Index F

Faà di Bruno’s formula
- for derivatives §1.4(iii)
Fabry’s transformation
- differential equations §2.7(ii)
factorials (rising or falling) §26.1
factorization
- of integers §27.19
  - via Weierstrass elliptic functions §23.20(iii)
Faddeeva (or Faddeyeva) function §7.21
- definition (7.2.3), §7.2(i)
fast Fourier transform §3.11(v)
Favard’s theorem §18.2(viii)
Fay’s trisecant identity
- Riemann theta functions with characteristics §21.7(ii)
  - generalizations §21.7(ii)
Fejér kernel
- Fourier integral §1.15(v)
- Fourier series §1.15(iii)
Fermat numbers
- number theory §27.18
Fermat’s last theorem
- Bernoulli and Euler numbers and polynomials §24.17(iii)
Fermi–Dirac integrals
- approximations 4th item
- computation §25.18(i)
- definition §25.12(iii)
- relation to polylogarithms §25.12(iii)
- tables 2nd item
- uniform asymptotic approximation §25.12(iii)
Ferrers board §26.15
Ferrers function
- of the first kind
  - integral equation for Lamé functions §29.8
Ferrers functions §14.1
- addition theorems §14.18(ii)
- analytic continuation §14.23
- applications
  - spherical harmonics §14.30—§14.30(iv)
  - spheroidal harmonics §14.30
- asymptotic approximations, see uniform asymptotic approximations.
- behavior at singularities §14.8—§14.8(iii)
- computation §14.32
- connection formulas §14.9(i), §14.9(ii)
- cross-products §14.2(iv)
- definitions §14.3—§14.3(iii)
- degree §14.1
- derivatives §14.10
  - with respect to degree or order §14.11
- differential equation, see associated Legendre equation.
- generating functions §14.7(iv)
- graphics Figure 14.4.16, Figure 14.4.16, Figure 14.4.16, §14.4(i)—§14.4(ii)
- integer degree and order §14.7—§14.7(iv)
- integer order §14.6—§14.6(ii)
- integral representations §14.12(i)—§14.12(i)
- integrals
  - definite §14.17(ii), §14.17(iii), §14.17(iv)
  - indefinite §14.17(i)
  - Laplace transforms §14.17(v)
  - Mellin transforms §14.17(vi)
  - orthogonality properties §14.17(iii)
- notation §14.1
- of the first kind §14.3(i)
- of the second kind §14.3(i)
- order §14.1
- orthogonality §14.17(iii)
- recurrence relations §14.10
- reflection formulas §14.7(iii)
- relations to other functions
  - elliptic integrals §14.5(v)
  - hypergeometric function §14.3(i), §14.3(iii), §15.9(iv)
  - Legendre polynomials §14.7(i)
  - ultraspherical polynomials §18.11(i)
- Rodrigues-type formulas §14.7(ii)
- special values §14.5—§14.5(v)
- sums §14.18—§14.18(iv)
- tables §14.33
- trigonometric expansions §14.13
- uniform asymptotic approximations
  - large degree §14.15(iii)—§14.15(v)
  - large order §14.15(i)—§14.15(ii)
- Wronskians §14.2(iv), §14.2(iv)—§14.2(iv)
- zeros §14.16(ii)
Ferrers graph §26.9(i)
Feynman diagrams
- Appell functions §16.24(ii)
Feynman path integrals
- theta functions §20.13
Fibonacci numbers §24.15(iv), §26.11
fine structure constant
- Coulomb functions §33.22(i)
finite Fourier series
- number theory §27.10
fixed point §3.8(i)
floating-point arithmetic
- bits §3.1(i)
  - format width §3.1(i)
  - significant §3.1(i)
- double precision §3.1(i)
- exponent §3.1(i)
- fractional part §3.1(i)
- IEEE standard §3.1(i)
- machine epsilon §3.1(i)
- machine number §3.1(i)
- machine precision §3.1(i)
- overflow §3.1(i)
- rounding
  - by chopping §3.1(i)
  - down §3.1(i)
  - symmetric §3.1(i)
  - to nearest machine number §3.1(i)
- significand §3.1(i)
- single precision §3.1(i)
- underflow §3.1(i)
Floquet solutions
- Hill’s equation §28.29(ii)
- Mathieu’s equation §28.2(iv)
Floquet’s theorem
- Hill’s equation §28.29(ii)
- Mathieu’s equation §28.2(iii)
fluid dynamics
- elliptic integrals §19.35(ii)
- Legendre polynomials §18.39(v)
- Riemann theta functions §21.9
- Struve functions §11.12
fold canonical integral §36.2(i), §36.7(i)
- bifurcation set §36.4(i)
- differential equation §36.10(i)
- integral identity §36.9
- relation to Airy function §36.2(ii)
- zeros §36.7(i)
fold catastrophe §36.2(i), §36.7(i)
formally self adjoint linear operator
- linear second order differential operator §1.18(iv), §1.18(iv)
- Schrödinger operator §1.18(iv)
formally self-adjoint differential operators
- Hilbert space §1.18(iii)
formally self-adjoint operators
- Hilbert space §1.18(iii)

Fourier cosine and sine transforms
- definition §1.14(ii)
- generalized §15.17(iii)
- inversion §1.14(ii)
- Parseval’s formula §1.14(ii)
- tables Table 1.14.2, Table 1.14.3, §1.14(viii)
Fourier eigenfunction expansion
- discrete spectrum §1.18(v)
Fourier integral
- asymptotic expansions §2.3(i), §2.3(iv)
- Dirac delta §1.17(ii)
- Fejér kernel §1.15(v)
- Poisson kernel §1.15(v)
- summability §1.15(v)
Fourier series §1.8—§1.8(v)
- coefficients §1.8(i)
- compendia §1.8(v)
- convergence §1.8(ii)
- definition §1.8(i)
- differentiation §1.8(iii)
- Dirac delta §1.17(ii)
- Fejér kernel §1.15(iii)
- finite
  - number theory §27.10
- integration §1.8(iii)
- Parseval’s Formula §1.8(i)
- Poisson kernel §1.15(v)
- Poisson’s summation formula §1.8(iv)—§1.8(iv)
- properties §1.8(i)
- summability §1.15(iii)—§1.15(iii)
- uniqueness §1.8(i)
Fourier transform §1.14(i)—§1.14(ii)
- convergence §1.14(i)
- convolution §1.14(i)
- definitions §1.14(i)
- discrete §3.11(v)
- distributions §1.16(vii)
- fast §3.11(v)
- group
  - hypergeometric function §15.17(iii)
- inversion §1.14(i)
- Parseval’s formula §1.14(i)
- special distributions §1.16(viii)
  - Dirac delta distribution §1.16(viii)
  - Heaviside function §1.16(viii)
  - sign function §1.16(viii)
- tables Table 1.14.1, §1.14(viii)
- tempered distributions §1.16(vii)
- uniqueness §1.14(i)
Fourier-series expansions
- nonuniformity of convergence §6.16(i)
- piecewise continuous functions §6.16(i)
Fourier–Bessel expansion
- Bessel functions §10.23(iii)
  - computation §10.74(vii)
fractals §3.8(viii)
fractional derivatives §1.15(vii)
fractional integrals §1.15(vi)
- asymptotic expansions §2.6(iii)—§2.6(iii)
- definition §2.6(iii)
fractional linear transformation, see bilinear transformation.
Fresnel integrals §7.2(iii)
- applications
  - Cornu’s spiral §7.20(ii)
  - interference patterns Figure 7.3.4, Figure 7.3.4, Figure 7.3.4
  - physics and astronomy §7.21
  - probability theory §7.20(iii)
  - statistics §7.20(iii)
- approximations 2nd item, §7.24(ii)
- asymptotic expansions §7.12(ii)
  - exponentially-improved §7.12(ii)
- auxiliary functions, see auxiliary functions for Fresnel integrals.
- computation §7.22(i)
- definition §7.2(iii)
- expansions in spherical Bessel functions §7.6(ii)
- graphics Figure 7.3.3, Figure 7.3.3, Figure 7.3.3
- integrals
  - Laplace transforms §7.14(ii)
- interrelations §7.5
- notation §7.1
- power-series expansions §7.6(i)
- relations to other functions
  - Anger–Weber functions §11.10(vi)
  - auxiliary functions §7.2(iv), §7.5
  - confluent hypergeometric functions §7.11
  - error functions §7.5
  - generalized hypergeometric functions §7.11
- symmetry §7.4
- tables §7.23(ii)
- values at infinity §7.2(iii)
- zeros §7.13(iii), §7.13(iv)
  - asymptotic expansions §7.13(iii), §7.13(iv)
  - tables Table 7.13.3, Table 7.13.4
Freud weight
- generalized §18.32
Freud weight function §18.32
Frobenius’ identity
- Riemann theta functions with characteristics §21.7(iii)
Fuchsian equation
- classification of parameters §31.14(i)
- definitions §31.14(i)
- normal form §31.14(i)
- polynomial solutions §31.15(i)
- relation to Heun’s equation §31.14(i)
functions
- analytic, see analytic function.
- analytically continued §1.10(ii)
- continuous, see continuous function.
- continuously differentiable §1.4(iii), §1.5(i)
- convex §1.4(viii)
- decreasing §1.4(i)
- defined by contour integrals §1.10(viii)
- differentiable §1.4(iii)
- entire, see entire functions.
- harmonic §1.9(ii)
- holomorphic, see analytic function.
- increasing §1.4(i)
- inverse §1.10(vii)
- limits, see limits of functions.
- many-valued, see multivalued function.
- meromorphic §1.10(iii)
- monotonic §1.4(i)
- multivalued, see multivalued function.
- nondecreasing §1.4(i)
- nonincreasing §1.4(i)
- of a complex variable §1.10—§1.10(x)
- of bounded variation §1.4(v)
- of compact support §1.16(i)
- of matrix argument, see functions of matrix argument.
- strictly decreasing §1.4(i)
- strictly increasing §1.4(i)
- strictly monotonic §1.4(i)
- support of §1.16(i)
- vector-valued §1.6(iii)—§1.6(iv)
functions of matrix argument §35.1
- Laplace transform §35.2
- orthogonal invariance §35.7(iii)
fundamental theorem of arithmetic §27.2(i)
fundamental theorem of calculus §1.4(v)

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov