Index C
-
calculus
- complex variable §1.9, ¶ ‣ §1.9(vii)
- one variable §1.4, §1.4(viii)
- two or more variables §1.5, ¶ ‣ §1.5(vi)
- calculus of finite differences ¶ ‣ §24.17(i)
-
canonical integrals ¶ ‣ §36.2(i)
-
applications
- acoustics §36.14(iv)
- caustics §36.14(i)
- integrals with coalescing critical points §36.12, §36.12(iii)
- optics §36.14(ii)
- quantum mechanics §36.14(iii)
- asymptotic approximations §36.11, §36.12(iii)
- computation Ch.36
- convergent series §36.8
- definitions §36.2(i)
- differential equations §36.10
- integral identities §36.9, §36.9
- notation §36.1
- relations to other functions
- special cases §36.2(ii), §36.2(iv)
- symmetries §36.2(iii)
- visualizations of modulus §36.3(i)
- visualizations of phase §36.3(ii)
- zeros §36.7, §36.7(iv)
-
applications
- cardinal function §3.3(vi)
- cardinal monosplines ¶ ‣ §24.17(ii), ¶ ‣ §24.17(ii)
- cardinal spline functions ¶ ‣ §24.17(ii)
-
Carmichael numbers
- number theory ¶ ‣ §27.12
-
Casimir forces
- Bernoulli polynomials §24.18
-
Casimir–Polder effect
- Riemann zeta function §25.17
-
Catalan numbers
- definitions §26.5(i)
- generating function §26.5(ii)
- identities §26.6(iv)
- limiting forms §26.5(iv)
- recurrence relations §26.5(iii)
- relation to lattice paths §26.5(i)
- table Table 26.5.1
-
Catalan’s constant
- Riemann zeta function §25.11(xi)
- Cauchy determinant ¶ ‣ §1.3(ii)
-
Cauchy principal values
- integrals ¶ ‣ §1.4(v)
- Cauchy–Riemann equations ¶ ‣ §1.9(ii)
- Cauchy–Schwarz inequalities for sums and integrals ¶ ‣ §1.7(i), ¶ ‣ §1.7(ii)
-
Cauchy’s integral formula ¶ ‣ §1.9(iii)
- for derivatives ¶ ‣ §1.9(iii)
- Cauchy’s theorem ¶ ‣ §1.9(iii)
- caustics
- Cayley’s identity for Schwarzian derivatives ¶ ‣ §1.13(iv)
- central differences in imaginary direction ¶ ‣ §18.1(i)
- Cesàro means ¶ ‣ §1.15(iii)
- Cesàro summability ¶ ‣ §1.15(iv), ¶ ‣ §1.15(i)
-
chain rule
- for derivatives ¶ ‣ §1.4(iii), ¶ ‣ §1.5(i)
- characteristic equation
- characteristics
- characters
- Charlier polynomials, see Hahn class orthogonal polynomials.
-
Chebyshev polynomials §18.3, see also Chebyshev-series expansions, and classical orthogonal polynomials.
-
applications
- approximation theory §18.38(i)
- solutions of differential equations ¶ ‣ §18.38(i)
- computation Ch.18
- continued fractions ¶ ‣ §18.13
- definition Table 18.3.1
- derivatives ¶ ‣ §18.9(iii)
- differential equations Table 18.8.1
- dilated §18.1(iii)
- expansions in series of ¶ ‣ §18.18(i), ¶ ‣ §18.18(viii), §3.11(ii)
- explicit representations ¶ ‣ §18.5(i), ¶ ‣ §18.5(iv)
- generating functions ¶ ‣ §18.12
- graphs Figure 18.4.3, Figure 18.4.3
- inequalities ¶ ‣ §18.14(i)
- integral representations Table 18.10.1
- integrals ¶ ‣ §18.17(viii)
- interrelations with other classical orthogonal polynomials §18.7, ¶ ‣ §18.7(iii)
- leading coefficients Table 18.3.1
- linearization formula ¶ ‣ §18.18(v)
- local maxima and minima ¶ ‣ §18.14(iii)
- normalization Table 18.3.1
- notation ¶ ‣ §18.1(ii)
- of the first, second, third, and fourth kinds Table 18.3.1
-
orthogonality properties
- with respect to integration Table 18.3.1, §3.11(ii)
- with respect to summation ¶ ‣ §18.3, §3.11(ii)
- recurrence relations Table 18.9.1, §3.11(ii)
-
relations to other functions
- hypergeometric function ¶ ‣ §15.9(i)
- Jacobi polynomials ¶ ‣ §18.7(i)
- trigonometric functions ¶ ‣ §18.5(i)
- Rodrigues formula Table 18.5.1
- scaled ¶ ‣ §18.38(i)
- shifted §18.1(iii), Table 18.3.1
- special values Table 18.6.1
- symmetry Table 18.6.1
-
tables §18.41(i)
- of coefficients §18.3
- upper bounds ¶ ‣ §18.14(iii)
- weight functions Table 18.3.1
- zeros §18.2(vi), ¶ ‣ §18.3, ¶ ‣ §3.5(v)
-
applications
-
Chebyshev
-function §25.16(i)
-
Chebyshev-series expansions
- complex variables ¶ ‣ §3.11(ii)
- computation of coefficients ¶ ‣ §3.11(ii)
- relation to minimax polynomials ¶ ‣ §3.11(ii)
- summation ¶ ‣ §3.11(ii)
-
chemical reactions
-
symbols §34.12
-
-
Chinese remainder theorem
- number theory §27.15
-
chi-square distribution function
- incomplete gamma functions §8.23
- Christoffel coefficients (or numbers), see Gauss quadrature, Christoffel coefficients (or numbers)
-
Christoffel-Darboux formula
- classical orthogonal polynomials §18.2(v)
- confluent form ¶ ‣ §18.2(v)
-
Chu–Vandermonde identity
- hypergeometric function ¶ ‣ §15.4(ii)
- circular trigonometric functions, see trigonometric functions.
- classical dynamics
-
classical orthogonal polynomials Ch.18
- addition theorems §18.18(ii)
-
applications
- approximation theory §18.38(i)
- Bieberbach conjecture ¶ ‣ §18.38(ii)
- integrable systems ¶ ‣ §18.38(ii)
- numerical solution of differential equations ¶ ‣ §18.38(i)
- physical §18.39
- quadrature ¶ ‣ §18.38(i)
- quantum mechanics §18.39(i)
- Radon transform ¶ ‣ §18.38(ii)
- random matrix theory ¶ ‣ §18.38(ii)
- Riemann–Hilbert problems ¶ ‣ §18.38(ii)
- asymptotic approximations §18.15, §18.15(vi)
- computation Ch.18
- connection formulas ¶ ‣ §18.18(iv), §18.18(iv)
- contiguous relations §18.9(ii)
- continued fractions §18.13, ¶ ‣ §18.13
- definitions Ch.18, Table 18.3.1
- derivatives ¶ ‣ §18.9(iii), §18.9(iii)
- differential equations Table 18.8.1
- expansions in series of §18.18, §18.18(ix)
- explicit representations §18.5, ¶ ‣ §18.5(iv)
- Fourier transforms ¶ ‣ §18.17(v), §18.17(v)
- generating functions §18.12
-
inequalities
- local maxima and minima ¶ ‣ §18.14(iii), §18.14(iii)
- Turan-type §18.14(ii)
- upper bounds §18.14(i)
-
integral representations §18.10, ¶ ‣ §18.10(iv)
- for products ¶ ‣ §18.17(ii), §18.17(ii)
-
integrals §18.17, §18.17(ix)
- compendia §18.17(ix)
-
interrelations
- limiting forms §18.7(iii)
- linear ¶ ‣ §18.7(i), §18.7(i)
- quadratic §18.7(ii), §18.7(ii)
- with other orthogonal polynomials Figure 18.21.1, Figure 18.21.1
- in two or more variables §18.37
- Laplace transforms §18.17(vi)
- leading coefficients Table 18.3.1
-
limiting forms
- Mehler–Heine type formulas §18.11(ii)
- linearization formulas §18.18(v)
- local maxima and minima ¶ ‣ §18.14(iii), §18.14(iii)
- Mellin transforms §18.17(vii)
- multiplication theorems §18.18(iii)
- normalization Table 18.3.1
- notations ¶ ‣ §18.1(ii)
- orthogonality properties Table 18.3.1, ¶ ‣ §18.5(iii)
- parameter constraints Table 18.3.1, ¶ ‣ §18.5(iii)
- Poisson kernels §18.18(vii)
- recurrence relations §18.9(i)
-
relations to other functions
- confluent hypergeometric functions §18.5(iii)
- generalized hypergeometric functions §18.5(iii)
- hypergeometric function ¶ ‣ §15.9(i), §15.9(i), §18.5(iii)
-
sums §18.18, §18.18(ix)
- Bateman-type §18.18(vi)
- compendia §18.18(ix)
-
tables §18.41(i)
- of coefficients §18.3
- upper bounds §18.14(i)
- weight functions Table 18.3.1
-
zeros
- asymptotic approximations ¶ ‣ §18.16(ii), §18.16(v)
- distribution §18.2(vi)
- inequalities ¶ ‣ §18.16(ii), ¶ ‣ §18.16(iv)
- classical theta functions, see theta functions.
- Clausen’s integral §25.12(i)
-
Clebsch–Gordan coefficients, see
symbols.
- relation to generalized hypergeometric functions §16.24(iii)
-
Clenshaw–Curtis quadrature formula §3.5(iv), §3.5(vii)
- comparison with Gauss quadrature §3.5(iv)
-
Clenshaw’s algorithm
- Chebyshev series ¶ ‣ §3.11(ii)
- classical orthogonal polynomials §18.40
- closed point set §1.6(iv), ¶ ‣ §1.9(ii)
-
closure
- of interval ¶ ‣ §1.4(v)
- of point sets in complex plane ¶ ‣ §1.9(ii)
- coalescing saddle points §36.12(i), §36.12(iii)
-
coaxial circles
- symmetric elliptic integrals §19.34
-
coding theory
- combinatorics §26.19
-
Krawtchouk and
-Racah polynomials ¶ ‣ §18.38(iii)
- cofactor, see determinants.
-
coherent states
-
generalized
- confluent hypergeometric functions §13.28(iii)
-
generalized
- cols, see saddle points.
- combinatorial design §26.19
-
combinatorics §26.1
- applications §26.19, §26.20
- generalized hypergeometric functions §16.23(iv)
- hypergeometric identities §15.17(iv)
- Painlevé transcendents §32.14
- compact set ¶ ‣ §1.9(vii)
- complementary error function, see error functions.
- complementary exponential integral, see exponential integrals.
- completely multiplicative functions §27.3
-
complex numbers
- arithmetic operations ¶ ‣ §1.9(i)
- complex conjugates ¶ ‣ §1.9(i)
- DeMoivre’s theorem ¶ ‣ §1.9(i)
- imaginary part ¶ ‣ §1.9(i)
- modulus ¶ ‣ §1.9(i)
- phase ¶ ‣ §1.9(i)
- polar representation ¶ ‣ §1.9(i)
- powers ¶ ‣ §1.9(i)
- real part ¶ ‣ §1.9(i)
- triangle inequality ¶ ‣ §1.9(i)
-
complex physical systems
- incomplete gamma functions §8.23
-
complex tori
- theta functions §20.12(ii)
-
computer-aided design
- Cornu’s spiral §7.21
-
computer arithmetic
- generalized exponentials and logarithms §4.44
-
conductor
- generalized Bernoulli polynomials §24.16(ii)
-
confluent Heun equation ¶ ‣ §31.12
- applications §31.17(ii)
- properties of solutions ¶ ‣ §31.12
- special cases ¶ ‣ §31.12
-
confluent hypergeometric functions, see also Kummer functions, and Whittaker functions.
-
of matrix argument §35.6
- asymptotic approximations §35.6(iv)
- computation §35.10
- definition §35.6(i)
- first kind §35.1
- Laguerre form ¶ ‣ §35.6(i)
- notation §35.1
- properties §35.6(ii)
- relations to Bessel functions of matrix argument §35.6(iii)
- second kind §35.1
-
relations to other functions
- Airy functions §9.6(iii)
- Bessel and Hankel functions ¶ ‣ §10.16
- classical orthogonal polynomials §18.5(iii)
- Coulomb functions §33.14(iii), §33.2(ii), §33.2(iii)
- error functions ¶ ‣ §7.11
- exponential integrals ¶ ‣ §6.11
- generalized Bessel polynomials §18.34(i)
- generalized exponential integral §8.19(vi)
- Hahn class orthogonal polynomials ¶ ‣ §18.23
- modified Bessel functions ¶ ‣ §10.39
- parabolic cylinder functions ¶ ‣ §12.14(vii), §12.7(iv)
- repeated integrals of error functions ¶ ‣ §7.18(iv)
- sine and cosine integrals ¶ ‣ §6.11
-
of matrix argument §35.6
-
conformal mapping ¶ ‣ §1.9(iv), §1.9(iv)
- generalized hypergeometric functions §16.23(iii)
- hypergeometric function §15.17(ii)
- Jacobian elliptic functions §22.18(ii)
- modular functions ¶ ‣ §23.20(i)
- symmetric elliptic integrals §19.32
- Weierstrass elliptic functions ¶ ‣ §23.20(i), §23.20(i)
- congruence of rational numbers §24.10(i)
-
conical functions §14.20(i)
- applications §14.31(ii)
-
asymptotic approximations
- large degree §14.20(vii), §14.20(viii)
- large order §14.20(ix)
- behavior at singularities §14.20(iii)
- connection formulas §14.20(i), §14.20(i)
- definitions §14.20(i)
- degree §14.1
- differential equation §14.20(i)
- generalized Mehler–Fock transformation §14.20(vi)
- graphics §14.20(ii)
- integral representation §14.20(iv)
- integrals with respect to degree §14.20(x)
- notation §14.1, §14.20(i)
- order §14.1
- tables §14.33, §14.33
- trigonometric expansion §14.20(v)
- Wronskians §14.20(i), §14.20(i)
- zeros §14.20(x)
- connected point set ¶ ‣ §1.9(ii)
-
constants
- roots of §1.11(iv)
-
continued fractions §1.12, §1.12(vi)
- applications §1.12(vi)
- approximants §1.12(ii)
- canonical denominator (or numerator) §1.12(ii)
- contraction §1.12(iv)
- convergence §1.12(v)
-
convergents §1.12(ii)
- existence of §1.12(iii)
- determinant formula ¶ ‣ §1.12(ii)
- equivalent ¶ ‣ §1.12(ii)
- even part §1.12(iv)
- extension §1.12(iv)
- fractional transformations ¶ ‣ §1.12(ii)
-
Jacobi fraction ¶ ‣ §3.10(ii)
- associated ¶ ‣ §3.10(ii)
-
-fraction ¶ ‣ §3.10(ii)
- notation §1.12(i)
-
numerical evaluation
- backward recurrence ¶ ‣ §3.10(iii)
- forward recurrence ¶ ‣ §3.10(iii)
- forward series recurrence ¶ ‣ §3.10(iii)
- odd part §1.12(iv)
- Pringsheim’s theorem ¶ ‣ §1.12(v)
- quotient-difference algorithm ¶ ‣ §3.10(ii)
- recurrence relations ¶ ‣ §1.12(ii)
- relation to power series ¶ ‣ §3.10(ii), §3.10(ii)
- series ¶ ‣ §1.12(ii)
-
-fraction ¶ ‣ §3.10(ii)
- Stieltjes fraction ¶ ‣ §3.10(ii)
- Van Vleck’s theorem ¶ ‣ §1.12(v)
- continuous dual Hahn polynomials, see Wilson class orthogonal polynomials.
-
continuous dynamical systems and mappings
- Painlevé transcendents §32.16
-
continuous function
- at a point §1.4(ii), §1.5(i), ¶ ‣ §1.9(ii)
- notation §1.4(ii)
- of two variables §1.5(i), ¶ ‣ §1.9(ii)
- on an interval §1.4(ii)
- on a point set §1.5(i)
- on a region ¶ ‣ §1.9(ii)
- on the left (or right) §1.4(ii)
- piecewise §1.4(ii), §1.5(i)
- removable discontinuity §1.4(ii)
- sectionally §1.4(ii)
- simple discontinuity §1.4(ii)
- continuous Hahn polynomials, see Hahn class orthogonal polynomials.
-
continuous
-Hermite polynomials §18.28(vi), §18.28(vii)
- asymptotic approximations to zeros §18.29
-
continuous
-ultraspherical polynomials §18.28(v)
- contour §1.9(iii)
- convergence
- convex functions §1.4(viii)
-
coordinate systems
- cylindrical ¶ ‣ §1.5(ii)
- ellipsoidal §23.21(iii), §29.18(ii)
- elliptic §31.17(i)
- elliptical §28.32(i), §28.32(i)
- oblate spheroidal §30.14(i)
- parabolic cylinder §12.17
- paraboloidal §13.28(i), §28.32(ii)
- paraboloid of revolution §12.17
- polar ¶ ‣ §1.5(ii)
- projective §23.20(ii)
- prolate spheroidal §30.13(i)
- spherical (or spherical polar) ¶ ‣ §1.5(ii)
- sphero-conal §29.18(i)
- toroidal §14.19(i), §14.31(ii)
- Cornu’s spiral §7.20(ii)
- cosecant function, see trigonometric functions.
- cosine function, see trigonometric functions.
-
cosine integrals §6.2(ii)
- analytic continuation §6.4
- applications §6.17
- approximations §6.20(i)
-
asymptotic expansions §6.12(ii)
- exponentially-improved §6.12(ii)
- auxiliary functions, see auxiliary functions for sine and cosine integrals.
- Chebyshev-series expansions §6.20(ii), §6.20(ii)
- computation §6.18
- definition §6.2(ii)
- expansion in spherical Bessel functions §6.10(ii)
- generalized §8.21, §8.21(viii)
- graphics Figure 6.3.2, Figure 6.3.2
-
hyperbolic analog ¶ ‣ §6.2(ii)
- analytic continuation §6.4
- integral representations §6.7(ii)
- integrals §6.14(i), §6.14(ii)
- Laplace transform §6.14(i)
- notation §6.1
- power series §6.6
- principal value §6.2(ii)
- relations to exponential integrals §6.5
- sums §6.15
- tables §6.19(ii), §6.19(ii)
- value at infinity ¶ ‣ §6.2(ii)
-
zeros §6.13
- asymptotic expansion §6.13
- computation §6.18(iii)
-
cosmology
- confluent hypergeometric functions §13.28(iii)
- incomplete beta functions §8.24(ii)
- cotangent function, see trigonometric functions.
- Coulomb excitation of nuclei ¶ ‣ §33.22(ii)
- Coulomb field §33.22(iv)
-
Coulomb functions
- Dirac delta ¶ ‣ §1.17(ii)
-
Coulomb functions: variables
Ch.33
- analytic properties §33.14(i), §33.14(ii), §33.14(iii)
- applications Ch.33, §33.22(vii)
-
asymptotic approximations and expansions for large
§33.21(i), §33.21(ii)
-
asymptotic expansions as
§33.20(iii)
- uniform §33.20(iv)
-
case
§33.20(i)
- complex variables and parameters §33.22(vii)
- computation §33.23
- conversions between variables and parameters §33.22(iii)
- definitions §33.14(ii), §33.14(iii), §33.14(iv)
- derivatives §33.17
- expansions in Airy functions §33.20(iv)
- expansions in Bessel functions §33.20(i), §33.20(iii), §33.20(iv)
- expansions in modified Bessel functions §33.20(i), §33.20(iii)
-
functions
§33.14(ii), §33.14(iii)
-
functions
§33.14(iv)
- graphics §33.15, §33.15(ii)
- integral representations for Dirac delta §33.14(iv)
-
limiting forms for large
§33.18
-
power-series expansions in
§33.20(ii)
-
power-series expansions in
§33.19
- recurrence relations §33.17
-
relations to other functions
- confluent hypergeometric functions §33.14(ii)
-
Coulomb functions with variables
§33.16(ii), §33.16(iv)
- Whittaker functions §33.14(ii), §33.14(iii), §33.16(iii), §33.16(v)
- scaling of variables and parameters ¶ ‣ §33.22(ii), ¶ ‣ §33.22(ii), ¶ ‣ §33.22(ii)
- tables §33.24
- Wronskians §33.14(v)
-
Coulomb functions: variables
Ch.33
- analytic properties §33.2(i), §33.2(ii), §33.2(iii)
- applications Ch.33, §33.22(vii)
-
asymptotic expansions
-
large
§33.12
-
large
§33.11
- uniform expansions §33.12(ii), §33.12(ii)
-
large
-
case
§33.5(ii)
- complex variable and parameters §33.13, §33.22(vii)
- computation §33.23
- continued fractions §33.8
- conversions between variables and parameters §33.22(iii)
- cross-product §33.2(iv)
- definitions §33.2(ii), §33.2(iii)
- derivatives §33.4
- expansions in Airy functions §33.12(i)
- expansions in Bessel functions §33.9(ii)
- expansions in modified Bessel functions §33.9(ii)
- expansions in spherical Bessel functions §33.9(i)
-
functions
§33.2(ii), §33.2(iii)
- graphics §33.3, §33.3(ii)
- integral representations §33.7
-
limiting forms
-
large
§33.5(iv)
-
large
§33.10(ii), §33.10(iii)
-
large
§33.10(i), §33.11
-
small
§33.5(iii)
-
small
§33.5(i)
-
large
- normalizing constant §33.2(ii)
- phase shift (or phase) §33.2(iii), §33.25
-
power-series expansions in
§33.6
- recurrence relations §33.4
-
relations to other functions
- confluent hypergeometric functions §33.2(ii), §33.2(iii)
-
Coulomb functions with variables
§33.16(i)
- Whittaker functions §33.2(ii), §33.2(iii)
- scaling of variables and parameters ¶ ‣ §33.22(ii), ¶ ‣ §33.22(ii), ¶ ‣ §33.22(ii)
- tables §33.24
- transition region §33.12(i)
- WKBJ approximations §33.23(vii)
- Wronskians §33.2(iv)
- Coulomb phase shift §33.2(iii), §33.23, §33.25, ¶ ‣ §5.20
- Coulomb potential barriers §33.22(vi)
-
Coulomb potentials Ch.33, ¶ ‣ §33.22(ii)
-
-hypergeometric function §17.17
-
-
Coulomb radial functions, see Coulomb functions: variables
.
-
Coulomb spheroidal functions §30.12
- as confluent Heun functions ¶ ‣ §31.12
- Coulomb wave equation
-
Coulomb wave functions, see Coulomb functions: variables
, and Coulomb functions: variables
.
- counting techniques §26.18
-
critical phenomena
- elliptic integrals §19.35(ii)
- hypergeometric function §15.18
-
critical points §36.4(i)
- coalescing §36.12(i), §36.12(iii)
- cross ratio ¶ ‣ §1.9(iv)
-
cryptography §27.16
- Weierstrass elliptic functions §23.20(iii)
-
cubature
- for disks and squares §3.5(x)
-
cubic equation ¶ ‣ §1.11(iii)
- resolvent ¶ ‣ §1.11(iii)
-
cubic equations
- solutions as trigonometric and hyperbolic functions §4.43
- curve
-
cusp bifurcation set
- formula ¶ ‣ §36.4(i)
- picture §36.4(ii)
-
cusp canonical integral ¶ ‣ §36.2(i), §36.7(ii)
-
zeros §36.7(ii)
- table Table 36.7.1
-
zeros §36.7(ii)
- cusp catastophe ¶ ‣ §36.2(i), Figure 36.5.1, Figure 36.5.1
-
cuspoids
- normal forms ¶ ‣ §36.2(i)
-
cut ¶ ‣ §1.10(vi)
- domain ¶ ‣ §1.10(vi)
- neighborhood ¶ ‣ §1.10(vi)
- cycle ¶ ‣ §26.2
- cyclic identities
-
cyclotomic fields
- Bernoulli and Euler polynomials §24.17(iii)
-
cylinder functions
- addition theorems §10.23(ii)
- definition ¶ ‣ §10.2(ii)
- derivatives §10.6(ii), §10.6(ii)
- differential equations §10.13, §10.2(i)
- integrals ¶ ‣ §10.22(i), §10.22(i)
- multiplication theorem §10.23(i)
- recurrence relations §10.6(i)
- zeros, see zeros of cylinder functions.
- cylindrical coordinates ¶ ‣ §1.5(ii)
- cylindrical polar coordinates, see cylindrical coordinates.