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
- asymptotic expansions for large parameters
- 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
- approximations §8.27(i)
- asymptotic approximations and expansions
- basic properties §8.2
- Chebyshev-series expansions 3rd item
- computation Ch.8
- continued fraction §8.9
- definitions
- derivatives §8.8
- differential equations §8.2(iii)
- expansions in series of
- generalizations §8.16
- graphics
- inequalities §8.10
- integral representations
- 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
- special values §8.4
- sums §8.15
- tables §8.26(ii)
- zeros §8.13—§8.13(iii)
- incomplete Riemann zeta function §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
- 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
- infinite sequences
- infinite series, see also power series.
- inhomogeneous Airy functions, see Scorer functions.
-
initial-value problems
- Mathieu functions §28.33(iii)
- inner product
- integrable differential equations
- integrable equations, see integrable differential equations.
-
integral
-
Steiltjes
- measure with jumps §1.4(v)
-
Steiltjes
-
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
- 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
- interior points §1.9(ii)
- interpolation §3.3—§3.3(vi), §3.8(iii), see also Lagrange interpolation.
-
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)
-
Lagrange inversion theorem
§1.10(vii)
- inverse Gudermannian function §4.23(viii)
-
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
- 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
-
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
-
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
- 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)
