Index L

L’Hôpital’s rule for derivatives §1.4(iii)
$\ell^{2}$ space
- Hilbert space §1.18(i)
Lagrange interpolation §3.3(i)—§3.3(ii)
- abscissas §3.3(i)
- Barycentric form §3.3(i)
- coefficients §3.3(ii)
- equally-spaced nodes §3.3(ii)—§3.3(ii)
- error term §3.3(i)
- formula §3.3(i)
- Newton’s interpolation formula §3.3(iv)
- nodal polynomials §3.3(i)
- nodes §3.3(i)
- polynomial §3.3(i)
- remainder terms §3.3(i)—§3.3(ii)
- via divided differences §3.3(iii)
Lagrange inversion theorem §1.10(vii)
- extended §1.10(vii)
Lagrange’s formula for reversion of series §2.2
Laguere EOP’s §18.36(vi)
Laguerre functions
- associated §33.22(v)
Laguerre polynomials §18.3, see also classical orthogonal polynomials.
- addition theorem §18.18(ii)
- as Sheffer polynomials §18.2(xii)
- asymptotic approximations §18.15(iv)—§18.15(iv)
- computation Ch.18
- continued fraction §18.13
- derivatives §18.9(iii)
- differential equations Table 18.8.1
- Dirac delta §1.17(iii)
- expansions in series of §18.18(iii), §18.18(i), §18.18(ii)
- explicit representations §18.5—§18.5(iv)
- Fourier transforms §18.17(v)
- generalized §18.1(ii)
- generating functions §18.12
- graphics Figure 18.4.5, Figure 18.4.5, Figure 18.4.5, Figure 18.4.6, Figure 18.4.6, Figure 18.4.6
- Hille–Hardy formula §18.18(vii)
- inequalities §18.14(iii), §18.14(i), §18.14(ii)
  - positive sums §18.14(iv)
  - Turán-type §18.14(ii)
- integral representations §18.10(ii), §18.10(iv), Table 18.10.1
- integrals §18.17(iv), §18.17(vi), §18.17(i)
  - fractional §18.17(iv)
  - indefinite §18.17(i)
- interrelations with other orthogonal polynomials Figure 18.21.1, Figure 18.21.1, Figure 18.21.1, §18.21(ii), §18.21(ii), §18.7(iii), §18.7(iii)
- Laplace transform §18.17(vi)
- leading coefficients Table 18.3.1
- limiting form as a Bessel function §18.11(ii)
- limits to monomials §18.6(ii)
- local maxima and minima §18.14(iii)
- Mellin transform §18.17(vii)
- monic §3.5(v)
- multiplication theorem §18.18(iii)
- notation §18.1(ii)
- orthogonality properties Table 18.3.1
- parameter constraint Table 18.3.1, §18.5(iii)
- Poisson kernels §18.18(vii)
- recurrence relations Table 18.9.1, Table 18.9.2
- relation to confluent hypergeometric functions §13.18(v), §13.6(v), §18.11(i), §18.5(iii)
- Rodrigues formula Table 18.5.1
- standardization Table 18.3.1
- tables §18.41(i)
  - of coefficients §18.3
  - of zeros Table 3.5.6
- tables of zeros Table 3.5.7, Table 3.5.8, Table 3.5.9
- upper bounds §18.14(i)
- value at $z=0$ §18.6(i)
- weight function Table 18.3.1
- zeros §18.16(iv), §18.2(vi)
  - asymptotic behavior §18.16(iv)
  - tables Table 3.5.6
Lah numbers
- definitions §5.2(iii)
Lambert series
- number theory §27.7
Lambert $W$ -function §4.13
- applications §4.44
- asymptotic expansions §4.13
- branch cuts Figure 4.13.2, Figure 4.13.2, Figure 4.13.2
- computation §4.45(iii)
- definition §4.13
- graphs Figure 4.13.1, Figure 4.13.1, Figure 4.13.1
- notation §4.13
- principal branch §4.13
  - other branches Figure 4.13.1, Figure 4.13.1, Figure 4.13.1, §4.13
- properties §4.13
Lamé functions Ch.29
- algebraic §29.17(ii)
- applications
  - conformal mapping §29.18(iv)
  - physical §29.19
  - rotation group §29.18(iv)
  - sphero-conal coordinates §29.18(i)
- asymptotic expansions §29.7(ii)
- computation §29.20(i)
- definition §29.3(iv)
- differential equation, see Lamé’s equation.
- eigenvalues
  - asymptotic expansions §29.7(i)
  - coalescence §29.3(ii)
  - computation §29.20(i)
  - continued-fraction equation §29.3(iii)
  - definition §29.3(i)
  - distribution §29.3(ii)
  - graphics §29.4(i)—§29.4(ii)
  - interlacing §29.3(ii)
  - limiting forms §29.5
  - notation §29.1
  - parity Table 29.3.1
  - periods Table 29.3.1
  - power-series expansions §29.3(vii)
  - special cases §29.5
  - tables §29.21
- Fourier series §29.6—§29.6(iv)
- graphics §29.4(iii)—§29.4(iv)
- integral equations §29.8
- limiting forms §29.5
- normalization §29.3(v)
- notation §29.1
- order §29.3(iv)
- orthogonality §29.3(vi)
- parity Table 29.3.2
- period Table 29.3.2
- relations to Heun functions §31.7(ii), §31.8
- relations to Lamé polynomials §29.12(i), §29.6(i)
- special cases §29.5
- with imaginary periods §29.10
- with real periods §29.3(iv)
- zeros §29.3(iv)
Lamé polynomials Ch.29
- algebraic form §29.12(ii)
- applications
  - ellipsoidal harmonics §29.18(iii)
  - physical §29.12(iii), §29.19
  - spherical harmonics §29.18(iii)
- asymptotic expansions §29.16
- Chebyshev series §29.15(ii)
- coefficients §29.20(ii)
- computation §29.20(ii)
- definition §29.12
- eigenvalues
  - asymptotic expansions §29.16
  - computation §29.20(ii)
  - graphics §29.13(i)—§29.13(i)
- elliptic-function form §29.12(i)
- explicit formulas §29.15(ii)
- Fourier series §29.15
- graphics §29.13(ii)—§29.13(iii)
- notation §29.1, §29.12(i), Table 29.12.1
- orthogonality §29.14
- periodicity §29.12(i)
- relation to Lamé functions §29.12(i), §29.6(i)
- tables 1st item
- zeros §29.12(i)
  - computation §29.20(iii)
  - electrostatic interpretation §29.12(iii)
Lamé wave equation §29.11
Lamé–Wangerin functions §29.17(iii)
Lamé’s equation §29.2(i)
- algebraic form §29.2(i)
- eigenfunctions Table 29.3.2
- eigenvalues, see Lamé functions, eigenvalues.
- Jacobian elliptic-function form §29.2(ii)
- other forms §29.2(ii)—§29.2(ii)
- relation to Heun’s equation §29.2(ii)
- second solution §29.17(i)
- singularities §29.2(i)
- stability §29.9
- trigonometric form §29.2(ii)
- Weierstrass elliptic-function form §29.2(ii)
Lanczos tridiagonalization of a symmetric matrix §3.2(vi)
Lanczos vectors §3.2(vi)
Landen transformations
- Jacobian elliptic functions §22.17(ii), §22.20(iii), §22.7(i), §22.7(ii)
- theta functions §20.7(vi)
Laplace equation
- $\mathit{3j},\mathit{6j},\mathit{9j}$ symbols §34.12
Laplace transform
- analyticity §1.14(iii)
- asymptotic expansions for large parameters §2.3(i), §2.3(ii), §2.4(i)
- asymptotic expansions for small parameters §2.5(iii)—§2.5(iii)
- convergence §1.14(iii)
- convolution §1.14(iii)
- definition §1.14(iii)
- derivatives §1.14(iii)
- differentiation §1.14(iii)
- for functions of matrix argument §35.2
  - analytic properties §35.2
  - convolution theorem §35.2
  - definition §35.2
  - inversion formula §35.2
- integration §1.14(iii)
- inversion §1.14(iii)
- notation §1.14(iii)
- numerical inversion §3.11(iv), §3.5(viii)—§3.5(ix)
- of periodic functions §1.14(iii)
- tables Table 1.14.4, §1.14(viii)
- translation §1.14(iii)
- uniqueness §1.14(iii)
Laplace’s equation
- Bessel functions §10.73(i)
- for elliptical cones §29.19(i)
- spherical coordinates §14.30(iv)
- symmetric elliptic integrals §19.18(ii)
- toroidal coordinates §14.31(ii)
Laplace’s method for asymptotic expansions of integrals §2.3(iii), §2.4(iii)
Laplacian §1.5(ii)
- cylindrical coordinates §1.5(ii)
- ellipsoidal coordinates §23.21(iii)
- numerical approximations §3.4(iii)
- oblate spheroidal coordinates §30.14(iii)
- parabolic cylinder coordinates §12.17
- polar coordinates §1.5(ii)
- prolate spheroidal coordinates §30.13(iii)
- spherical coordinates §1.5(ii), §18.39(ii)
lattice
- for elliptic functions, see Weierstrass elliptic functions, lattice.
lattice models of critical phenomena
- elliptic integrals §19.35(ii)
lattice parameter
- theta functions §20.1
lattice paths §26.2—§26.6(iv)
- definition §26.2
- $k$ -dimensional §26.2
lattice walks
- Appell functions §16.24(i)
- generalized hypergeometric functions §16.24(i)
Laurent polynomial
- symmetric §18.38(iii)
Laurent series §1.10(iii)
- asymptotic approximations for coefficients §2.10(iv)
Lauricella’s function
- relation to symmetric elliptic integrals §19.25(vii)
Lax pairs
- classical orthogonal polynomials §18.38(ii)
- Painlevé transcendents §32.4(i)
layered materials
- elliptic integrals §19.35(ii)

least squares approximations §3.11(v), §3.11(v)—§3.11(v)
- conditioning §3.11(v)
- normal equations §3.11(v)
- orthogonal functions with respect to weighted summation §3.11(v)
$L^{2}$
- inner product §1.18(ii)
Lebesgue constants §1.8(i), §3.11(ii)
- asymptotic behavior §1.8(i)
Lebesgue–Stieltjes measure §18.2(i)
Legendre functions §14.1, see also associated Legendre functions and Ferrers functions.
- complex degree §14.31(iii)
Legendre functions on the cut, see Ferrers functions.
Legendre polynomials §18.3, see also classical orthogonal polynomials.
- addition theorem §18.18(ii)
- asymptotic approximations §18.15(iii)
- computation Ch.18
- continued fraction §18.13
- definition Table 18.3.1
- differential equation Table 18.8.1
- Dirac delta §1.17(iii)
- expansions in series of §18.18(viii), §18.18(i)
- explicit representations §18.5—§18.5(iv)
- Fourier transforms §18.17(v)
- generating functions §18.12
- graphs Figure 18.4.4, Figure 18.4.4, Figure 18.4.4
- inequalities §18.14(ii)
  - Turán-type §18.14(ii)
- integral representations §18.10(i), Table 18.10.1
  - for products §18.17(ii)
- integrals §18.17(viii), §18.17(iii)
  - Nicholson-type §18.17(iii)
- interrelations with other orthogonal polynomials §18.7(i)
- large degree §2.10(iv)
- leading coefficients Table 18.3.1
- Mellin transforms §18.17(vii)
- monic §3.5(v)
- notation §18.1(ii)
- orthogonality properties Table 18.3.1
- recurrence relations Table 18.9.1, Table 18.9.2
- relations to other functions
  - associated Legendre functions §14.7(i)
  - Ferrers functions §14.7(i)
  - hypergeometric function §15.9(i)
  - $\mathit{3j}$ symbols §34.3(vii)—§34.3(vii)
- Rodrigues formula Table 18.5.1
- shifted §18.1(ii), 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
  - of zeros Table 3.5.2
- tables of zeros Table 3.5.3, Table 3.5.4, Table 3.5.5
- weight function Table 18.3.1
- zeros §18.16(iii), §18.2(vi)
  - tables Table 3.5.2
Legendre symbol
- prime numbers §27.9
Legendre’s elliptic integrals §19.2(ii)
- addition theorem §19.11
- applications
  - mathematical Ch.19—§19.35(i)
  - physical §19.35(ii)
- approximations (except asymptotic) §19.38
- arithmetic-geometric mean §19.8(i)
- asymptotic approximations §19.12
- change of amplitude §19.7(ii)
- change of modulus §19.7(ii)
- change of parameter §19.7(iii)
- circular cases §19.2(ii), §19.7(iii)
- complete §19.2(ii)
- computation §19.36(iv), §19.36(i)—§19.36(i)
- connection formulas §19.7
- derivatives §19.4(i)
- differential equations §19.4(ii)
- duplication formulas §19.11(iii)
- first, second, and third kinds §19.1
- Gauss transformation §19.8(iii)
- graphics §19.3—§19.3(ii)
- hyperbolic cases §19.2(ii), §19.7(iii)
- imaginary-argument transformations §19.7(ii)
- imaginary-modulus transformations §19.7(ii)
- incomplete §19.1
- inequalities
  - complete integrals §19.9(i)
  - incomplete integrals §19.9(ii)
- integration
  - with respect to amplitude §19.13(ii)
  - with respect to modulus §19.13(i)
- Landen transformations
  - ascending §19.8(ii)
  - descending §19.8(ii)
- Laplace transforms §19.13(iii)
- limiting values §19.6
- notation §19.1
- power-series expansions §19.5—§19.5
- quadratic transformations §19.8
- reciprocal-modulus transformation §19.7(ii)
- reduction of general elliptic integrals §19.14—§19.14(ii)
- relations to other functions
  - $\operatorname{am}$ function §22.16(i)
  - Appell functions §19.5
  - inverse Gudermannian function §19.6(ii)
  - inverse Jacobian elliptic functions §22.15(ii)
  - Jacobi’s epsilon function §22.16(ii)
  - Jacobi’s zeta function §22.16(iii)
  - Jacobian elliptic functions §19.10(i)
  - symmetric elliptic integrals §19.25(i), §19.25(iii)
  - theta functions §19.10(i)
  - Weierstrass elliptic functions §19.10(i)
- special cases §19.6
- tables §19.37—§19.37(iii)
Legendre’s equation §14.2(i)
- standard solutions §14.2(i)
Legendre’s relation
- Legendre’s elliptic integrals §19.7(iii)
Legendre’s relation for the hypergeometric function
- generalized §15.16
Leibniz’s formula for derivatives §1.4(iii)
lemniscate arc length §22.18(i)
lemniscate constants §19.20(i), §19.20(iv)
lengths of plane curves
- Bernoulli’s lemniscate §19.30(iii)
- ellipse §19.30(i)
- hyperbola §19.30(ii)
Lerch’s transcendent
- definition §25.14(i)
- properties §25.14(ii)
- relation to Hurwitz zeta function §25.14(i)
- relation to polylogarithms §25.14(i)
level-index arithmetic §3.1(iv)
Levi-Civita symbol for vectors §1.6(ii)
Levin’s transformations
- application to asymptotic expansions §2.11(vi)
- for sequences §3.9(v)
Lie algebras
- $q$ -series §17.16
light absorption
- Voigt functions §7.21
limit circle
- second order linear differential operator §1.18(ix)
limit point
- second order linear differential operator §1.18(ix)
limit point and limit circle boundary conditions
- mathematical background §1.18(ix)
limit points (or limiting points) §1.9(ii)
limits of functions
- of a complex variable §1.9(ii)
- of one variable §1.4(ii)
- of two complex variables §1.9(ii)
- of two variables §1.5(i)
line broadening function §7.19(i)
linear algebra §3.2—§3.2(vii), see also Gaussian elimination.
- condition numbers §3.2(iii), §3.2(v)
- conditioning of linear systems §3.2(iii)
- error bounds §3.2(iii)
  - a posteriori §3.2(iii)
- norms
  - Euclidean §3.2(iii)
  - of arbitrary order §3.2(iii)
  - of matrices §3.2(iii)
  - of vectors §3.2(iii)
linear functional §1.16(i)
linear operators
- Hilbert space §1.18(iii)
linear second order differential operator
- formally self adjoint linear operator §1.18(iv), §1.18(iv)
- Liouville normal form §1.18(iv)
linear transformation §1.9(iv)
Liouville normal form
- linear second order differential operator §1.18(iv)
- Schrödinger operator §1.18(iv)
Liouville transformation for differential equations §1.13(iv), §2.8(i)
Liouville–Green (or WKBJ) approximation §2.7(iii)—§2.7(iii)
- for difference equations §2.9(iii)
Liouville’s function
- number theory §27.2(i)
Liouville’s theorem for entire functions §1.9(iii)
little $q$ -Jacobi polynomials §18.27(iv)
local maxima and minima §18.14(iii)
locally analytic §32.2(i)
locally integrable §2.5(i)
logarithm function Ch.4
- analytic properties §4.2(i)
- approximations §4.47
- branch cut Figure 4.2.1, Figure 4.2.1, Figure 4.2.1
- Briggs §4.2(ii)
- Chebyshev-series expansions §4.47(i)
- common §4.2(ii)
- computation §4.45(i)
- conformal maps §4.3(ii)
- continued fractions §4.9(i)
- definition §4.2(i)
- derivatives §4.7(i)
- differential equations §4.7(i)
- general base §4.2(ii)
- general value §4.2(i)
- generalized §4.12
- graphics
  - complex argument Figure 4.3.3, Figure 4.3.3, Figure 4.3.3
  - real argument Figure 4.3.1, Figure 4.3.1, Figure 4.3.1
- hyperbolic §4.2(ii)
- identities §4.8(i)
- inequalities §4.5(i)
- integrals §4.10(i)
- limits §4.4(iii)
- Napierian §4.2(ii)
- natural §4.2(ii)
- notation §4.1
- power series §4.6(i)
- principal value §4.2(i)
- real and imaginary parts §4.2(i)
- special values §4.4(i)
- sums §4.11
- tables §4.46
- values on the cut §4.2(i)
- zeros §4.2(i)
logarithmic integral §6.2(i)
- asymptotic expansion §6.12(i)
- definition §6.2(i)
- graph Figure 6.16.2, Figure 6.16.2, Figure 6.16.2
- notation §6.1
- number-theoretic significance §6.16(ii)
- relation to exponential integrals §6.2(i)
Lommel functions §11.9
- asymptotic expansions for large argument §11.9(iii)
- computation §11.13(i)
- definitions §11.9—§11.9(i)
- differential equation §11.9(i)
- integral representations §11.9(iv)
- integrals §11.9(iv)
- notation §11.1
- power series §11.9(i)
- reflection formulas §11.9(i)
- relation to Anger–Weber functions §11.10(vi)—§11.10(vi)
- series expansions
  - Bessel functions §11.9(ii)
  - power series §11.9(i)
Lucas numbers §24.15(iv)

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Index L

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