Index F

♦A♦B♦C♦D♦E♦F♦G♦H♦I♦J♦K♦L♦M♦N♦O♦P♦Q♦R♦S♦T♦U♦V♦W♦Z♦

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)
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(ii)
- 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)
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 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)
- Bessel’s inequality §1.8(i)
- 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(iv)
- 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 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)