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Index B
Index
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B
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C
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D
♦
E
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F
♦
G
♦
H
♦
I
♦
J
♦
K
♦
L
♦
M
♦
N
♦
O
♦
P
♦
Q
♦
R
♦
S
♦
T
♦
U
♦
V
♦
W
♦
Z
♦
Abel means
§1.15(iii)
Abel summability
§1.15(i)
,
§1.15(iv)
Abel–Plana formula
§2.10(i)
Abelian functions
§21.8
absolute error
§3.1(v)
Absolutely continuous integration measure
§1.4(v)
acceleration of convergence
definition
§3.9(i)
for sequences
§3.9
—
§3.9(vi)
for series
§3.9
—
§3.9(vi)
limit-preserving
§3.9(i)
accumulation point
§1.9(ii)
acoustics
canonical integrals
§36.14(iv)
additive number theory
Ch.27
—
§27.14(vi)
Dedekind modular function
§27.14(iv)
Dedekind sum
§27.14(iii)
discriminant function
§27.14(vi)
Euler’s pentagonal number theorem
§27.14(ii)
Goldbach conjecture
§27.13(ii)
Jacobi’s identities
§27.13(iv)
notation
§27.1
partition function
§27.13(i)
unrestricted
§27.14(i)
Ramanujan’s identity
§27.14(v)
Ramanujan’s tau function
§27.14(vi)
representation by squares
§27.13(iv)
Waring’s problem
§27.13(iii)
aerodynamics
Struve functions
§11.12
affine Weyl groups
Painlevé equations
§32.7(viii)
Airy functions
§9.1
analytic properties
§9.2(i)
applications
mathematical
§9.15
physical
§9.16
—
§9.16
ship waves
§36.13
approximations
expansions in Chebyshev series
§9.19(ii)
in terms of elementary functions
§9.19(i)
in the complex plane
§9.19(iii)
asymptotic expansions
§9.7
—
§9.7(v)
error bounds
§9.7(iii)
,
§9.7(iv)
exponentially-improved
§9.7(v)
computation
§9.17
—
§9.17(v)
connection formulas
§9.2(v)
definitions
§9.2(i)
differential equation
§9.2(i)
for products
§9.11(i)
initial values
§9.2(ii)
numerically satisfactory solutions
§9.2(iii)
Riccati form
§9.2(vi)
Dirac delta
§1.17(ii)
envelope functions
§2.8(iii)
generalized
,
see
generalized Airy functions
.
graphics
§9.3
—
§9.3(ii)
incomplete
§9.14
integral identities
§36.9
integral representations
§9.11(iii)
,
§9.5(i)
,
§9.5(ii)
integrals
approximations
§9.19(i)
,
1st item
,
1st item
,
2nd item
,
§9.19(ii)
,
§9.19(iii)
asymptotic approximations
§9.10(ii)
definite
§9.10(iv)
indefinite
§9.10(i)
of products
§9.11(iv)
,
§9.11(v)
repeated
§9.10(viii)
tables
§9.18(v)
Laplace transforms
§9.10(v)
Maclaurin series
§9.4
Mellin transform
§9.10(vi)
modulus and phase
asymptotic expansions
§9.8(iv)
—
§9.8(iv)
definitions
§9.8(i)
graphs
§9.3(i)
identities
§9.8(ii)
monotonicity
§9.8(iii)
relation to Bessel functions
§9.8(i)
relation to zeros
§9.9(ii)
notation
§9.1
products
differential equation
§9.11(i)
integral representations
§9.11(iii)
integrals
§9.11(iv)
,
§9.11(v)
Wronskian
§9.11(ii)
relation to umbilics
§36.2(ii)
relations to other functions
Bessel functions
§9.6(i)
—
§9.6(ii)
confluent hypergeometric functions
§13.18(iii)
,
§13.6(iii)
,
§9.6(iii)
Hankel functions
§9.6(i)
—
§9.6(ii)
modified Bessel functions
§9.6(i)
—
§9.6(ii)
Stieltjes transforms
§9.10(vii)
tables
complex variables
§9.18(iii)
integrals
§9.18(v)
real variables
§9.18(ii)
zeros
§9.18(iv)
,
§9.9(v)
—
§9.9(v)
Wronskians
§9.2(iv)
zeros
asymptotic expansions
§9.9(iv)
computation
§3.3(v)
,
§9.17(v)
differentiation
§9.9(iii)
relation to modulus and phase
§9.9(ii)
tables
§9.18(iv)
,
§9.9(v)
—
§9.9(v)
Airy transform
§9.10(ix)
Airy’s equation
,
see
Airy functions, differential equation
.
Aitken’s
Δ
2
-process
for sequences
§3.9(iii)
iterated
§3.9(iii)
Al-Salam–Chihara polynomials
§18.28(iv)
algebraic curves
Riemann surface
2nd item
,
§21.7(i)
,
§21.7(iii)
algebraic equations
parametrization via Jacobian elliptic functions
§22.18(i)
spherical trigonometry
§22.18(iii)
uniformization
§22.18(iii)
algebraic Lamé functions
§29.17(ii)
almost Mathiew equation
singular continuous spectra
§1.18(vii)
alternant
determinant
§1.3(ii)
amplitude (
am
) function
§22.16(i)
applications
§22.19(i)
approximations
small
k
,
k
′
§22.16(i)
small
x
§22.16(i)
computation
§22.20(vi)
definition
§22.16(i)
Fourier series
§22.16(i)
integral representation
§22.16(i)
quasi-periodicity
§22.16(i)
relation to elliptic integrals
§22.16(i)
relation to Gudermannian function
§22.16(i)
special values
§22.16(i)
tables
§22.21
analytic continuation
§1.10(ii)
by reflection
§1.10(ii)
analytic continuation of matrix elements of the resolvent onto higher Riemann sheets
dilatation transformations
§1.18(viii)
analytic continuation onto higher Riemann sheets
matrix elements of the resolvent
§1.18(viii)
analytic function
§1.9(ii)
at infinity
§1.9(iv)
in a domain
§1.9(ii)
singularities
§1.10(iii)
zeros
§1.10(i)
Anderson localization
random potentials
§1.18(vii)
Anger function
,
see
Anger–Weber functions
.
Anger–Weber functions
§11.10
analytic properties
§11.10(i)
asymptotic expansions
large argument
§11.11(i)
—
§11.11(i)
large order
§11.11(ii)
computation
§11.13(i)
,
§3.6(vi)
definitions
§11.10(i)
derivatives
§11.10(ix)
differential equation
§11.10(ii)
graphics
Figure 11.10.1
,
Figure 11.10.1
,
Figure 11.10.1
,
Figure 11.10.2
,
Figure 11.10.2
,
Figure 11.10.2
,
Figure 11.10.3
,
Figure 11.10.3
,
Figure 11.10.3
,
Figure 11.10.4
,
Figure 11.10.4
,
Figure 11.10.4
incomplete
§11.14(v)
integral representations
§11.10(i)
integrals
§11.10(x)
interrelations
§11.10(v)
Maclaurin series
§11.10(iii)
notation
§11.1
order
§11.1
recurrence relations
§11.10(ix)
relations to other functions
Fresnel integrals
§11.10(vi)
Lommel functions
§11.10(vi)
—
§11.10(vi)
Struve functions
§11.10(vi)
series expansions
power series
§11.10(iii)
products of Bessel functions
§11.10(viii)
special values
§11.10(vii)
sums
§11.10(x)
tables
§11.14(iv)
angle between arcs
§1.9(iv)
angular momenta
§34.2
angular momentum
generalized hypergeometric functions
§16.24(iii)
angular momentum coupling coefficients
,
see
3
j
symbols
,
6
j
symbols
,
and
9
j
symbols
.
angular momentum operator
spherical coordinates
§14.30(iv)
annulus
§1.10(iii)
antenna research
Lamé functions
§29.19(i)
Appell functions
§16.13
analytic continuation
§16.15
applications
physical
§16.24
computation
Ch.16
definition
§16.13
—
§16.13
integral representations
§16.15
integrals
§16.15
inverse Laplace transform
§16.15
notation
§16.13
partial differential equations
§16.14(i)
relation to Legendre’s elliptic integrals
§19.5
relation to symmetric elliptic integrals
§19.25(vii)
relations to hypergeometric functions
§16.16(i)
transformations of variables
§16.16
—
§16.16(ii)
quadratic
§16.16(ii)
reduction formulas
§16.16(i)
approximation techniques
Chebyshev-series expansions
§3.11(ii)
least squares
§3.11(v)
,
§3.11(v)
—
§3.11(v)
minimax polynomials
§3.11(i)
minimax rational functions
§3.11(iii)
Padé
§3.11(iv)
—
§3.11(iv)
splines
§3.11(vi)
—
§3.11(vi)
arc length
Jacobian elliptic functions
§22.18(i)
arc(s)
§1.9(iii)
angle between
§1.9(iv)
area of triangle
§10.22(iv)
argument principle
,
see
phase principle
.
arithmetic Fourier transform
§27.17
arithmetic mean
§1.2(iv)
,
§1.7(iii)
arithmetic progression
§1.2(ii)
arithmetic-geometric mean
§19.8(i)
hypergeometric function
§15.17(iv)
integral representations
§19.8(i)
Jacobian elliptic functions
§22.20(ii)
Legendre’s elliptic integrals
§19.8(i)
—
§19.8(i)
symmetric elliptic integrals
§19.22(ii)
arithmetics
complex
§3.1(v)
exact rational
§3.1(iii)
floating-point
§3.1(i)
interval
§3.1(ii)
level-index
§3.1(iv)
Askey polynomials
§18.33(iv)
Askey scheme for orthogonal polynomials
Figure 18.21.1
,
Figure 18.21.1
,
Figure 18.21.1
Askey–Gasper inequality
§18.14(iv)
,
§18.38(ii)
Askey–Wilson class orthogonal polynomials
§18.28
—
§18.28(xi)
as eigenfunctions of a
q
-difference operator
§18.28(i)
asymptotic approximations
§18.29
interrelations with other orthogonal polynomials
Figure 18.21.1
,
Figure 18.21.1
,
Figure 18.21.1
limit relations
§18.28(x)
orthogonality properties
§18.28(i)
representation as
q
-hypergeometric functions
§18.28(i)
—
§18.28(xi)
Askey–Wilson polynomials
§18.28(ii)
,
see also
Askey–Wilson class orthogonal polynomials.
and Zhedanov algebra
§18.38(iii)
asymptotic approximations
§18.29
duality
§18.28(ii)
nonsymmetric
§18.38(iii)
q
-difference equation
§18.28(ii)
recurrence relation
§18.28(ii)
relation to
q
-hypergeometric functions
§18.28(ii)
—
§18.28(xi)
associated Anger–Weber function
,
see
Anger–Weber functions
.
associated Hermite polynomials
§18.30(iv)
associated Laguerre functions
§33.22(v)
associated Laguerre polynomials
§18.30(iii)
associated Legendre equation
§14.2(ii)
,
§14.21(i)
exponent pairs
§14.2(iii)
numerically satisfactory solutions
§14.2(iii)
,
§14.21(ii)
singularities
§14.2(iii)
standard solutions
§14.2(ii)
,
§14.21(i)
,
§14.3(ii)
associated Legendre functions
§14.1
,
see also
Ferrers functions
.
addition theorems
§14.18(ii)
,
§14.28(i)
analytic continuation
§14.24
analytic properties
§14.21(i)
applications
Ch.14
—
§14.31(iii)
asymptotic approximations
,
see
uniform asymptotic approximations
.
behavior at singularities
§14.21(iii)
,
§14.8(i)
computation
§14.32
connection formulas
§14.21(iii)
,
§14.9(iii)
continued fractions
§14.14
cross-products
§14.2(iv)
definitions
§14.21(i)
,
§14.3(ii)
—
§14.3(iii)
degree
§14.1
derivatives
§14.10
with respect to degree or order
§14.11
differential equation
,
see
associated Legendre equation
.
expansions in series of
§14.18(i)
generalized
§14.29
generating functions
§14.21(iii)
,
§14.7(iv)
graphics
§14.22
—
§14.4(iii)
,
§14.4(iv)
Heine’s formula
§14.28(ii)
hypergeometric representations
§14.21(iii)
,
§14.3
—
§14.3(iii)
integer degree and order
§14.21(iii)
,
§14.7
—
§14.7(iv)
integer order
§14.21(i)
,
§14.6
—
§14.6(ii)
integral representations
§14.12(ii)
,
§14.25
integrals
definite
§14.17(ii)
,
§14.17(iii)
,
§14.17(iv)
Laplace transforms
§14.17(v)
Mellin transforms
§14.17(vi)
products
§14.17(iv)
notation
§14.1
of the first kind
§14.3(ii)
of the second kind
§14.3(ii)
Olver’s
§14.21(i)
,
§14.3(ii)
order
§14.1
orthogonality
§14.17(iii)
principal values (or branches)
§14.21(i)
recurrence relations
§14.10
,
§14.21(iii)
relations to other functions
elliptic integrals
§14.5(v)
Gegenbauer function
§14.3(iv)
hypergeometric function
§14.3(ii)
,
§14.3(iii)
,
§15.9(iv)
Jacobi function
§14.3(iv)
Legendre polynomials
§14.7(i)
Rodrigues-type formulas
§14.7(ii)
special values
§14.5(iii)
,
§14.5(v)
sums
§14.18
—
§14.18(iv)
,
§14.28
tables
§14.33
uniform asymptotic approximations
large degree
§14.15(iii)
—
§14.15(v)
,
§14.26
large order
§14.15(i)
—
§14.15(ii)
,
§14.26
values on the cut
§14.23
Whipple’s formula
§14.9(iv)
Wronskians
§14.2(iv)
—
§14.2(iv)
,
§14.21(iii)
zeros
§14.16(iii)
,
§14.27
associated orthogonal polynomials
§18.2(x)
,
§18.30
—
§18.30
and corecursive OP’s
§18.2(x)
,
§18.30(vi)
Hermite
§18.30(iv)
Jacobi
§18.30(i)
Laguerre
§18.30(iii)
Legendre
§18.30(ii)
Meixner–Pollaczek
§18.30(v)
monic
§18.2(x)
,
§18.30(vii)
type 2 Pollaczek
§18.30(viii)
ultraspherical
§18.30(viii)
astrophysics
error functions and Voigt functions
§7.21
Heun functions and Heun’s equation
§31.17(ii)
asymptotic and order symbols
§2.1(i)
definition
§2.1(i)
differentiation
§2.1(ii)
integration
§2.1(ii)
asymptotic approximations and expansions
,
see also
asymptotic approximations of integrals
,
asymptotic approximations of sums and sequences
,
asymptotic solutions of difference equations
,
asymptotic solutions of differential equations
,
and
asymptotic solutions of transcendental equations
.
algebraic operations
§2.1(iii)
cases of failure
§2.11(i)
—
§2.11(i)
,
§2.6(i)
differentiation
§2.1(iii)
double asymptotic properties
Bessel functions
§10.41(v)
Hankel functions
§10.41(v)
Kelvin functions
§10.69
modified Bessel functions
§10.41(iv)
parabolic cylinder functions
§12.10(vi)
exponentially-improved expansions
§2.11(iii)
—
§2.11(v)
generalized
§2.1(v)
hyperasymptotic expansions
§2.11(v)
improved accuracy via numerical transformations
§2.11(vi)
—
§2.11(vi)
integration
§2.1(iii)
logarithms of
§2.1(iii)
null
§2.1(iii)
numerical use of
§2.11(i)
—
§2.11(vi)
Poincaré type
§2.1(iii)
powers of
§2.1(iii)
re-expansion of remainder terms
§2.11(iii)
—
§2.11(vi)
reversion of
§2.2
Stokes phenomenon
§2.11(iv)
substitution of
§2.1(iii)
uniform
§2.1(iv)
uniqueness
§2.1(iii)
via connection formulas
§2.11(ii)
asymptotic approximations of integrals
§2.2
—
§2.6(iv)
Bleistein’s method
§2.3(v)
Chester–Friedman–Ursell method
§2.4(v)
coalescing critical points
§2.4(v)
,
§2.4(vi)
coalescing peak and endpoint
§2.3(v)
coalescing saddle points
§2.4(v)
distributional methods
§2.6
—
§2.6(iv)
Fourier integrals
§2.3(i)
Haar’s method
§2.4(ii)
integration by parts
§2.3(i)
inverse Laplace transforms
§2.4(i)
—
§2.4(ii)
Laplace transforms
§2.3(i)
Laplace’s method
§2.3(iii)
—
§2.4(iii)
Mellin transform
§2.3(vi)
Mellin transform methods
§2.5
extensions
§2.5(ii)
—
§2.5(ii)
method of stationary phase
§2.3(iv)
extensions
§2.3(iv)
method of steepest descents
§2.4(iv)
multidimensional integrals
§2.5(ii)
Stieltjes transforms
§2.6(ii)
—
§2.6(ii)
generalized
§2.6(ii)
Watson’s lemma
§2.3(ii)
,
§2.4(i)
generalized
§2.3(ii)
asymptotic approximations of sums and sequences
§2.10
—
§2.10(iv)
Abel–Plana formula
§2.10(i)
—
§2.10(i)
Darboux’s method
§2.10(iv)
—
§2.10(iv)
entire functions
§2.10(iii)
Euler–Maclaurin formula
§2.10(i)
—
§2.10(i)
summation by parts
§2.10(ii)
asymptotic scale or sequence
§2.1(v)
asymptotic solutions of difference equations
§2.9
—
§2.9(iii)
characteristic equation
§2.9(i)
coincident characteristic values
§2.9(ii)
Liouville–Green (or WKBJ) type approximations
§2.9(iii)
transition points
§2.9(iii)
turning points
§2.9(iii)
with a parameter
§2.9(ii)
—
§2.9(iii)
asymptotic solutions of differential equations
§2.6(iv)
—
§2.8(vi)
characteristic equation
§2.7(ii)
coincident characteristic values
§2.7(ii)
error-control function
§2.7(iii)
Fabry’s transformation
§2.7(ii)
irregular singularities of rank 1
§2.7(ii)
Liouville–Green (or WKBJ) approximations
§2.7(iii)
—
§2.7(iii)
Liouville–Green approximation theorem
§2.7(iii)
numerically satisfactory solutions
§2.7(iv)
resurgence
§2.11(v)
,
§2.7(ii)
with a parameter
§2.8
—
§2.8(vi)
classification of cases
§2.8(i)
coalescing transition points
§2.8(vi)
connection formulas across transition points
§2.8(v)
in terms of Airy functions
§2.8(iii)
in terms of Bessel functions of fixed order
§2.8(iv)
—
§2.8(iv)
in terms of Bessel functions of variable order
§2.8(vi)
in terms of elementary functions
§2.8(ii)
Liouville transformation
§2.8(i)
transition points
§2.8(i)
turning points
§2.8(i)
asymptotic solutions of transcendental equations
§2.2
Lagrange’s formula
§2.2
atomic photo-ionization
Coulomb functions
§33.22(ii)
atomic physics
Coulomb functions
§33.22(ii)
error functions
§7.21
atomic spectra
Coulomb functions
§33.22(ii)
atomic spectroscopy
3
j
,
6
j
,
9
j
symbols
§34.12
attractive potentials
Coulomb functions
§33.22(ii)
,
§33.22(ii)
,
§33.22(ii)
auxiliary functions for Fresnel integrals
approximations
§7.24(i)
asymptotic expansions
§7.12(ii)
computation
§7.22(i)
definitions
§7.2(iv)
derivatives
§7.10
integral representations
§7.7(ii)
Mellin–Barnes integrals
§7.7(ii)
symmetry
§7.4
auxiliary functions for sine and cosine integrals
analytic continuation
§6.4
approximations
4th item
asymptotic expansions
§6.12(ii)
exponentially-improved
§6.12(ii)
Chebyshev-series expansions
5th item
computation
§6.18(ii)
definition
§6.2(iii)
integral representations
§6.7(iii)
principal values
§6.4
relation to confluent hypergeometric functions
§6.11
tables
§6.19(ii)
axially symmetric potential theory
§19.18(ii)