Index D

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

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 §7.24(i), §7.24(ii)
- computation §7.22(i)
- definition §7.2(ii)
- generalized §7.16
- graphics 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
Debye functions §8.22(ii)
Dedekind modular function §27.14(iv)
- functional equation §27.14(iv)
Dedekind’s eta function, see modular functions.
Dedekind’s modular function, see modular functions.
Dedekind sums
- number theory §27.14(iii)
Dellanoy 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
del operator ¶ ‣ §1.6(iii)
delta sequence §1.17(i)
delta wing equation
- Lamé polynomials §29.19(ii)
De Moivre’s theorem
- trigonometric functions ¶ ‣ §4.21(iii)
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)
- left-hand §1.8(ii)
- Leibniz’s formula ¶ ‣ §1.4(iii)
- L’Hôpital’s rule ¶ ‣ §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.
- numerically satisfactory solutions §2.7(iv)
- 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)
  - stability §3.7(ii)
  - Sturm–Liouville eigenvalue problems §3.7(iv)
  - Taylor-series methods §3.7(ii), §3.7(iii)
- 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 ¶ ‣ §18.38(i), §18.40
  - in series of classical orthogonal polynomials §18.39(i), §18.39(i), §18.39(i)
  - linearly independent ¶ ‣ §1.13(i)
  - products §1.13(v)
  - Wronskian ¶ ‣ §1.13(i)
- subdominant solutions, see recessive solutions.
- with a parameter §1.13(ii)
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)

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
diffraction problems
- Mathieu functions §28.33(ii)
diffusion equations
- theta functions §20.13
diffusion problems
- Mathieu functions §28.33(ii)
digamma function, see psi function.
dilogarithms
- analytic properties §25.12(i)
- approximations §25.20
- computation §25.18(i)
- definition §25.12(i)
- graphics §25.12(i)
- principal branch (or value) §25.12(i)
- tables §25.19, §25.19, §25.19
Dirac delta §1.17, §1.17(iv)
- delta sequences ¶ ‣ §1.17(iii), §1.17(i)
- integral representations
  - Airy functions ¶ ‣ §1.17(ii)
  - Bessel functions ¶ ‣ §1.17(ii)
  - Coulomb functions ¶ ‣ §1.17(ii)
  - Fourier §1.17(ii), §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 §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)
- tables §25.19
- zeros §25.15(ii)
Dirichlet problem
- with toroidal symmetry §14.31(i)
Dirichlet product (or convolution) §27.5
Dirichlet’s divisor problem
- number theory §27.11
Dirichlet series §25.2(ii), §27.4
- generating function §27.4
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)
discriminant
- 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)
distributional derivative §1.16(ii)
distribution function
- Painlevé transcendents §32.14
distribution functions
- connection with
  - incomplete beta functions §8.23
  - incomplete gamma functions §8.23
distributions §1.16, §1.16(vii)
- convergence §1.16(i), §1.16(v)
- convolutions §2.6(iii), §2.6(iii)
- derivatives §1.16(ii)
- Dirac delta §1.16(iii)
- distributional derivative §1.16(ii)
- Fourier transforms §1.16(vii)
- 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 functions §1.16(i)
  - convergence §1.16(i)
- test function space §1.16(i)
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 $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits(1)$ 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)
- 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 bilateral sum ¶ ‣ §15.4(ii)
Dougall’s $\mathop{{{}_{{7}}F_{{6}}}\/}\nolimits(1)$ sum
- F. H. Jackson’s $q$ -analog ¶ ‣ §17.7(iii)
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)
dynamical systems
- Mathieu functions §28.33(iii)
- Painlevé transcendents ¶ ‣ §32.16
Dyson’s integral
- gamma function ¶ ‣ §5.14