Index M

$M$ -test for uniform convergence
- infinite products §1.10(ix)
- infinite series §1.9(v)
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.
Markov’s theorem §18.2(x)
- numerator and denominator polynomials §18.30(vi)
mathematical background
- boundary conditions and the Weyl alternative §1.18(ix)
- essentially self-adjoint operator §1.18(ix)
- limit point and limit circle boundary conditions §1.18(ix)
- self-adjoint extensions of a symmetric operator §1.18(ix)
- self-adjoint operator §1.18(ix)
- spectrum of an operator §1.18(ix)
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 $q$ , 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 $q$ §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 $q$ §28.4(v)
  - tables §28.35—§28.35(ii)
  - values at $q=0$ §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 $q$ §28.15(ii), §28.6(ii)
- pseudoperiodicity §28.12(ii), §28.2(iv)
- reflection properties in $q$ §28.12(ii), §28.12(iii), §28.2(vi)
- reflection properties in $z$ §28.12(ii)
- reflection properties in $\nu$ §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 $q=0$ §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 $q$ §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 $q$ §28.15(i), §28.6(i)—§28.6(i)
  - reflection properties in $q$ §28.12(i), §28.2(v)
  - reflection properties in $\nu$ §28.12(i)
  - 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
  - definition §28.1
  - stability chart §28.17
  - stable pairs §28.17
  - stable regions §28.17—§28.17
  - unstable pairs §28.17
- 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 $q$ §28.5(i)
  - values at $q=0$ §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)
  - characteristic polynomial §3.2(iv)
  - computation §3.2(vii)
  - condition numbers §3.2(v)
  - conditioning §3.2(v)
  - multiplicity §3.2(iv)
- eigenvectors
  - left §3.2(iv)
  - normalized §3.2(iv)
  - right §3.2(iv)
- 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)
matrix elements of the resolvent
- analytic continuation onto higher Riemann sheets §1.18(viii)
- eigenfunction expansions §1.18(viii)
matrix exponential
- definition §1.2(vi)
matrix, index notation for m by n §1.2(v)
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)
mean value property for harmonic functions §1.9(iii)
mean value theorems
- differentiable functions §1.4(iii)
- integrals §1.4(v), §1.4(v)
means, see Abel means, arithmetic mean, Cesàro means, geometric mean, harmonic mean, and weighted means.
measure §18.2(i)
- Stieltjes
  - absolutely convergent §1.4(v)
- Stieltjes with jumps §1.4(v)
- theory §18.2(i)
Mehler functions, see conical functions.
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–Sonine integrals
- Bessel and Hankel functions §10.9(i)
Meijer $G$ -function §16.17
- approximations §16.26
- asymptotic expansions §16.22
- differential equation §16.21
- identities §16.19
- integral representations §16.17
- integrals §16.20
- notation §16.17
- relation to generalized hypergeometric function §16.17—§16.18
- special cases §16.18
- sums §16.20
Meixner polynomials, see Hahn class orthogonal polynomials.
- relation to hypergeometric function §15.9(i)
Meixner–Pollaczek polynomials, see Hahn class orthogonal polynomials.
- associated §18.30(v)
- relation to hypergeometric function §15.9(i)
- relation to type 2 Pollaczek polynomials §18.35(iii)
Mellin transform
- analytic properties §2.5(i)
- analyticity §1.14(iv)
- 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 Table 1.14.5, §1.14(viii)
Mellin–Barnes integrals §5.19(ii)
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)
Mill’s ratio for complementary error function §7.8
- inequalities §7.8
Miller’s algorithm
- difference equations §3.6(iii)—§3.6(vi)
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)
mixed spectra
- eigenfunction expansions §1.18(viii)
- Schrödinger Operators §1.18(viii)
- Schrödinger–Coulomb problem §1.18(viii)
Möbius function
- number theory §27.2(i)
Möbius inversion formulas
- number theory §27.5, §27.5
Möbius transformation, see bilinear transformation.
modified Bessel functions Ch.10
- addition theorems §10.44(ii)
- analytic continuation §10.34
- applications
  - asymptotic solutions of differential equations §10.72(i)
  - wave equation §10.73(i)
- 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 10.26.10, Figure 10.26.7, Figure 10.26.7, Figure 10.26.7, Figure 10.26.8, Figure 10.26.8, Figure 10.26.8, Figure 10.26.9, 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
- Poisson kernel for Laguerre polynomials §18.18(vii)
- 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 $q$ §28.26
  - for large $\Re z$ §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
  - Bessel functions §28.23—§28.23
  - cross-products of Bessel functions and modified Bessel functions §28.24
- 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)—§28.22(i)
  - tables 6th item
- 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)
- Dedekind modular function §23.15(ii)
- Dedekind’s eta function §23.15(ii)
- 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
- Klein’s complete invariant §23.15(ii)
- 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, Figure 20.3.2, §20.9(ii), §23.15—§23.15(ii)
- special values §23.17(i)
- Weierstrass $\wp$ -function (23.2.4)
- Weierstrass sigma function (23.2.6)
- Weierstrass zeta function (23.2.5)
modular theorems
- generalized elliptic integrals §19.35(i)
molecular spectra
- Coulomb functions §33.22(ii)
molecular spectroscopy
- $\mathit{3j},\mathit{6j},\mathit{9j}$ symbols §34.12
mollified error §3.1(v)
moments §18.2(ix)
monic polynomial §1.11(ii), §3.5(v)
monodromy groups
- Heun functions §31.16(i)
- hypergeometric function §15.17(v)
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)
Morse oscillator §18.39(i)
- Romanovski–Bessel polynomials §18.39(i)
- Whittaker functions §18.39(i)
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
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
- completely multiplicative functions §27.3
- Dirichlet series §27.4
- Euler product §27.4
- fundamental theorem of arithmetic §27.2(i)
- multiplicative functions §27.3
- notation §27.1
- primitive roots §27.2(i)
multiplicity
- eigenvalue §1.18(iv)
multivalued function §1.10(vi)
- branch §1.10(vi), §4.2(i)
- branch cut §4.2(i)
- principal value §4.2(i)
  - closed definition §4.2(i)
multivariate beta function
- definition (35.3.3)
- notation §35.1
- properties §35.3(ii)
multivariate gamma function
- definition §35.3(i)
- notation §35.1
- properties §35.3(ii)
multivariate hypergeometric function §19.16(ii)—§19.16(ii)
mutual inductance of coaxial circles
- elliptic integrals §19.34

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