Index E

ecological systems
- incomplete gamma functions §8.24(i)
eigenfunction expansion
- cosine transform §1.18(vi)
- Hermite differential operator §1.18(v)
- sine transform §1.18(vi)
eigenfunction expansions
- continuous spectra §1.18(vi)
- differential operators §1.18
- discrete spectra §1.18(v)
- matrix elements of the resolvent §1.18(viii)
- mixed spectra §1.18(viii)
eigenfunctions
- completeness relation §1.18(viii)
- continuous spectrum
  - completeness relation §1.18(vi)
  - ortho-normality relation §1.18(vi)
- discrete spectrum
  - completeness relation §1.18(ii), §1.18(v)
  - ortho-normality relation §1.18(ii)
- ortho-normality relation §1.18(viii)
- Sturm-Liouville theory §1.13(viii)
eigenvalue
- multiplicity §1.18(iv)
- Sturm-Liouville theory §1.13(viii)
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(v)
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
EOP’s (exceptional orthogonal polynomials) §18.36(vi)
epsilon function, see Jacobi’s epsilon function.
equation of Ince, see Hill’s equation, equation of Ince.
equiconvergence theorem §18.2(xi)
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.
essentially self-adjoint operator
- mathematical background §1.18(ix)
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
- as Sheffer polynomials §18.2(xii)
- 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)
expansion of arbitrary function
- Hilbert space §1.18(ii)
expansion of arbitrary vector
- Hilbert space §1.18(i)
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
exponential of the trace
- definition §1.2(vi)
extended complex plane §1.9(iv)

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