Index M
-
magic squares
- number theory §27.17
-
magnetic monopoles
- Riemann theta functions §21.9
-
Mangoldt’s function
- number theory §27.2(i)
-
many-body systems
- confluent hypergeometric functions §13.28(iii)
- many-valued function, see multivalued function.
- mathematical constants §3.12
-
Mathieu functions Ch.28, Ch.28, see also Mathieu’s equation, modified Mathieu functions, and radial Mathieu functions.
- analytic properties §28.12(iii), §28.2(ii), §28.7
- antiperiodicity §28.2(vi)
-
applications
- mathematical §28.32, §28.32(ii)
- physical §28.33, §28.33(iii)
-
asymptotic expansions for large , see also uniform asymptotic approximations for large parameters.
- Goldstein’s §28.8(iii)
- Sips’ §28.8(ii)
- computation Ch.28, §28.34(iv)
- connection formulas §28.12(iii)
- definitions §28.12
- differential equation §28.2(i)
- expansions in series of §28.11, ¶ ‣ §28.11, §28.19
-
Fourier coefficients
- asymptotic forms for small §28.15, §28.4(vi)
- asymptotic forms of higher coefficients §28.4(vii)
- normalization §28.14, §28.4(iii)
- recurrence relations §28.14, §28.4(ii)
- reflection properties in §28.4(v)
- tables §28.35, §28.35(ii)
- values at §28.4(iv)
- Fourier series §28.14, §28.2(iv), §28.4
- graphics §28.13, §28.3, §28.3(ii)
-
integral equations
- compendia §28.10(iii)
- variable boundaries §28.10(iii)
- with Bessel-function kernels §28.10(ii)
- with elementary kernels §28.10(i), §28.28(i)
-
integral representations §28.28(i)
- compendia §28.28(v)
-
integrals
- compendia §28.28(v)
- of products §28.28(iv), §28.28(iv)
- of products with Bessel functions §28.28(ii), §28.28(ii)
- irreducibility §28.7
- limiting forms as order tends to integers §28.12(ii), §28.12(iii)
- normalization §28.12(ii), §28.2(vi)
- notation §28.1
- of integer order §28.2(vi)
- of noninteger order §28.12(ii)
- orthogonality §28.12(ii), §28.2(vi)
- parity §28.2(vi)
- periodicity §28.12(ii), §28.2(vi)
- power series in §28.15(ii), §28.6(ii)
- pseudoperiodicity §28.12(ii), §28.2(iv)
- reflection properties in §28.12(ii)
- reflection properties in §28.12(ii), §28.12(iii), §28.2(vi)
- reflection properties in §28.12(ii)
-
relations to other functions
- basic solutions of Mathieu’s equation §28.2(vi)
- confluent Heun functions ¶ ‣ §31.12
- modified Mathieu functions §28.20(ii)
- tables §28.35, §28.35(ii)
-
uniform asymptotic approximations for large parameters
- Barrett’s ¶ ‣ §28.8(iv)
- Dunster’s ¶ ‣ §28.8(iv), ¶ ‣ §28.8(iv)
- values at §28.2(vi)
- Wronskians ¶ ‣ §28.5(i)
-
zeros §28.9
- tables §28.35(iii)
-
Mathieu’s equation §28.2(i)
- algebraic form §28.2(i)
-
basic solutions §28.2(ii)
- relation to eigenfunctions §28.2(vi)
- characteristic equation §28.2(iii)
-
characteristic exponents §28.2(iii)
- computation §28.34(i)
- eigenfunctions, see Mathieu functions.
-
eigenvalues (or characteristic values) §28.2(v)
- analytic continuation §28.7, §28.7
- analytic properties §28.7, §28.7
- asymptotic expansions for large §28.16, §28.8
- branch points §28.7
- characteristic curves §28.17
- computation §28.34(ii), §28.34(ii)
- continued-fraction equations §28.15(i), §28.6(i)
- distribution §28.12(i), ¶ ‣ §28.2(v)
- exceptional values §28.7
- graphics §28.13(i), §28.2(v)
- normal values §28.12(i), §28.7
- notation §28.1, §28.12(i), §28.2(v)
- power-series expansions in §28.15(i), §28.6(i), §28.6(i)
- reflection properties in §28.12(i)
- reflection properties in §28.12(i), ¶ ‣ §28.2(v)
- tables §28.35, §28.35(ii)
-
Floquet solutions §28.2(iv)
- computation §28.34(iii)
- Fourier-series expansions §28.2(iv)
- uniqueness §28.2(iv)
- Floquet’s theorem §28.2(iii)
- parameters
-
second solutions
- antiperiodicity ¶ ‣ §28.5(i)
- definitions ¶ ‣ §28.5(i)
- expansions in Mathieu functions ¶ ‣ §28.5(i)
- Fourier series ¶ ‣ §28.5(i)
- graphics §28.5(ii)
- normalization ¶ ‣ §28.5(i)
- notation §28.1
- periodicity §28.5(i)
- reflection properties in ¶ ‣ §28.5(i)
- values at ¶ ‣ §28.5(i)
- singularities §28.2(i)
- standard form §28.2(i)
- Theorem of Ince §28.2(iv), §28.5(i)
- transformations §28.2(ii)
-
matrix, see also linear algebra.
- augmented §3.2(i)
- characteristic polynomial §3.2(iv)
- condition number §3.2(iii)
- eigenvalues §3.2(iv), §3.2(vii)
- eigenvectors
- equivalent §21.6(i)
- factorization §3.2(i)
- Jacobi §3.5(vi)
- nondefective §3.2(iv)
- norms §3.2(iii)
- Riemann §21.1
-
symmetric
- tridiagonalization §3.2(vi)
- symplectic §21.5(i)
- triangular decomposition §3.2(i)
- tridiagonal §3.2(ii)
-
maximum §1.4(vii)
- local ¶ ‣ §1.4(iii), §1.5(iii)
-
maximum-modulus principle
- analytic functions ¶ ‣ §1.10(v)
- harmonic functions ¶ ‣ §1.10(v)
- Schwarz’s lemma ¶ ‣ §1.10(v)
- McKean and Moll’s theta functions ¶ ‣ §20.1
-
McMahon’s asymptotic expansions
-
zeros of Bessel functions §10.21(vi)
- error bounds §10.21(vi)
-
zeros of Bessel functions §10.21(vi)
- means, see Abel means, arithmetic mean, Cesàro means, geometric mean, harmonic mean, and weighted means.
- mean value property for harmonic functions ¶ ‣ §1.9(iii)
-
mean value theorems
- differentiable functions ¶ ‣ §1.4(iii)
- integrals ¶ ‣ §1.4(v), ¶ ‣ §1.4(v)
-
measure ¶ ‣ §18.2(i)
- theory ¶ ‣ §18.2(i)
-
Mehler–Dirichlet formula
- Ferrers functions §14.12(i)
-
Mehler–Fock transformation §14.20(vi), §14.31(ii)
- generalized §14.20(vi), §14.31(ii)
- Mehler functions, see conical functions.
-
Mehler–Sonine integrals
- Bessel and Hankel functions ¶ ‣ §10.9(i)
- Meijer -function §16.17
-
Meixner–Pollaczek polynomials, see Hahn class orthogonal polynomials.
- relation to hypergeometric function ¶ ‣ §15.9(i)
-
Meixner polynomials, see Hahn class orthogonal polynomials.
- relation to hypergeometric function ¶ ‣ §15.9(i)
- Mellin–Barnes integrals §5.19(ii)
-
Mellin transform
- analyticity §1.14(iv)
- analytic properties §2.5(i)
- convergence §1.14(iv)
- convolution ¶ ‣ §1.14(iv)
- convolution integrals §2.5(i)
- definition §1.14(iv), §2.5(i)
- inversion ¶ ‣ §1.14(iv), §2.5(i)
- notation §1.14(iv)
- Parseval-type formulas ¶ ‣ §1.14(iv)
- tables §1.14(viii), Table 1.14.5
- meromorphic function §1.10(iii)
-
Mersenne numbers
- number theory §27.18
-
Mersenne prime
- number theory ¶ ‣ §27.12
-
method of stationary phase
- asymptotic approximations of integrals §2.3(iv)
-
metric coefficients
- for oblate spheroidal coordinates §30.14(ii)
- for prolate spheroidal coordinates §30.13(ii)
-
Miller’s algorithm
- difference equations ¶ ‣ §3.6(vi), §3.6(iii)
-
Mill’s ratio for complementary error function §7.8
- inequalities §7.8
-
minimax polynomial approximations §3.11(i)
- computation of coefficients §3.11(i)
-
minimax rational approximations §3.11(iii)
- computation of coefficients §3.11(iii)
- type §3.11(iii)
- weight function §3.11(iii)
-
minimum §1.4(vii)
- local ¶ ‣ §1.4(iii), §1.5(iii)
- Minkowski’s inequalities for sums and series ¶ ‣ §1.7(i), ¶ ‣ §1.7(ii)
- minor, see determinants.
- Mittag-Leffler function §10.46
-
Mittag-Leffler’s expansion
- infinite partial fractions ¶ ‣ §1.10(x)
-
Möbius function
- number theory §27.2(i)
- Möbius inversion formulas
- Möbius transformation, see bilinear transformation.
-
modified Bessel functions Ch.10
- addition theorems §10.44(ii)
- analytic continuation §10.34
- applications
- approximations §10.76(ii)
-
asymptotic expansions for large argument §10.40, §10.40(iv)
- error bounds §10.40(ii), §10.40(iii)
- exponentially-improved §10.40(iv)
- for derivatives with respect to order ¶ ‣ §10.40(i)
- for products ¶ ‣ §10.40(i)
-
asymptotic expansions for large order §10.41, §10.41(v)
- asymptotic forms §10.41(i)
- double asymptotic properties §10.41(iv), §10.41(v)
- in inverse factorial series §10.41(iii)
- uniform §10.41(ii), §10.41(iii)
- branch conventions ¶ ‣ §10.25(ii)
- computation Ch.10, §10.74(v)
- connection formulas §10.27
- continued fractions §10.33
- cross-products §10.28
- definitions §10.25(i)
-
derivatives
- asymptotic expansions for large argument ¶ ‣ §10.40(i), §10.40(i)
- explicit forms §10.29(ii)
- uniform asymptotic expansions for large order §10.41(ii), §10.41(iii)
-
derivatives with respect to order §10.38
- asymptotic expansion for large argument ¶ ‣ §10.40(i)
- differential equations §10.25(i), §10.36, see also modified Bessel’s equation.
- generating function §10.35
- graphics §10.26(i)
- hyperasymptotic expansions §10.74(i)
- incomplete §10.46
- inequalities §10.37
-
integral representations
- along real line ¶ ‣ §10.32(i), §10.32(i)
- compendia §10.32(iv)
- contour integrals §10.32(ii)
- Mellin–Barnes type ¶ ‣ §10.32(ii)
- products §10.32(iii)
- integrals, see integrals of modified Bessel functions.
- limiting forms §10.30(i)
- monotonicity §10.37
- multiplication theorem §10.44(i)
- notation §10.1
-
of imaginary order
- approximations ¶ ‣ §10.76(ii)
- computation §10.74(viii)
- definitions §10.45
- graphs Figure 10.26.10, Figure 10.26.10, Figure 10.26.7, Figure 10.26.7, Figure 10.26.8, Figure 10.26.8, Figure 10.26.9, Figure 10.26.9
- limiting forms §10.45
- numerically satisfactory pairs §10.45
- tables §10.75(viii)
- uniform asymptotic expansions for large order §10.45
- zeros §10.45
- power series §10.31
- principal branches (or values) §10.25(ii)
- recurrence relations §10.29(i)
-
relations to other functions
- Airy functions §9.6(ii)
- confluent hypergeometric functions ¶ ‣ §10.39, §13.18(iii), §13.6(iii)
- elementary functions ¶ ‣ §10.39
- generalized Airy functions §9.13(i)
- generalized hypergeometric functions ¶ ‣ §10.39
- parabolic cylinder functions ¶ ‣ §10.39, §12.7(iii)
-
sums
- addition theorems §10.44(ii)
- compendia §10.44(iv)
- expansions in series of §10.44(iii)
- multiplication theorem §10.44(i)
- tables §10.75(v)
- Wronskians §10.28
-
zeros §10.42, §10.42
- computation §10.74(vi)
- tables §10.75(vi)
-
modified Bessel’s equation §10.25(i)
- inhomogeneous forms ¶ ‣ §11.2(ii), §11.9(iii)
- numerically satisfactory solutions §10.25(iii)
- singularities §10.25(i)
- standard solutions §10.25(ii)
-
modified Korteweg–de Vries equation
- Painlevé transcendents §32.13(i)
-
modified Mathieu functions Ch.28, see also radial Mathieu functions.
- addition theorems §28.27
- analytic continuation §28.20(iii)
-
applications
- mathematical §28.32(i)
- physical §28.33, §28.33(iii)
-
asymptotic approximations, see also uniform asymptotic approximations for large parameters.
- for large §28.26
- for large §28.20(iii), §28.25
- computation §28.34(iv)
- connection formulas §28.20(ii), §28.22, §28.22(ii)
- definitions §28.20(ii)
- differential equation §28.20(i)
- expansions in series of
- graphics §28.21, ¶ ‣ §28.21
-
integral representations §28.28(i), §28.28(ii)
- compendia §28.28(v)
- of cross-products §28.28(iv), §28.28(iv)
-
integrals §28.28(i)
- compendia §28.28(v)
-
joining factors §28.1, §28.22(i)
- tables §28.35(i)
- notation §28.1
- relation to Mathieu functions §28.20(ii)
- shift of variable §28.20(vii)
- tables §28.35, §28.35(ii)
- uniform asymptotic approximations for large parameters §28.26(ii), ¶ ‣ §28.8(iv)
- Wronskians §28.20(vi)
-
zeros
- tables §28.35(iii)
-
modified Mathieu’s equation §28.20(i)
- algebraic form §28.20(i)
- modified spherical Bessel functions, see spherical Bessel functions.
- modified Struve functions, see Struve functions and modified Struve functions.
- modified Struve’s equation, see Struve functions and modified Struve functions, differential equations.
-
modular equations
- modular functions §23.20(iv)
-
modular functions §23.15
- analytic properties §23.15(i)
-
applications
- mathematical §23.20, §23.20(v)
- physical §23.21, §23.21(iv)
- computation §23.22(i)
- cusp form §23.15(i)
- definitions §23.15, ¶ ‣ §23.15(ii)
- elementary properties §23.17
- general §23.15(i)
- graphics §23.16
- infinite products §23.17(iii)
- interrelations §23.19
- Laurent series §23.17(ii)
- level §23.15(i)
- modular form §23.15(i)
- modular transformations §23.18
- notation §23.1, §23.15(ii)
- power series §23.17(ii)
- relations to theta functions Figure 20.3.2, Figure 20.3.2, §20.9(ii), §23.15, ¶ ‣ §23.15(ii)
- special values §23.17(i)
-
modular theorems
- generalized elliptic integrals §19.35(i)
-
molecular spectra
- Coulomb functions ¶ ‣ §33.22(ii)
-
molecular spectroscopy
- symbols §34.12
- mollified error §3.1(v)
- moment functionals §18.34(ii)
- monic polynomial §1.11(ii), ¶ ‣ §3.5(v)
- monodromy groups
-
monosplines
- Bernoulli ¶ ‣ §24.17(ii)
- cardinal ¶ ‣ §24.17(ii)
- monotonicity §1.4(i)
-
Monte-Carlo methods
- for multidimensional integrals §3.5(x)
- Monte Carlo sampling §8.24(ii)
-
Mordell’s theorem §23.20(ii)
- elliptic curves §23.20(ii)
-
Motzkin numbers
- definition ¶ ‣ §26.6(i)
- generating function §26.6(ii)
- identities §26.6(iv)
- recurrence relation §26.6(iii)
- relation to lattice paths ¶ ‣ §26.6(i)
- table Table 26.6.2
-
-test for uniform convergence
- infinite products ¶ ‣ §1.10(ix)
- infinite series ¶ ‣ §1.9(v)
- multidimensional theta functions, see Riemann theta functions, and Riemann theta functions with characteristics.
-
multinomial coefficients
- definitions §26.4(i)
- generating function §26.4(ii)
- recurrence relation §26.4(iii)
- relation to lattice paths §26.4(i)
- table Table 26.4.1
- multiple orthogonal polynomials §18.36(iii)
- multiplicative functions §27.3
- multiplicative number theory Ch.27, ¶ ‣ §27.12
- multivalued function §1.10(vi)
- multivariate beta function
- multivariate gamma function
- multivariate hypergeometric function §19.16(ii), §19.16(ii), §19.16(ii)
-
mutual inductance of coaxial circles
- elliptic integrals §19.34