Index J

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

Jacobi–Abel addition theorem
- Jacobian elliptic functions §22.18(iv), §22.18(iv)
Jacobian §1.5(vi)
Jacobian elliptic functions §22.2
- addition theorems §22.8, §22.8(iii)
- analytic properties §22.17(ii), §22.2
- applications
  - mathematical §22.18, §22.18(iv)
  - physical §22.19(i), §22.19(v)
- change of modulus §22.17
- computation §22.20, §22.20(vii)
- congruent points §22.4(i)
- coperiodic §22.4(i)
- copolar §22.4(i)
- cyclic identities
  - notation §22.9(i)
  - points §22.9(i)
  - rank §22.9(i)
  - simultaneously permuted ¶ ‣ §22.9(ii)
- definitions §22.2
- derivatives §22.13(i)
- differential equations
  - first-order §22.13(ii)
  - second-order §22.13(iii)
- double argument §22.6(ii)
- Eisenstein series §22.12, §22.12
- elementary identities §22.6, §22.6(v)
- equianharmonic case §22.5(ii)
- expansions in doubly-infinite partial fractions §22.12, §22.12
- Fourier series §22.11
  - for squares §22.11
- fundamental unit cell §22.4(ii)
- Glaisher’s notation §22.1, §22.4(ii)
- graphical interpretation via Glaisher’s notation §22.4(ii)
- graphics
  - complex modulus §22.3(iv)
  - complex variable §22.3(iii)
  - real variable §22.3(i), §22.3(ii)
- half argument §22.6(iii)
- hyperbolic series for squares §22.11
- integrals
  - definite §22.14(iv)
  - indefinite §22.14(i), §22.14(iii)
  - of squares ¶ ‣ §22.16(ii), ¶ ‣ §22.16(ii)
- interrelations §19.25(v)
- inverse, see inverse Jacobian elliptic functions.
- Jacobi’s imaginary transformation §22.6(iv)
- Landen transformations
  - ascending §22.17(ii), §22.20(iii), §22.7(ii)
  - descending §22.17(ii), §22.20(iii), §22.7(i)
  - generalized §22.7(iii)
  - theta functions §20.7(vi)
- lattice §22.4(ii)
  - computation §22.20(iv)
- lemniscatic case §22.5(ii)
- limiting forms as $k\to 0$ or $k\to 1$ §22.5(ii), §22.5(ii)
- Maclaurin series
  - in $k,k^{{\prime}}$ §22.10(ii), §22.10(ii)
  - in $z$ §22.10(i)
- modulus §22.2
  - change of §22.17, §22.17(ii)
  - complex §22.17(ii), Figure 22.3.24, Figure 22.3.24, Figure 22.3.25, Figure 22.3.25, Figure 22.3.26, Figure 22.3.26, Figure 22.3.27, Figure 22.3.27, Figure 22.3.28, Figure 22.3.28, Figure 22.3.29, Figure 22.3.29
  - limiting values §22.5(ii)
  - outside the interval $[0,1]$ §22.17, §22.17(ii)
  - purely imaginary §22.17(i)
  - real §22.17(i)
- nome §22.2
- notation §22.1
- periods §22.2, §22.4(i)
- poles §22.4(i)
- poristic polygon constructions §22.8(iii)
- principal §22.1
- relations to other functions
  - symmetric elliptic integrals §19.25(v)
  - theta functions §22.2
  - Weierstrass elliptic functions §23.6(ii)
- rotation of argument §22.6(iv)
- special values of the variable §22.5, §22.5(i), §22.8(iii)
- subsidiary §22.1
- sums of squares §22.6(i)
- tables §22.21
- translation of variable §22.4(iii)
- trigonometric series expansions §22.11, §22.12
- zeros §22.4(i), §22.4(i)
Jacobi fraction ( $J$ -fraction) ¶ ‣ §3.10(ii)
Jacobi function
- applications §15.17(iii)
- definition §15.9(ii)
- relations to other functions
  - associated Legendre functions §14.3(iv)
  - conical functions §14.31(ii)
  - hypergeometric function §15.9(ii)

Jacobi polynomials §18.3, see also classical orthogonal polynomials.
- applications
  - Bieberbach conjecture ¶ ‣ §18.38(ii)
- associated ¶ ‣ §18.30
- asymptotic approximations §18.15(i)
- Bateman-type sums ¶ ‣ §18.18(vi)
- computation Ch.18
- definition Table 18.3.1
- derivatives ¶ ‣ §18.9(iii)
- differential equations Table 18.8.1
- expansions in series of §18.18, ¶ ‣ §18.18(vi)
- Fourier transform ¶ ‣ §18.17(v)
- generating functions ¶ ‣ §18.12
- graphs Figure 18.4.1, Figure 18.4.1, Figure 18.4.2, Figure 18.4.2
- inequalities ¶ ‣ §18.14(i), ¶ ‣ §18.14(iii), ¶ ‣ §18.14(ii), ¶ ‣ §18.14(iii)
  - Szegö–Szász ¶ ‣ §18.14(iii)
  - Turan-type ¶ ‣ §18.14(ii)
- integral representations ¶ ‣ §18.10(ii), Table 18.10.1
- integrals ¶ ‣ §18.17(i), ¶ ‣ §18.17(vi), ¶ ‣ §18.17(iv), §18.17(ix)
  - fractional ¶ ‣ §18.17(iv)
  - indefinite ¶ ‣ §18.17(i)
- interrelations with other orthogonal polynomials Figure 18.21.1, Figure 18.21.1, ¶ ‣ §18.21(ii), §18.7, ¶ ‣ §18.7(iii)
- Laplace transform ¶ ‣ §18.17(vi)
- leading coefficients Table 18.3.1
- limiting form
  - as Bessel functions ¶ ‣ §18.11(ii)
  - as Bessel polynomials §18.34(iii)
- limits to monomials §18.6(ii)
- local maxima and minima ¶ ‣ §18.14(iii), ¶ ‣ §18.14(iii)
- Mellin transform ¶ ‣ §18.17(vii)
- monic ¶ ‣ §3.5(v)
- normalization Table 18.3.1
- notation ¶ ‣ §18.1(ii)
- orthogonality properties Table 18.3.1
- parameter constraint Table 18.3.1, ¶ ‣ §18.5(iii)
- recurrence relations §18.9(i)
- relations to other functions
  - hypergeometric function ¶ ‣ §15.9(i), §18.5(iii)
  - orthogonal polynomials on the triangle ¶ ‣ §18.37(ii)
- Rodrigues formula Table 18.5.1
- shifted §18.1(iii)
- special values Table 18.6.1
- symmetry Table 18.6.1
- tables of coefficients §18.3
- upper bounds ¶ ‣ §18.14(i)
- weight function Table 18.3.1
- zeros §18.16(ii), §18.2(vi)
  - asymptotic approximations ¶ ‣ §18.16(ii)
Jacobi’s amplitude function, see amplitude ( $\mathop{\mathrm{am}\/}\nolimits$ ) function.
Jacobi’s epsilon function §22.16(ii)
- applications ¶ ‣ §22.18(i)
- computation §22.20(vi)
- definition ¶ ‣ §22.16(ii)
- graphs §22.16(iv)
- integral representations ¶ ‣ §22.16(ii), ¶ ‣ §22.16(ii)
- quasi-addition formula ¶ ‣ §22.16(ii)
- quasi-periodicity ¶ ‣ §22.16(ii)
- relation to Legendre’s elliptic integrals ¶ ‣ §22.16(ii)
- relation to theta functions ¶ ‣ §22.16(ii)
- tables §22.21
Jacobi’s identities
- number theory §27.13(iv)
Jacobi’s imaginary transformation §22.6(iv)
Jacobi’s inversion problem for elliptic functions §20.9(ii)
Jacobi’s nome
- power-series expansion §19.5
Jacobi’s theta functions, see theta functions.
Jacobi’s triple product ¶ ‣ §20.4(i)
- $q$ -version ¶ ‣ §17.8
Jacobi symbol
- number theory §27.9
Jacobi’s zeta function §22.16(iii)
- computation §22.20(vi)
- definition ¶ ‣ §22.16(iii)
- graphs §22.16(iv)
- quasi-addition formula ¶ ‣ §22.16(iii)
- tables §22.21
Jacobi transform §14.31(ii), §15.9(ii)
- inversion §15.9(ii)
Jacobi-type polynomials §18.36(i)
Jensen’s inequality for integrals §1.7(iv)
Jonquière’s function, see polylogarithms.
Jordan curve theorem ¶ ‣ §1.9(iii)
Jordan’s function
- number theory §27.2(i)
Jordan’s inequality
- sine function ¶ ‣ §4.18
Julia sets §3.8(viii)