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)
-
of integers
§27.19
- Faddeeva (or Faddeyeva) function §7.21
- fast Fourier transform §3.11(v)
- Favard’s theorem §18.2(viii)
- Fay’s trisecant identity
-
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
-
of the first kind
-
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
- 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
-
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)
- fold catastrophe §36.2(i), §36.7(i)
- formally self adjoint linear operator
-
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
-
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
-
Fourier–Bessel expansion
-
Bessel functions
§10.23(iii)
- computation §10.74(vii)
-
Bessel functions
§10.23(iii)
- fractals §3.8(viii)
- fractional derivatives §1.15(vii)
- fractional integrals §1.15(vi)
- 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
- 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
-
applications
-
Freud weight
- generalized §18.32
- Freud weight function §18.32
-
Frobenius’ identity
- Riemann theta functions with characteristics §21.7(iii)
- Fuchsian equation
-
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)
