  - Runge–Kutta method §3.7(v), §3.7(v)—§3.7(v)
  - stability §3.7(ii)
  - Sturm–Liouville eigenvalue problems §3.7(iv)
  - Taylor-series methods §3.7(ii)—§3.7(iii)
- numerically satisfactory solutions §2.7(iv)—§2.7(iv)
- of arbitrary order §16.8, §16.8(i)
- ordinary point §16.8(i), §2.7(i)
- rank of singularity §2.7(ii)
- recessive solutions §2.7(iii)
- regular singularity §2.7(i)
- Schwarzian derivative §1.13(iv)
- solutions
  - existence §1.13(i)
  - fundamental pair §1.13(i)
  - in series of Chebyshev polynomials §18.40(i)
  - linearly independent §1.13(i)
  - products §1.13(v)
  - pseudo spectral methods §18.38(i)
  - spectral solutions §18.38(i)
  - Wronskian §1.13(i)
- subdominant solutions, see recessive solutions.
- with a parameter §1.13(ii)
differential operators
- eigenfunction expansions §1.18
- linear
  - second order §1.18
differentiation
- Cauchy–Riemann equations §1.9(ii)
- numerical
  - analytic functions §3.4(ii)
  - Lagrange’s formula for equally-spaced nodes §3.4(i)—§3.4(i)
  - partial derivatives §3.4(iii)—§3.4(iii)
- of integrals §1.10(viii), §1.5(iv)—§1.5(iv)
- partial §1.5(i)
differentiation formulas
- degree lowering and raising §18.2(xii)
  - classical orthogonal polynomials item 2., §18.9(iii)
diffraction catastrophes §36.12(i), §36.2(i)
- notation §36.1
- scaling laws §36.6

diffraction of light
- Fresnel integrals and Cornu’s spiral §7.21, Figure 7.3.3, Figure 7.3.3, Figure 7.3.3
diffraction problems
- Mathieu functions 1st item
diffusion equations
- theta functions §20.13
diffusion problems
- Mathieu functions 1st item
digamma function, see psi function.
dilatation transformations
- analytic continuation of matrix elements of the resolvent onto higher Riemann sheets §1.18(viii)
dilogarithms
- analytic properties §25.12(i)
- approximations 3rd item
- computation §25.18(i)
- definition §25.12(i)
- graphics §25.12(i)
- principal branch (or value) §25.12(i)
- tables 1st item, 3rd item, §25.19
Dirac delta §1.17—§1.17
- delta sequences §1.17(i)
- integral representations
  - Airy functions §1.17(ii)
  - Bessel functions §1.17(ii)
  - Cosines §1.17(ii)
  - Coulomb functions §1.17(ii)
  - Fourier §1.17(ii)—§1.17(ii)
  - Sines §1.17(ii)
  - spherical Bessel functions §1.17(ii)
- mathematical definitions §1.17(iv)
- series representations
  - Fourier §1.17(iii)
  - Hermite polynomials §1.17(iii)
  - Laguerre polynomials §1.17(iii)
  - Legendre polynomials §1.17(iii)
  - spherical harmonics §1.17(iii)
Dirac delta distribution §1.16(iii), §2.6(ii)
Dirac delta function, see Dirac delta.
Dirac equation
- Coulomb functions §33.22(iv)
Dirichlet characters §27.8
- Gauss sum §27.10
Dirichlet $L$ -functions
- analytic properties §25.15(i)
- definition §25.15(i)
- functional equation §25.15(i)
- infinite products §25.15(i), §25.15(i)—§25.15(i)
- tables 3rd item
- zeros §25.15(ii)
Dirichlet problem
- with toroidal symmetry §14.31(i)
Dirichlet product (or convolution) §27.5
Dirichlet series §25.2(ii), §27.4
- generating function §27.4
Dirichlet’s divisor problem
- number theory §27.11
Dirichlet’s theorem
- prime numbers in arithmetic progression §27.11
discontinuity §1.4(ii)
discrete Fourier transform §3.11(v)
discrete $q$ -Hermite I and II polynomials §18.27(vii)
discrete spectra
- eigenfunction expansions §1.18(v)
- in one and two dimensions §1.18(viii)
- second order differential operators §1.18(v)
discrete spectrum
- Fourier eigenfunction expansion §1.18(v)
discriminant §18.16(vii), §18.2(vi)
- of a polynomial §1.11(ii)
discriminant function
- number theory §27.14(vi)
  - functional equation §27.14(vi)
disk
- around infinity §1.9(iv)
- open §1.9(ii)
disk polynomials §18.37(i)
dislocation theory
- Heun functions §31.17(ii)
distribution
- definition §1.16(i)
distribution function
- Painlevé transcendents §32.14
distribution functions
- connection with
  - incomplete beta functions §8.23
  - incomplete gamma functions §8.23
distributional derivative §1.16(ii)
distributions §1.16(i)—§1.16(viii)
- convergence §1.16(i), §1.16(v)
- convolutions §2.6(iii), §2.6(iii)
- derivatives §1.16(ii)
- Dirac delta distribution §1.16(iii)
- distributional derivative §1.16(ii)
- Fourier transforms §1.16(vii), §1.16(viii)
- Heaviside function §1.16(iv)
- linear functionals §1.16(i)
- of derivatives §2.6(ii)
- regular §1.16(i)
- regularization §2.6(iv)
- several variables §1.16(vi)—§1.16(vii)
- singular §1.16(i)
- support §1.16(i)
- tempered §2.6(ii), see tempered distributions.
- test function space §1.16(i), §1.16(v)
- test functions §1.16(i), §1.16(v)
  - convergence §1.16(i), §1.16(v)
divergence theorem, see Gauss’s theorem for vector-valued functions.
divergent integrals §2.6(i)
- regularization §2.6(iv)
divided differences
- definition §3.3(iii)
- integral representation §3.3(iii)
divisor function
- number theory §27.2(i)
Dixon’s ${{}_{3}F_{2}}(1)$ sum
- $q$ -analog §17.7(iii)
Dixon’s sum
- F. H. Jackson’s $q$ -analog §17.7(ii)
domain §1.9(ii)
- closed §1.9(ii)
- cut §1.10(vi)
- simply-connected §1.13(i)
dominated convergence theorem
- infinite series §1.9(vii)
double gamma function, see Barnes’ $G$ -function.
double integrals §1.5(vi), §1.5(v)—§1.5(v)
- change of order of integration §1.5(v)
- change of variables §1.5(vi)
- infinite §1.5(v)
double sequence §1.9(vii)
- convergence §1.9(vii)
double series §1.9(vii)
- convergence §1.9(vii)
doubly-confluent Heun equation §31.12
Dougall’s ${{}_{7}F_{6}}(1)$ sum
- F. H. Jackson’s $q$ -analog §17.7(iii)
Dougall’s bilateral sum §15.4(ii)
Dougall’s expansion
- associated Legendre functions §14.18(iii)
dual Hahn polynomials, see Wilson class orthogonal polynomials.
Duffing’s equation
- Jacobian elliptic functions §22.19(ii)
Dunkl operator §18.38(iii)
- and Bessel functions §18.38(iii)
Dunkl type operator §18.38(iii)
DVR (discrete variable representations)
- of differential operators §18.38(i)
DVR weight functions
- non-classical OP’s Table 18.39.1
dynamical systems
- Mathieu functions 1st item
- Painlevé transcendents §32.16
Dyson’s integral
- gamma function §5.14

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