Index B
-
Bäcklund transformations
- classical orthogonal polynomials ¶ ‣ §18.38(ii)
- Painlevé transcendents §32.7, §32.7(viii)
- backward recursion §3.6(ii)
-
Bailey’s bilateral summations
- bilateral -hypergeometric function ¶ ‣ §17.8
-
Bailey’s sum
- -analog ¶ ‣ §17.7(i)
-
Bailey’s sum
- -analogs (first and second) ¶ ‣ §17.7(iii)
-
Bailey’s transformations
- bilateral -hypergeometric function §17.10
- bandlimited functions §30.15(iii)
- Barnes’ beta integral ¶ ‣ §5.13
- Barnes’ -function
-
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 -hypergeometric function, and -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
- 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
- inequalities §24.9
- infinite series expansions
- 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
- 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)
-
applications
- Bernoulli’s lemniscate §19.30(iii)
- Bernstein–Szegö polynomials §18.31
-
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
- 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)
- 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)
- basic properties §10.18(ii)
- definitions §10.18(i)
- graphics §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
-
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, second, and third kinds §10.2(ii), §10.2(iii)
- 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’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)
- Bessel transform, see Hankel transform.
- 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.
- 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 -Jacobi polynomials §18.27(iii)
-
biharmonic operator
- numerical approximation ¶ ‣ §3.4(iii)
- bilateral basic hypergeometric function, see bilateral -hypergeometric function.
- bilateral hypergeometric function §16.4(v)
- bilateral -hypergeometric function
- bilateral series §16.4(v)
-
bilinear transformation ¶ ‣ §1.9(iv)
- cross ratio ¶ ‣ §1.9(iv)
- SL §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
- definitions §26.3(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)
- binomials §1.2(i)
- binomial theorem ¶ ‣ §1.2(i)
-
black holes
- Heun functions §31.17(ii)
- Bohr-Mollerup theorem
-
Bohr radius
- Coulomb functions ¶ ‣ §33.22(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
- boundary points §1.6(iv), ¶ ‣ §1.9(ii)
- boundary-value methods or problems
- bounded variation ¶ ‣ §1.4(v)
- bound-state problems
-
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)
-
of multivalued function §1.10(vi), §4.2(i)
- 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)