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
  - Airy function §36.2(ii)
  - Pearcey integral §36.2(ii)
- special cases §36.2(ii), §36.2(iv)
- symmetries §36.2(iii)
- visualizations of modulus §36.3(i)—§36.3(i)
- visualizations of phase §36.3(ii)—§36.3(ii)
- zeros §36.7—§36.7(iv)
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 operator §18.38(iii)
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
- Airy functions §9.16
- canonical integrals §36.14(i)
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
- difference equations §2.9(i)
- differential equations §2.7(ii)
characteristics
- Riemann theta functions §21.2(ii), §21.3(ii)
characters
- number theory
  - Dirichlet §27.8
  - induced modulus §27.8
  - orthogonality relation §27.8
  - primitive §27.8
  - principal §27.8
  - quadratic Jacobi symbol §27.9
  - quadratic Legendre symbol §27.9
  - real §27.8
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)
  - numerical 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, 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)
- monic §18.38(i)
- 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, Table 18.9.2, §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
- shifted §18.1(iii), Table 18.3.1
- special values Table 18.6.1
- standardization Table 18.3.1
- 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 §18.16(iii), §18.2(vi), §18.3, §3.5(v)—§3.5(v)
Chebyshev $\psi$ -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
- $\mathit{3j},\mathit{6j},\mathit{9j}$ symbols §34.12
chi-square distribution function
- incomplete gamma functions §8.23
Chinese remainder theorem
- number theory §27.15
Christoffel coefficients (or numbers), see Gauss quadrature, Christoffel coefficients (or numbers)
Christoffel–Darboux formula §18.2(v)
- confluent form §18.2(v)
Chu–Vandermonde identity
- hypergeometric function §15.4(ii)
circular trigonometric functions, see trigonometric functions.
classical dynamics
- Jacobian elliptic functions §22.19(v)
- Weierstrass elliptic functions §23.21
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)
  - quadrature §18.38(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)
- characterizations §18.3
- 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
- discriminant §18.16(vii)
- expansions in series of §18.18—§18.18(ix)
- explicit representations §18.5—§18.5(iv)
- Fourier transforms §18.17(v), §18.17(v)—§18.17(v)
- generating functions §18.12
- in two or more variables §18.37
- inequalities
  - local maxima and minima §18.14(iii)—§18.14(iii)
  - positive sums §18.14(iv)
  - Turán-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, Figure 18.21.1
- Laplace transforms §18.17(vi)
- leading coefficients Table 18.3.1
- limiting forms §18.7(iii)
  - 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)
- 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)—§18.9(i), §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)
- standardization Table 18.3.1
- 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)
- tables 1st item, 3rd item
Clebsch–Gordan coefficients, see $\mathit{3j}$ 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(i)
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 $q$ -Racah polynomials §18.38(iii)
cofactor, see determinants.
coherent states
- generalized
  - confluent hypergeometric functions §13.28(iii)
cols, see saddle points.
combinatorial design §26.19
combinatorics §26.1
- applications §26.19, §26.20
  - mathematical §26.19
  - physical §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
completeness relation
- eigenfunctions
  - continuous spectrum §1.18(vi)
completness relation
- eigenfunctions §1.18(viii)
  - discrete spectrum §1.18(ii), §1.18(v)
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 arithmetic
- generalized exponentials and logarithms §4.44
computer-aided design
- Cornu’s spiral §7.21
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

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)—§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 4th item, 5th item
- 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)
contiguous relations §18.2(v)
continued fractions §1.12—§1.12(vi), §18.13, §18.2(x)
- 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)
- $J$ -fraction §3.10(ii)
- Jacobi fraction §3.10(ii)
  - associated §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)
- $S$ -fraction §3.10(ii)
- series §1.12(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 a point set §1.5(i)
- on a region §1.9(ii)
- on an interval §1.4(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 $q$ -Hermite polynomials §18.28(vi), §18.28(vii)
- asymptotic approximations to zeros §18.29
continuous $q$ -ultraspherical polynomials §18.28(v)
continuous spectra
- eigenfunction expansions §1.18(vi)
- second order differential operators §1.18(vi)
contour §1.9(iii)
- simple §1.9(iii)
- simple closed §1.9(iii)
convergence
- acceleration, see acceleration of convergence.
- cubic §3.8(i)
- geometric §3.8(i)
- linear §3.8(i)
- local §3.8(i)
- of the $p$ th order §3.8(i)
- quadratic §3.8(i)
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
- paraboloid of revolution §12.17
- paraboloidal §13.28(i), §28.32(ii)
- polar §1.5(ii)
- projective §23.20(ii)
- prolate spheroidal §30.13(i)
- spherical (or spherical polar) §1.5(ii), §18.39(ii)
- sphero-conal §29.18(i)
- toroidal §14.19(i), §14.31(ii)
corecursive orthogonal polynomials §18.2(x), §18.30(vi), §18.30(vii)
Cornu’s spiral §7.20(ii)
- applications §7.21
- connection with Fresnel integrals §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 4th item
- asymptotic expansions §6.12(ii)
  - exponentially-improved §6.12(ii)
- auxiliary functions, see auxiliary functions for sine and cosine integrals.
- Chebyshev-series expansions 2nd item—4th item
- 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, 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 1st item, 2nd item
- value at infinity §6.2(ii)
- zeros §6.13
  - asymptotic expansion §6.13
  - computation §6.18(iii)
cosine transform
- eigenfunction expansion §1.18(vi)
Cosines
- Dirac delta §1.17(ii)
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 $r,\epsilon$ 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 $\left|r\right|$ §33.21(i), §33.21(ii)
- asymptotic expansions as $\epsilon\rightarrow 0$ §33.20(iii)
  - uniform §33.20(iv)
- case $\epsilon=0$ §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 $f(\epsilon,\ell;r),h(\epsilon,\ell;r)$ §33.14(ii), §33.14(iii)
- functions $s(\epsilon,\ell;r),c(\epsilon,\ell;r)$ §33.14(iv)
- graphics §33.15—§33.15(ii)
- integral representations for Dirac delta §33.14(iv)
- limiting forms for large $\ell$ §33.18
- power-series expansions in $r$ §33.19
- power-series expansions in $\epsilon$ §33.20(ii)
- recurrence relations §33.17
- relations to other functions
  - confluent hypergeometric functions §33.14(ii)
  - Coulomb functions with variables $\rho,\eta$ §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 $\rho,\eta$ Ch.33
- analytic properties §33.2(i), §33.2(ii), §33.2(iii)
- applications Ch.33—§33.22(vii)
- asymptotic expansions
  - large $\eta$ §33.12
  - large $\rho$ §33.11
  - uniform expansions §33.12(ii)—§33.12(ii)
- case $\eta=0$ §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 $F_{\ell}(\eta,\rho),G_{\ell}(\eta,\rho),{H^{\pm}_{\ell}}(\eta,\rho)$ §33.2(ii), §33.2(iii)
- graphics §33.3—§33.3(ii)
- integral representations §33.7
- limiting forms
  - large $\ell$ §33.5(iv)
  - large $\left|\eta\right|$ §33.10(ii), §33.10(iii)
  - large $\rho$ §33.10(i), §33.11
  - small $\left|\eta\right|$ §33.5(iii)
  - small $\rho$ §33.5(i)
- normalizing constant §33.2(ii)
- phase shift (or phase) §33.2(iii), §33.25
- power-series expansions in $\rho$ §33.6
- recurrence relations §33.4
- relations to other functions
  - confluent hypergeometric functions §33.2(ii), §33.2(iii)
  - Coulomb functions with variables $r,\epsilon$ §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)
- $q$ -hypergeometric function §17.17
Coulomb Problem in 3D §18.39(ii)
Coulomb radial functions, see Coulomb functions: variables $\rho,\eta$ .
Coulomb spheroidal functions §30.12
- as confluent Heun functions §31.12
Coulomb wave equation
- irregular solutions §33.14(i), §33.2(i)
- regular solutions §33.14(i), §33.2(i)
- singularities §33.14(i), §33.2(i)
- turning points §33.14(i), §33.2(i), §33.22(vi)
Coulomb wave functions, see Coulomb functions: variables $\rho,\eta$ and Coulomb functions: variables $r,\epsilon$ .
Coulomb–Pollaczek polynomials
- and Coulomb Schrödinger equation §18.39(iv)
- and Coulomb wave functions §18.39(iv)
- attractive §18.39(iv)
- continuous weight function on $[-1,1]$ §18.39(iv)
- discrete weights §18.39(iv)
- repulsive §18.39(iv)
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)—§3.5(x)
cubic equation §1.11(iii)
- resolvent §1.11(iii)
cubic equations
- solutions as trigonometric and hyperbolic functions §4.43
curve
- piecewise differentiable §1.6(iv)
- simple closed §1.6(iv)
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
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
- Riemann surface §21.7(i), §21.7(i)
cyclic identities
- Jacobian elliptic functions §22.9—§22.9(v)
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.

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