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Index C
Index E
Index D
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A
♦
B
♦
C
♦D♦
E
♦
F
♦
G
♦
H
♦
I
♦
J
♦
K
♦
L
♦
M
♦
N
♦
O
♦
P
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Q
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R
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S
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T
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U
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V
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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
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
.
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)
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
§18.38(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
,
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
.
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(iv)
delta sequences
§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
§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)
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)
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
F
2
3
(
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
F
6
7
(
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)
dynamical systems
Mathieu functions
1st item
Painlevé transcendents
§32.16
Dyson’s integral
gamma function
§5.14