Index E

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

ecological systems
- incomplete gamma functions §8.24(i)
Einstein functions §4.44
Einstein summation convention for vectors §1.6(ii)—§1.6(ii)
Eisenstein convention §23.8(ii)
Eisenstein series
- Jacobian elliptic functions §22.12—§22.12
electric particle field
- Stieltjes electrostatic interpretation §31.15(ii)
electromagnetic scattering
- Bessel functions and spherical Bessel functions §10.73(i)
electromagnetic theory
- sine and cosine integrals §6.17
electromagnetic waves
- Mathieu functions 1st item, 2nd item, 6th item
electron-ion collisions
- Coulomb functions §33.22(ii)
electronic structure of heavy elements
- Coulomb functions §33.22(iv)
electrostatics
- Jacobian elliptic functions §22.19(v)
- zeros of classical orthogonal polynomials §18.39(ii)
elementary functions, see exponential function, hyperbolic functions, inverse hyperbolic functions, inverse trigonometric functions, Lambert $W$ -function, logarithm function, power function, and trigonometric functions.
- relation to $R_{C}$ -function §19.10(ii)
elementary particle physics
- conical functions §14.31(ii)
ellipse
- elliptic integrals §19.30(i)
ellipse arc length
- Jacobian elliptic functions §22.18(i)
ellipsoid
- capacity §19.33(ii)
- depolarization §19.15, §19.33(iii)
- potential §19.33(ii)
- self-energy §19.33(iv)
- surface area §19.33(i)
- triaxial §19.33
ellipsoidal coordinates §29.18(ii)
- Weierstrass elliptic functions §23.21(iii)
ellipsoidal harmonics
- Lamé polynomials §29.18(iii)
ellipsoidal wave equation, see Lamé wave equation.
elliptic coordinates §31.17(i)
elliptic crack and punch problems
- Lamé polynomials §29.19(ii)
elliptic curves §22.18(iv)
- addition law §23.20(ii)
- definition §23.20(ii)
- Jacobi–Abel addition theorem §22.18(iv)—§22.18(iv)
- Jacobian normal form §22.18(iv)
- Mordell’s theorem §23.20(ii)
- tables §23.20(ii)
- Weierstrass elliptic functions §23.20(ii)
elliptic functions, see also Jacobian elliptic functions and Weierstrass elliptic functions.
- general §23.2(iii)
  - representation as Weierstrass elliptic functions §23.6(iii)
- Weierstrass, see Weierstrass elliptic functions.
elliptic integrals, see basic elliptic integrals, Bulirsch’s elliptic integrals, general elliptic integrals, generalizations of elliptic integrals, Legendre’s elliptic integrals, and symmetric elliptic integrals.
- complete
  - quasiconformal mapping §15.17(ii)
- relations to other functions
  - associated Legendre functions §14.5(v)
  - Ferrers functions §14.5(v)
  - Weierstrass elliptic functions §23.6(iv)
elliptic modular function, see modular functions.
elliptic umbilic bifurcation set
- formula §36.4(i)
- picture §36.4(ii)
elliptic umbilic canonical integral §36.2(i)
- asymptotic approximations §36.11—§36.12(iii)
- convergent series §36.8
- differential equations §36.10(iii)
- formulas for Stokes set §36.5(iii)
- integral identity (36.9.9)
- picture of Stokes set Figure 36.5.8, Figure 36.5.8, Figure 36.5.8
- pictures of modulus Figure 36.3.6, Figure 36.3.6, Figure 36.3.6, Figure 36.3.7, Figure 36.3.7, Figure 36.3.7, Figure 36.3.8, Figure 36.3.8, Figure 36.3.8
- pictures of phase Figure 36.3.15, Figure 36.3.15, Figure 36.3.16, Figure 36.3.16, Figure 36.3.17, Figure 36.3.17
- scaling laws §36.6
- zeros §36.7(iii)
elliptic umbilic catastrophe §36.2(i), Figure 36.5.5, Figure 36.5.5
elliptical coordinates
- Mathieu functions §28.32(i)—§28.32(i)
entire functions §1.9(ii)
- asymptotic expansions §2.10(iii)
- Liouville’s theorem §1.9(iii)
enumerative topology
- Painlevé transcendents §32.16
epsilon function, see Jacobi’s epsilon function.
equation of Ince, see Hill’s equation, equation of Ince.
equiconvergent §30.4(iv)
Erlang loss function
- incomplete gamma functions §8.23
error-control function
- differential equations §2.7(iii)
error functions §7.2(i)
- applications
  - asymptotic approximation of integrals §7.20(i)
  - physics §7.21—§7.21
  - statistics §7.20(iii)
  - Stokes phenomenon §7.20(i)
- approximations §7.24(i), 1st item, 2nd item, 3rd item, §7.24(ii)
- asymptotic expansions §7.12(i)
  - exponentially-improved §7.12(i)
- computation §3.5(ix)—§7.22(i)
- continued fractions §7.9
- definitions §7.2(i)
- derivatives §7.10
- expansions in spherical Bessel functions §7.6(ii)
- generalized §7.16
- graphics Figure 7.3.1, Figure 7.3.1, Figure 7.3.1, Figure 7.3.5, Figure 7.3.5, Figure 7.3.5, Figure 7.3.6, Figure 7.3.6, Figure 7.3.6
- inequalities §7.8
- integral representations §7.7(i)—§7.7(i)
- integrals
  - Fourier transform §7.14(i)
  - Laplace transforms §7.14(i)
- interrelations §7.5
- inverse functions §7.17(i)
  - asymptotic expansions §7.17(iii)
  - computation §7.22(i)
  - power-series expansions §7.17(ii)
- notation §7.1
- power-series expansions §7.6(i)
- relations to other functions
  - confluent hypergeometric functions §13.18(ii), §13.6(ii), §7.11
  - Dawson’s integral §7.5
  - Fresnel integrals §7.5
  - generalized exponential integrals §7.11
  - incomplete gamma functions §7.11
  - parabolic cylinder functions §12.7(ii)
  - probability functions §7.1
  - Voigt functions §7.19(i)
- repeated integrals of, see repeated integrals of the complementary error function.
- sums §7.15
- tables 3rd item, 1st item, §7.23(ii), §7.23(iii)
- values at infinity §7.2(i)
- zeros §7.13(i)—§7.13(ii)
  - asymptotic expansions §7.13(i)
  - tables Table 7.13.1, Table 7.13.2, 1st item, §7.23(iv)
error measures
- absolute error §3.1(v)
- complex arithmetic §3.1(v)
- mollified error §3.1(v)
- relative error §3.1(v)
- relative precision §3.1(v)
error term §2.3(i)
essential singularity §1.10(iii), see also isolated essential singularity.
eta function, see Dedekind’s eta function.
Euler numbers §24.1
- arithmetic properties §24.10
- asymptotic approximations §24.11
- computation §24.19(i)
- definition §24.2(ii)
- explicit formulas §24.6—§24.6
- factors §24.10(iv)
- finite expansions §24.4(iv)
- generalizations §24.16, §24.16(iii)
- generating function §24.2(ii)
- identities §24.5(iii)
- inequalities §24.9
- integral representation §24.7(i)
- inversion formulas §24.5(iii)
- Kummer congruences §24.10(ii)
- notation §24.1
- recurrence relations
  - linear §24.5
  - quadratic and higher order §24.14(i), §24.14(ii)
- sums §24.14
- tables Table 24.2.1, §24.20

Euler polynomials §24.1
- applications
  - mathematical §24.17—§24.17(iii)
  - physical §24.18
- asymptotic approximations §24.11
- computation §24.19(i)
- definition §24.2(ii)
- derivative §24.4(vii)
- difference equation §24.4(i)
- explicit formulas §24.6
- finite expansions §24.4(iv)
- generalized §24.16, §24.16(iii)
- generating function §24.2(ii)
- graphs Figure 24.3.2, Figure 24.3.2, Figure 24.3.2
- inequalities §24.9
- infinite series expansions
  - Fourier §24.8(i)
  - other (24.8.9)
- integral representations §24.7(ii)
- integrals §24.13(ii)
  - compendia §24.13(iii)
- Laplace transforms §24.13(iii)
- multiplication formulas §24.4(v)
- notation §24.1
- recurrence relations
  - linear §24.5
  - quadratic §24.14(i)
- representations as sums of powers §24.4(iii)
- special values §24.4(vi)
- sums §24.14
- symbolic operations §24.4(viii)
- symmetry §24.4(ii)
- tables Table 24.2.2
- zeros
  - complex §24.12(iii)
  - multiple §24.12(iv)
  - real §24.12(ii)
Euler product
- number theory §27.4
Euler splines §24.17(ii)
Euler sums
- Riemann zeta function §25.16(ii)
  - reciprocity law §25.16(ii)
Euler–Fermat theorem
- number theory §27.16, §27.2(i)
Euler–Maclaurin formula §2.10(i)
- extensions §2.10(i)
- generalization §24.17(i)
Euler–Poisson differential equations §19.18(ii)
Euler–Poisson–Darboux equation
- symmetric elliptic integrals §19.18(ii)
Euler–Tricomi equation
- Airy functions §9.16
Euler’s beta integral §5.12
Euler’s constant §5.2(ii)
- integral representations §5.9(ii)
Euler’s homogeneity relation
- symmetric elliptic integrals §19.18(ii)
Euler’s integral
- gamma function §5.2(i)
Euler’s pentagonal number theorem
- number theory §27.14(ii)
Euler’s sums
- $q$ -hypergeometric function §17.5, §17.5
Euler’s totient
- number theory §27.2(i)
Euler’s transformation
- applied to asymptotic expansions §2.11(vi)—§2.11(vi)
- of series §3.9(ii)
Eulerian numbers
- definition §26.14(i)
- generating functions §26.14(ii)
- identities §26.14(iii)
- notation §26.14(i)
- relation to Bernoulli numbers §24.4(ix)
- relation to permutations §26.14(i)
- special values §26.14(iv)
- table Table 26.14.1
evolution equations
- Lamé polynomials §29.19(ii)
exact rational arithmetic §3.1(iii)
exponential function §4.2(iii)
- analytic properties §4.2(iii)
- approximations §4.47
- Chebyshev-series expansions §4.47(i)
- computation §4.45(i)
- conformal maps §4.3(ii)
- continued fractions §4.9(ii)
- definition §4.2(iii)
- derivatives §4.7(ii)
- differential equation §4.7(ii)
- generalized §4.12
- graphics
  - complex argument Figure 4.3.4, Figure 4.3.4, Figure 4.3.4
  - real argument Figure 4.3.1, Figure 4.3.1, Figure 4.3.1
- identities §4.8(ii)
- inequalities §4.5(ii)
- integrals §4.10(ii)
- limits §4.4(iii)
- notation §4.1
- periodicity §4.2(iii)
- power series §4.2(iii)
- special values §4.4(ii)
- sums §4.11
- tables §4.46
- zeros §4.2(iii)
exponential growth §1.14(iii)
exponential integrals §6.2(i)
- analytic continuation §6.4
- applications §6.17
- approximations 1st item, 2nd item, 3rd item
- asymptotic expansions §6.12(i)
  - exponentially-improved §6.12(i)
  - re-expansion of remainder term §6.12(i)
- Chebyshev-series expansions 1st item, 2nd item, 3rd item, 4th item, 5th item
- computation §6.18
- continued fraction §6.9
- definition §6.2(i)
- expansion in inverse factorials §6.10(i)
- expansions in modified spherical Bessel functions §6.10(ii)
- generalized §8.19
- graphics Figure 6.3.1, Figure 6.3.1, Figure 6.3.3, Figure 6.3.3, Figure 6.3.3
- inequalities §6.8
- integral representations §6.7—§6.7(i)
- integrals §6.14(i), §6.14(ii)
- interrelations §6.2(i), §6.5
- Laplace transform §6.14(i)
- notation §6.1
- power series §6.6
- principal value §6.2(i)
- relations to other functions
  - confluent hypergeometric functions §6.11
  - incomplete gamma function §6.11
  - logarithmic integral §6.2(i)
  - sine and cosine integrals §6.5
- small argument §2.5(iii)
- tables 1st item, 2nd item, 1st item, 2nd item
- zeros §6.13
extended complex plane §1.9(iv)