Index D
-
Darboux’s method
- asymptotic approximations of sums and sequences ¶ ‣ §2.10(iv), §2.10(iv)
- Dawson’s integral §7.2(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
- 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
-
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
- 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)
-
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
-
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 -functions
-
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
- discontinuity §1.4(ii)
- discrete Fourier transform ¶ ‣ §3.11(v)
- discrete -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)
-
number theory §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
-
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
-
divisor function
- number theory §27.2(i)
-
Dixon’s sum
- -analog ¶ ‣ §17.7(iii)
-
Dixon’s sum
- F. H. Jackson’s -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’ -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 sum
- F. H. Jackson’s -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