Index R

Raabe’s theorem
- Bernoulli polynomials §24.4(v)
Racah polynomials §16.4(iii), see Wilson class orthogonal polynomials.
radial Mathieu functions §28.20(iv), see also modified Mathieu functions.
- definitions §28.20(iv)
- expansions in series of Bessel functions §28.23—§28.23
- expansions in series of cross-products of Bessel functions and modified Bessel functions §28.24—§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)
- joining factors §28.1, §28.22(i)—§28.22(i)
- notation §28.1
- relation to modified Mathieu functions §28.20(iv)
- shift of variable §28.20(vii)
radial spheroidal wave functions §30.11
- applications §30.14(v)
- asymptotic behavior for large variable §30.11(iii)
- computation §30.16(iii)
- connection formulas §30.11(i)
- connection with spheroidal wave functions §30.11(v)
- definitions §30.11(i)
- graphics §30.11(ii)—§30.11(ii)
- integral representation §30.11(vi)
- tables §30.17
- Wronskian §30.11(iv)
radiative equilibrium
- generalized exponential integral §8.24(iii)
Radon transform
- classical orthogonal polynomials §18.38(ii)
railroad track design
- Cornu’s spiral §7.21
rainbow
- Airy functions §9.16
Ramanujan’s ${{}_{1}\psi_{1}}$ summation
- bilateral $q$ -hypergeometric function §17.8
Ramanujan’s beta integral §5.13
Ramanujan’s cubic transformation
- hypergeometric function §15.8(v)
Ramanujan’s partition identity
- number theory §27.14(v)
Ramanujan’s sum
- number theory §27.10
Ramanujan’s tau function
- number theory §27.14(vi)—§27.14(vi)
random graphs
- generalized hypergeometric functions §16.23(ii)
random matrix theory
- Hermite polynomials §18.38(ii)
- Painlevé transcendents §32.14
random potentials
- Anderson localization §1.18(vii)
random walks §16.24(i)
range and domain
- self-adjoint operator §1.18(ix)
rational arithmetics §3.1(iii)
- exact §3.1(iii)
rational functions
- summation §5.19(i)
Rayleigh function §10.21(xiii)
- applications §10.73(v)
$R_{C}$ -function §19.2(iv)
- asymptotic approximations §19.12
- limiting values §19.6(v)
- relation to elementary functions §19.10(ii)
- relation to Gudermannian function §19.10(ii)
- relation to inverse Gudermannian function §19.6(ii)
- special values §19.6(v)
reduced Planck’s constant §14.30(iv), §33.22(i)
reduced residue system
- number theory §27.2(i)
reductions of partial differential equations
- Painlevé transcendents §32.13
Regge poles
- Coulomb functions §33.22(vii)
Regge symmetries
- $\mathit{3j}$ symbols §34.3(ii)
- $\mathit{6j}$ symbols §34.5(ii)
region §1.9(ii)
regularization
- distributional methods §2.6(iv)
relative error §3.1(v)
relative precision §3.1(v)
relativistic Coulomb equations 5th item
relaxation times for proteins
- incomplete gamma functions §8.24(i)
Remez’s second algorithm
- minimax rational approximations §3.11(iii)
removable singularity §1.10(iii)
repeated integrals of the complementary error function §7.18
- applications §7.21
- asymptotic expansions §7.18(vi)
- computation §7.22(iii)
- continued fractions §7.18(v)
- definition §7.18(i)
- derivatives §7.18(iii)
- differential equation §7.18(iii)
- graphics Figure 7.18.1, Figure 7.18.1, Figure 7.18.1
- power-series expansion §7.18(iii)
- recurrence relations §7.18(iii)
- relations to other functions
  - confluent hypergeometric functions §7.18(iv)
  - Hermite polynomials §7.18(iv)
  - parabolic cylinder functions §12.7(ii), §7.18(iv)
  - probability functions §12.7(ii), §7.18(iv)
- scaled §7.18(ii)
- tables §7.23(ii)
representation theory
- partitions §26.19
repulsive potentials
- Coulomb functions §33.22(ii), §33.22(ii), §33.22(ii)
residue §1.10(iii)
- theorem §1.10(iv)
resistive MHD instability theory
- Struve functions §11.12
resolvent cubic equation §1.11(iii)
resonances
- Coulomb functions §33.22(vii)
restricted integer partitions
- Bessel-function expansion §26.10(vi)
- conjugate §26.9(i)
- generating functions §26.10(ii), §26.9(ii)
- identities §26.10(iv)
- limiting form §26.10(v), §26.9(iv)
- notation §26.10(i), §26.9(i)
- recurrence relations §26.10(iii), §26.9(iii)
- relation to lattice paths §26.9(i)
- tables Table 26.10.1, Table 26.9.1
resurgence
- asymptotic solutions of differential equations §2.7(ii)
reversion of series §2.2
Riccati–Bessel functions §10.21(xi)
- zeros §10.21(xi)
Riemann hypothesis §25.10(i)
- equivalent statements §25.16(i), §25.17, §27.12
Riemann identity
- Riemann theta functions §21.6(i)
- Riemann theta functions with characteristics §21.6(i)

Riemann matrix §21.1
- computation §21.10(ii)
Riemann surface §21.7
- connection with Riemann theta functions §21.10(ii)—§21.10(ii), §21.7(i)
- cycles §21.7(i)
- definition §21.7(i)
- genus §21.7(i)
- handle §21.7(i)
- holomorphic differentials §21.7(i)
- hyperelliptic §21.7(iii)
- intersection indices §21.1, §21.7(i)
- prime form §21.7(ii)
- representation via Hurwitz system 1st item
- representation via plane algebraic curve 2nd item
- representation via Schottky group 1st item
Riemann theta functions §21.2(i)
- analytic properties §21.2(i)
- applications Ch.21—§21.9
- components §21.2(i)
- definition §21.2(i)
- dimension §21.2(i)
- genus §21.2(i)
- graphics §21.4—§21.4
- modular group §21.5(i)
- modular transformations §21.5(i)—§21.5(ii)
- notation §21.1
- period lattice §21.3(i)
- products §21.6
- quasi-periodicity §21.3(i)
- relation to classical theta functions §21.2(iii)
- Riemann identity §21.6(i)
- scaled §21.2(i)
- symmetry §21.3(i)
Riemann theta functions with characteristics §21.2(ii)
- addition formulas §21.6(ii)
- applications
  - Abelian functions §21.8
- characteristics §21.2(ii)
  - half-period §21.2(ii)
- modular transformations §21.5(ii)
- notation §21.1
- quasi-periodicity §21.3(ii)
- Riemann identity §21.6(i)
- symmetry §21.3(ii)
Riemann zeta function §25.1
- analytic properties §25.2
- applications
  - mathematical §25.16
  - physical §25.17
- approximations 1st item, 2nd item, §25.20
  - asymptotic §25.9
- Chebyshev series 1st item, 2nd item, §25.20
- computation §25.18(i)
- connection with incomplete gamma functions §8.22(ii)
- critical line §25.10(i)
- critical strip §25.10(i)
- definition §25.2(i)
- derivatives §25.2(ii)
  - integer arguments §25.6(ii)
  - series expansions §25.2(ii)
- Euler-product representation §27.4
- graphics §25.3—§25.3
- incomplete §8.22(ii)
- infinite products §25.2(iv)
- integer argument §25.6(i)
- integral representations
  - along the real line §25.5—§25.5(ii)
  - contour integrals §25.5(iii)
- integrals §25.7
- notation §25.1
- recursion formulas §25.6(iii)
- reflection formulas §25.4
- relations to other functions
  - Bernoulli and Euler numbers and polynomials §24.17(iii), §25.6(i)
  - Hurwitz zeta function §25.11(i)
  - polylogarithms §25.12(ii)
- representations by Euler–Maclaurin formula §25.2(iii)
- series expansions §25.2(ii)
- sums §25.8
- tables 3rd item, §25.19
- zeros
  - computation §25.18(ii)
  - counting §25.10(ii), §25.18(i)
  - distribution §25.10(i)
  - on critical line or strip §25.10(i), §25.18(ii)
  - relation to quantum eigenvalues §25.17
  - Riemann hypothesis §25.10(i)
  - trivial §25.10(i)
Riemann–Hilbert problems
- classical orthogonal polynomials §18.38(ii)
Riemann–Lebesgue lemma §1.8(i)
Riemann–Siegel formula §25.10(ii)
- coefficients 3rd item
Riemann’s differential equation
- general form §15.11—§15.11(ii)
- reduction to hypergeometric differential equation §15.11(ii)
- singularities §15.11(i)
- solutions
  - $P$ -symbol notation §15.11(i)
  - transformations §15.11(ii)
Riemann’s $\xi$ -function §25.4
- approximations 2nd item
Riemann’s $P$ -symbol §15.11(i)
ring functions, see toroidal functions.
Ritt’s theorem
- differentiation of asymptotic approximations §2.1(ii)
robot trajectory planning
- Cornu’s spiral §7.21
Rodrigues formulas
- classical orthogonal polynomials item 3., Table 18.5.1
- Hahn class orthogonal polynomials §18.20(i)
Rogers polynomials, see continuous $q$ -ultraspherical polynomials.
Rogers–Ramanujan identities §17.12, §17.2(vi)
- constant term §17.14
- partitions §26.10(iv)
Rogers–Szegő polynomials §18.33(iv)
rolling of ships
- Mathieu functions 2nd item
Romanovski–Bessel polynomials §18.34(ii)
- Morse oscillator §18.39(i)
rook polynomial §26.15
roots
- of equations §3.8
Rossby waves
- biconfluent Heun functions §31.17(ii)
rotation matrices
- relation to $\mathit{3j}$ symbols §34.3(vii)
Rouché’s theorem §1.10(iv), §3.8(v)
round-robin tournaments §27.17
Runge–Kutta methods
- ordinary differential equations §3.7(v), §3.7(v)—§3.7(v)
Rutherford scattering
- Coulomb functions §33.22(ii)
- gamma function §5.20
Rydberg constant
- Coulomb functions §33.22(ii)

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