Index I

implicit function theorem §1.5(i)
Ince polynomials §28.31(ii)
- normalization §28.31(ii)
- zeros §28.31(ii)
Ince’s equation, see Hill’s equation, equation of Ince.
Ince’s theorem, see Theorem of Ince.
incomplete Airy functions §9.14
incomplete beta functions §8.17
- applications
  - physical §8.24
  - statistical §8.23
- asymptotic expansions for large parameters
  - general case §8.18(ii)
  - inverse function §8.18(ii)
  - symmetric case §8.18(ii)
- basic properties §8.17(i), §8.17(vii)
- continued fraction §8.17(v)
- historical profile §8.17(i)
- integral representation §8.17(iii)
- inverse function §8.18(ii)
- notation §8.1
- recurrence relations §8.17(iv)
- relation to hypergeometric function §8.17(ii)
- sums §8.17(vi)
- tables §8.26(iii)
incomplete gamma functions Ch.8
- analytic continuation §8.2(ii)
- applications
  - mathematical §8.22
  - physical §8.24
  - statistical §8.23
- approximations §8.27(i)
- asymptotic approximations and expansions
  - exponentially-improved §8.11(i), §8.12
  - for inverse function §8.12
  - large variable and/or large parameter §8.11—§8.11(v), §8.12
  - uniform for large parameter §8.12, §8.12
- basic properties §8.2
- Chebyshev-series expansions 3rd item
- computation Ch.8
- continued fraction §8.9
- definitions
  - general values §8.2(i)
  - principal values §8.2(i)
- derivatives §8.8
- differential equations §8.2(iii)
- expansions in series of
  - Bessel functions §8.7
  - Laguerre polynomials §8.7
  - modified spherical Bessel functions §8.7
- generalizations §8.16
- graphics
  - complex argument §8.3(ii)—§8.3(ii)
  - real variables §8.3(i)—§8.3(i)
- inequalities §8.10
- integral representations
  - along real line §8.6(i)
  - compendia §8.6(iii)
  - contour integrals §8.6(ii)
  - Mellin–Barnes type §8.6(ii)
- integrals §8.14
- monotonicity properties §8.3(i)
- normalized §8.2(i)
- notation §8.1
- of imaginary argument §8.6(ii)
- Padé approximant §8.10
- power-series expansions §8.7
- principal values §8.2(i)
- recurrence relations §8.8
- relations to other functions
  - confluent hypergeometric functions §13.18(ii), §13.6(ii), §8.5
  - error functions §7.11
  - exponential integrals §6.11
  - generalized exponential integral §8.19(i)
  - incomplete Riemann zeta function §8.22(ii)
- special values §8.4
- sums §8.15
- tables §8.26(ii)
- zeros §8.13—§8.13(iii)
incomplete Riemann zeta function §8.22(ii)
- asymptotic approximations §8.22(ii)
- expansions in series of incomplete gamma functions §8.22(ii)
- zeros §8.22(ii)
inductance
- symmetric elliptic integrals §19.34
inequalities
- Bernstein-type §18.14(i)
- general orthogonal polynomials §18.2(xii)
- Hermite polynomials §18.14(i)
- Jacobi polynomials §18.14(i)
- Laguerre polynomials §18.14(i)
- means §1.7(iii)
- sums and integrals
  - Cauchy–Schwarz §1.7(i), §1.7(ii)
  - Hölder’s §1.7(i), §1.7(ii)
  - Jensen’s §1.7(iv)
  - Minkowski’s §1.7(i), §1.7(ii)
- Turán-type §18.14(ii)
- ultraspherical polynomials §18.14(i)
infinite partial fractions §1.10(x)
- Mittag-Leffler’s expansion §1.10(x)
infinite products
- convergence §1.10(ix)
  - absolute §1.10(ix)
  - uniform §1.10(ix)
- $M$ -test for uniform convergence §1.10(ix)
- relation to infinite partial fractions §1.10(x)
- Weierstrass product §1.10(ix)
infinite sequences
- convergence §1.9(v)
  - absolute §1.9(v)
  - pointwise §1.9(v)
  - uniform §1.9(v)
- double §1.9(vii)
  - convergence §1.9(vii)
  - relation to infinite double series §1.9(vii)
infinite series, see also power series.
- convergence §1.9(v)
  - absolute §1.9(v)
  - pointwise §1.9(v)
  - uniform §1.9(v)
  - Weierstrass $M$ -test §1.9(v)
- divergent §1.9(v)
- dominated convergence theorem §1.9(vii)
- double §1.9(vii)
- doubly-infinite §1.9(v)
- summability methods §1.15—§1.15(iv)
- term-by-term integration §1.9(vii)
inhomogeneous Airy functions, see Scorer functions.
initial-value problems
- Mathieu functions §28.33(iii)
inner product
- $L^{2}$ §1.18(ii)
- space §1.18(i)
integrable differential equations
- Riemann theta functions §21.9—§21.9
integrable equations, see integrable differential equations.
integral
- Steiltjes
  - measure with jumps §1.4(v)
integral equations
- Painlevé transcendents §32.5
integral transforms §1.14, see also Fourier cosine and sine transforms, Fourier transform, Jacobi transform, Hankel transform, Hilbert transform, Kontorovich–Lebedev transform, Laplace transform, Mellin transform, spherical Bessel transform, and Stieltjes transform.
- compendia §1.14(viii)
- in terms of parabolic cylinder functions §12.16
- in terms of Whittaker functions §13.23(iv)
integrals
- asymptotic approximations, see asymptotic approximations of integrals.
- Cauchy principal values §1.4(v)
- change of variables §1.4(v)
- convergence §1.4(v)
  - absolute §1.4(v), §1.4(v)
  - uniform §1.10(viii), §1.5(iv)
- convolution product §2.6(iii)
- definite §1.4(v)
- differentiation §1.4(v), §1.5(iv)—§1.5(iv)
- double, see double integrals.
- fundamental theorem of calculus §1.4(v)
- generalized §2.6(i)
- indefinite §1.4(iv)
- infinite §1.4(v), §1.5(v), §1.9(iii)
- Jensen’s inequality §1.7(iv)
- Lebesgue §1.4(v)
- Lebesgue-Stieltjes §1.4(v)
- line §1.6(iv)
- mean value theorems
  - first §1.4(v)
  - second §1.4(v)
- multiple §1.5(v)
- over parametrized surface §1.6(v)
- path §1.6(iv)
- repeated §1.4(v)
- square-integrable §1.4(v)
- Stieltjes §1.4(v)
- summability methods §1.15(iv)—§1.15(viii)
- tables §1.4(iv)
- with coalescing saddle points §36.12—§36.12(iii)

integrals of Bessel and Hankel functions
- compendia §10.22(vi)
- convolutions §10.22(ii)
- fractional §10.22(ii)
- Hankel transform §10.22(v)
- indefinite §10.22(i)—§10.22(i)
- orthogonal properties §10.22(ii), §10.22(iv)
- over finite intervals §10.22(ii)—§10.22(ii)
- over infinite intervals §10.22(iii)—§10.22(iv), §13.4(i)
- products §10.22(i)—§10.22(iv)
  - triple §10.22(iv)
- trigonometric arguments §10.22(ii)
- Weber transform §10.22(v)
integrals of modified Bessel functions
- compendia §10.44(iv)
- computation §10.74(vii)
- fractional §10.43(iii)—§10.43(iv)
- indefinite §10.43(i)
- Kontorovich–Lebedev transform §10.43(v)
- over finite intervals §10.43(ii)
- over infinite intervals §10.43(ii)—§10.43(iv), §13.16(i), §13.4(i), §9.12(vii)
- products §10.43(iv)
- tables §10.75(v)
integration §1.4(iv), §1.4(v), §1.9(iii)
- by parts §1.4(iv)
- numerical, see cubature, Gauss quadrature, Monte Carlo methods, and quadrature.
- term by term §1.9(vii)
interaction potentials
- hypergeometric function §15.18
interior Dirichlet problem
- for oblate spheroids §30.14(v)
- for prolate spheroids §30.13(v)
interior points §1.9(ii)
interpolation §3.3—§3.3(vi), §3.8(iii), see also Lagrange interpolation.
- based on Chebyshev points §3.3(vi)
- based on Sinc functions §3.3(vi)
- bivariate §3.3(vi)
- convergence properties §3.3(vi)
- Hermite §3.3(vi)
- inverse §3.3(v)
- inverse linear §3.8(iii)
- linear §3.3(ii)
- rational §3.3(vi)
- spline §3.3(vi)
- trigonometric §3.3(vi)
interval
- closure §1.4(v)
interval arithmetic §3.1(ii)
inverse function §1.10(vii)
- Lagrange inversion theorem §1.10(vii)
  - extended §1.10(vii)
inverse Gudermannian function §4.23(viii)
- relation to Legendre’s elliptic integrals §19.6(ii)
- relation to $R_{C}$ -function §19.6(ii)
inverse hyperbolic functions §4.37
- addition formulas §4.38(iii)
- analytic properties §4.37(i)
- approximations §4.47
- branch cuts Figure 4.37.1, Figure 4.37.1, Figure 4.37.1
- branch points §4.37(i)
- Chebyshev-series expansions §4.47(i)
- computation §4.45(i)
- conformal maps §4.29(ii)
- continued fractions §4.39
- definitions §4.37(i), §4.37(ii)
- derivatives §4.38(ii)
- fundamental property §4.37(v)
- general values §4.37(i)
- graphics
  - complex argument §4.29(ii)
  - real argument §4.29(i)—§4.29(i)
- integrals §4.40
- interrelations §4.37(vi)
- logarithmic forms §4.37(iv)—§4.37(iv)
- notation §4.1
- power series §4.38(i)
- principal values §4.37(ii)
- reflection formulas §4.37(iii)
- tables §4.46
- values on the cuts §4.37(iv)—§4.37(iv)
inverse incomplete beta function §8.18(ii)
inverse incomplete gamma function §8.12
inverse Jacobian elliptic functions §22.15
- applications §22.18(i)
- as Legendre’s elliptic integrals §22.15(ii)
- as symmetric elliptic integrals §22.15(ii)
- computation §22.20(v)
- definitions §22.15(i)
- equivalent forms §22.15(ii)—§22.15(ii)
- normal forms §22.15(ii)
- notation §22.15(i)
- power-series expansions §22.15(ii)
- principal values §22.15(i)
inverse Laplace transforms
- asymptotic expansions §2.4(i)—§2.4(ii)
inverse trigonometric functions §4.23
- addition formulas §4.24(iii)
- analytic properties §4.23(i)
- approximations §4.47
- branch cuts Figure 4.23.1, Figure 4.23.1, Figure 4.23.1
- branch points §4.23(i)
- Chebyshev-series expansions §4.47(i)
- computation §4.45(i)
- conformal maps Figure 4.15.7, Figure 4.15.7, Figure 4.15.7
- continued fractions §4.25
- definitions §4.23(i)
- derivatives §4.24(ii)
- fundamental property §4.23(v)
- general values §4.23(i)
- graphics
  - complex argument §4.15(iii)—§4.15(iii)
  - real argument §4.15(i)
- integrals §4.26(iv)
- interrelations §4.23(vii)
- logarithmic forms §4.23(iv)—§4.23(iv)
- notation §4.1
- power series §4.24(i)
- principal values §4.23(ii)
- real and imaginary parts §4.23(vi)
- reflection formulas §4.23(iii)
- special values §4.23(vii)
- sums §4.27
- tables §4.46
- values on the cuts §4.23(iv)—§4.23(iv)
Ising model
- Appell functions §16.24(i)
- combinatorics §26.20
- generalized hypergeometric functions §16.24(i)
- Painlevé transcendents §32.16
isolated essential singularity §1.10(iii), see also essential singularity.
- movable §32.2(i)
isolated singularity §1.10(iii)
iterative methods
- Bairstow’s method (for zeros of polynomials) §3.8(iv)
- bisection method §3.8(iii)
- convergence
  - cubic §3.8(i)
  - geometric §3.8(i)
  - linear §3.8(i)
  - local §3.8(i)
  - logarithmic §3.9(v)
  - of the $p$ th order §3.8(i)
  - quadratic §3.8(i)
- eigenvalue methods §3.8(iii)
- fixed-point methods §3.8(vii), §3.8(viii)—§3.8(viii)
- Halley’s rule §3.8(v)
- Newton’s rule (or method) §3.8(ii)
- regula falsi §3.8(iii)
- secant method §3.8(iii)
- Steffensen’s method §3.8(iii)

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