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Index
Index C
Index B
♦
A
♦B♦
C
♦
D
♦
E
♦
F
♦
G
♦
H
♦
I
♦
J
♦
K
♦
L
♦
M
♦
N
♦
O
♦
P
♦
Q
♦
R
♦
S
♦
T
♦
U
♦
V
♦
W
♦
Z
♦
Bäcklund transformations
classical orthogonal polynomials
§18.38(ii)
Painlevé transcendents
§32.7
—
§32.7(viii)
backward recursion
§3.6(ii)
Bailey’s
F
1
2
(
-
1
)
sum
q
-analog
§17.7(i)
Bailey’s
ψ
2
2
transformations
bilateral
q
-hypergeometric function
§17.10
Bailey’s
F
3
4
(
1
)
sum
q
-analogs (first and second)
§17.7(iii)
Bailey’s bilateral summations
bilateral
q
-hypergeometric function
§17.8
bandlimited functions
§30.15(iii)
Barnes’ beta integral
§5.13
Barnes’
G
-function
asymptotic expansion
§5.17
definition
§5.17
infinite product
§5.17
integral representation
§5.17
recurrence relation
§5.17
Barnes’ integral
Ferrers functions
§14.17(ii)
Bartky’s transformation
Bulirsch’s elliptic integrals
§19.2(iii)
symmetric elliptic integrals
§19.22(i)
basic elliptic integrals
§19.29(ii)
basic hypergeometric functions
,
see
bilateral
q
-hypergeometric function
and
q
-hypergeometric function
.
Basset’s integral
modified Bessel functions
§10.32(i)
Bell numbers
asymptotic approximations
§26.7(iv)
definition
§26.7(i)
generating function
§26.7(ii)
recurrence relation
§26.7(iii)
table
Table 26.7.1
Bernoulli monosplines
§24.17(ii)
Bernoulli numbers
§24.1
arithmetic properties
§24.10
asymptotic approximations
§24.11
computation
§24.19(i)
definition
§24.2
degenerate
§24.16(i)
explicit formulas
§24.6
—
§24.6
factors
§24.10(iv)
finite expansions
§24.4(iv)
generalizations
§24.16
,
§24.16(iii)
generating function
§24.2
identities
§24.5(ii)
,
§24.5(iii)
inequalities
§24.9
integral representations
§24.7(i)
inversion formulas
§24.5(iii)
irregular pairs
§24.19(ii)
Kummer congruences
§24.10(ii)
notation
§24.1
of the second kind
§24.16(i)
recurrence relations
linear
§24.5
quadratic and higher order
§24.14(i)
,
§24.14(ii)
relations to
Eulerian numbers
§24.4(ix)
Genocchi numbers
§24.15(i)
Stirling numbers
§24.15(iii)
tangent numbers
§24.15(ii)
sums
§24.14
tables
Table 24.2.1
,
§24.20
Bernoulli polynomials
§24.1
applications
mathematical
§24.17
—
§24.17(iii)
physical
§24.18
asymptotic approximations
§24.11
computation
§24.19(i)
definitions
§24.2(i)
derivative
§24.4(vii)
difference equation
§24.4(i)
explicit formulas
§24.6
finite expansions
§24.4(iv)
generalized
§24.16
,
§24.16(iii)
generating function
§24.2(i)
graphs
Figure 24.3.1
,
Figure 24.3.1
,
Figure 24.3.1
inequalities
§24.9
infinite series expansions
Fourier
§24.8(i)
other
§24.8(ii)
integral representations
§24.7(ii)
integrals
§24.13(i)
compendia
§24.13(iii)
Laplace transforms
§24.13(iii)
multiplication formulas
§24.4(v)
notation
§24.1
recurrence relations
linear
§24.5
quadratic
§24.14(i)
relation to Eulerian numbers
§24.4(ix)
relation to Riemann zeta function
§24.4(ix)
representation as sums of powers
§24.4(iii)
special values
§24.4(vi)
sums
§24.14
symbolic operations
§24.4(viii)
symmetry
§24.4(ii)
tables
Table 24.2.2
zeros
complex
§24.12(iii)
multiple
§24.12(iv)
real
§24.12(i)
Bernoulli’s lemniscate
§19.30(iii)
Bernstein–Szegő polynomials
§18.31
Bessel Functions
Jacobi–Anger expansions
§10.35
Bessel functions
§10.1
,
see also
cylinder functions
,
Hankel functions
,
Kelvin functions
,
modified Bessel functions
,
and
spherical Bessel functions
.
addition theorems
§10.23(ii)
analytic continuation
§10.11
applications
asymptotic solutions of differential equations
§10.72(i)
—
§10.72(iii)
electromagnetic scattering
§10.73(i)
Helmholtz equation
§10.73(i)
—
§10.73(i)
Laplace’s equation
§10.73(i)
—
§10.73(i)
oscillation of chains
§10.73(i)
oscillation of plates
§10.73(i)
wave equation
§10.73(i)
approximations
§10.76
asymptotic expansions for large argument
§10.17
—
§10.17(v)
error bounds
§10.17(iii)
—
§10.17(v)
exponentially-improved
§10.17(v)
asymptotic expansions for large order
§10.19
—
§10.20(iii)
asymptotic forms
§10.19(i)
Debye’s expansions
§10.19(ii)
—
§10.19(ii)
double asymptotic properties
§10.20(iii)
,
§10.41(v)
resurgence properties of coefficients
§10.20(ii)
transition region
§10.19(iii)
uniform
§10.20
—
§10.20(iii)
branch conventions
§10.2(ii)
computation
Ch.10
—
§10.74(v)
computation by quadrature
§3.5(viii)
computation by recursion
§3.6(vi)
connection formulas
§10.4
contiguous
§10.21(i)
continued fractions
§10.10
cross-products
§10.5
,
§10.6(iii)
zeros
§10.21(x)
definite integrals
§9.11(iii)
definitions
§10.2
—
§10.2(ii)
derivatives
asymptotic expansions for large argument
§10.17(ii)
asymptotic expansions for large order
§10.19(ii)
—
§10.19(iii)
explicit forms
§10.6(ii)
uniform asymptotic expansions for large order
§10.20(i)
with respect to order
§10.15
—
§10.15
zeros
,
see
zeros of Bessel functions
.
differential equations
§10.13
,
§10.2(i)
,
see also
Bessel’s equation
.
Dirac delta
§1.17(ii)
envelope functions
§2.8(iv)
expansions in partial fractions
§10.23(ii)
expansions in series of
§10.23(iii)
—
§10.23(iv)
Fourier–Bessel expansion
§10.23(iii)
generalized
§10.46
generating functions
§10.12
graphics
§10.3
—
§10.3(iii)
incomplete
§10.46
inequalities
§10.14
infinite integrals
§18.10(iv)
infinite products
§10.21(iii)
integral representations
along the real line
§10.9(i)
—
§10.9(i)
compendia
§10.9(iv)
contour integrals
§10.9(ii)
—
§10.9(ii)
Mellin–Barnes type
§10.9(ii)
products
§10.9(iii)
—
§10.9(iii)
integrals
,
see also
integrals of Bessel and Hankel functions
and
Hankel transform
.
approximations
§10.76(ii)
computation
§10.74(vii)
tables
§10.75(iv)
,
§10.75(vii)
limiting forms
§10.7
minimax rational approximation
§3.11(iii)
modulus and phase functions
asymptotic expansions for large argument
§10.18(iii)
—
§10.18(iii)
basic properties
§10.18(ii)
definitions
§10.18(i)
graphics
§10.3(i)
—
§10.3(i)
relation to zeros
§10.21(ii)
monotonicity
§10.14
multiplication theorem
§10.23(i)
notation
§10.1
of imaginary argument
,
see
modified Bessel functions
.
of imaginary order
applications
§10.24
definitions
§10.24
graphs
§10.3(iii)
—
§10.3(iii)
limiting forms
§10.24
—
§10.24
numerically satisfactory pairs
§10.24
uniform asymptotic expansions for large order
§10.24
zeros
§10.24
of matrix argument
§35.5
applications
§35.9
asymptotic approximations
§35.5(iii)
definitions
§35.5(i)
notation
§35.1
of the first and second kinds
§35.1
properties
§35.5(ii)
relations to confluent hypergeometric functions of matrix argument
§35.6(iii)
of the first kind
§10.2(ii)
of the first, second, and third kinds
§10.2(ii)
—
§10.2(iii)
of the second kind
§10.2(ii)
of the third kind
§10.2(ii)
orthogonality
§10.22(ii)
,
§10.22(iv)
power series
§10.8
principal branches (or values)
§10.2(ii)
—
§10.2(iii)
recurrence relations
§10.6(i)
—
§10.6(iii)
relations to other functions
Airy functions
§9.6(i)
—
§9.6(ii)
confluent hypergeometric functions
§10.16
elementary functions
§10.16
generalized Airy functions
§9.13(i)
generalized hypergeometric functions
§10.16
parabolic cylinder functions
§10.16
,
§12.14(vii)
sums
§10.23(i)
—
§10.23(iv)
addition theorems
§10.23(ii)
—
§10.23(ii)
compendia
§10.23(iv)
expansions in series of Bessel functions
§10.23(iii)
—
§10.23(iii)
multiplication theorem
§10.23(i)
tables
§10.75
—
§10.75(iii)
Wronskians
§10.5
zeros
,
see
zeros of Bessel functions
.
Bessel polynomials
§10.49(ii)
,
§18.34
—
§18.34(iii)
asymptotic expansions
§18.34(iii)
definition
§18.34(i)
differential equations
§18.34(iii)
generalized
§18.34(i)
orthogonality properties
§18.34(ii)
recurrence relations
§18.34(i)
relations to other functions
complex orthogonal polynomials
§3.5(viii)
confluent hypergeometric functions
§18.34(i)
generalized hypergeometric functions
§18.34(i)
Jacobi polynomials
§18.34(iii)
Bessel transform
,
see
Hankel transform
.
Bessel’s equation
§10.2(i)
inhomogeneous forms
§11.10(ii)
,
§11.2(ii)
,
§11.9(i)
numerically satisfactory solutions
§10.2(iii)
singularities
§10.2(i)
standard solutions
§10.2(ii)
—
§10.2(ii)
Bessel’s inequality
Fourier series
§1.8(i)
Bessel’s integral
Bessel functions
§10.9(i)
best uniform polynomial approximation
§3.11(i)
best uniform rational approximation
§3.11(iii)
beta distribution
incomplete beta functions
§8.23
beta function
§5.12
,
see also
incomplete beta functions
.
applications
physical
§5.20
—
§5.20
definition
§5.12
integral representations
§5.12
multidimensional
§5.14
integrals
§5.13
multivariate
,
see
multivariate beta function
.
beta integrals
§5.12
—
§5.13
Bickley function
§10.43(iii)
—
§10.43(iv)
applications
§10.73(iv)
approximations
§10.76(iii)
biconfluent Heun equation
§31.12
application to Rossby waves
§31.17(ii)
Bieberbach conjecture
§16.23(iii)
,
§18.38(ii)
Jacobi polynomials
§18.38(ii)
bifurcation sets
§36.4
visualizations
§36.4(ii)
big
q
-Jacobi polynomials
§18.27(iii)
biharmonic operator
numerical approximation
§3.4(iii)
bilateral basic hypergeometric function
,
see
bilateral
q
-hypergeometric function
.
bilateral hypergeometric function
§16.4(v)
bilateral
q
-hypergeometric function
Bailey’s
ψ
2
2
transformations
§17.10
Bailey’s bilateral summations
§17.8
computation
§17.18
definition
§17.4(ii)
notation
§17.1
Ramanujan’s
ψ
1
1
summation
§17.8
special cases
§17.8
—
§17.8
transformations
§17.18
bilateral series
§16.4(v)
bilinear transformation
§1.9(iv)
cross ratio
§1.9(iv)
SL
(
2
,
ℤ
)
§23.15(i)
binary number system
§3.1(i)
binary quadratic sieve
number theory
§27.18
Binet’s formula
gamma function
§5.9(i)
binomial coefficients
definition
§1.2(i)
generating functions
§26.3(ii)
identities
§26.3(iv)
limiting form
§26.3(v)
recurrence relations
§26.3(iii)
relation to lattice paths
§26.3(i)
tables
§26.21
,
Table 26.3.1
,
Table 26.3.2
binomial expansion
§4.6(ii)
binomial theorem
§1.2(i)
binomials
§1.2(i)
black holes
Heun functions
§31.17(ii)
Bohr radius
Coulomb functions
§33.22(ii)
Bohr-Mollerup theorem
gamma function
§5.5(iv)
q
-gamma function
§5.18(ii)
Boole summation formula
§24.17(i)
Borel summability
§1.15(i)
Borel transform theory
applications to asymptotic expansions
§2.11(v)
Bose–Einstein condensates
Lamé functions
§29.19(i)
Bose–Einstein integrals
computation
§25.18(i)
definition
§25.12(iii)
relation to polylogarithms
§25.12(iii)
Bose–Einstein phase transition
§25.17
bound-state problems
hydrogenic
§33.22(v)
Whittaker functions
§33.22(v)
boundary points
§1.6(iv)
,
§1.9(ii)
boundary-value methods or problems
difference equations
§3.6(iv)
,
§3.6(vii)
ordinary differential equations
§3.7(iii)
parabolic cylinder functions
§12.17
bounded variation
§1.4(v)
Boussinesq equation
Painlevé transcendents
§32.13(iii)
box
plane partitions
§26.12(i)
branch
of multivalued function
§1.10(vi)
,
§4.2(i)
construction
§1.10(vi)
—
§1.10(vi)
example
§1.10(vi)
branch cut
§4.2(i)
branch point
§1.10(vi)
movable
§32.2(i)
Bromwich integral
§3.5(viii)
Bulirsch’s elliptic integrals
§19.2(iii)
computation
§19.36(ii)
first, second, and third kinds
§19.1
notation
§19.1
relation to symmetric elliptic integrals
§19.25(ii)