Index J

J-Matrix Theory of Scattering §18.39(iv)
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 matrix §18.2(iv)
- truncated §18.2(vi)
Jacobi polynomials §18.3, see also classical orthogonal polynomials.
- applications
  - Bieberbach conjecture §18.38(ii)
- Askey–Gasper inequality §18.38(ii)
- associated §18.30(i)
- asymptotic approximations §18.15(i)—§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 18.4.1, Figure 18.4.2, Figure 18.4.2, Figure 18.4.2
- inequalities §18.14(i), §18.14(iii), §18.14(ii), §18.14(iii)
  - Bernstein-type §18.14(i)
  - positive sums §18.14(iv)
  - Szegő–Szász §18.14(iii)
  - Turán-type §18.14(ii)
- integral representations §18.10(ii), Table 18.10.1
- integrals §18.17(ix), §18.17(i), §18.17(vi), §18.17(iv)
  - fractional §18.17(iv)
  - indefinite §18.17(i)
- interrelations with other orthogonal polynomials Figure 18.21.1, 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)—§18.14(iii)
- Mellin transform §18.17(vii)
- monic §3.5(v)
- notation §18.1(ii)
- orthogonality properties Table 18.3.1
- parameter constraint Table 18.3.1, §18.5(iii)
- recurrence relations §18.9(i), §18.9(i)
- relations to other functions
  - generalized hypergeometric functions §18.38(ii)
  - 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
- standardization Table 18.3.1
- 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 symbol
- number theory §27.9
Jacobi transform §14.31(ii), §15.9(ii)
- inversion §15.9(ii)—§15.9(ii)
Jacobi-type polynomials §18.36(i)
Jacobi–Abel addition theorem
- Jacobian elliptic functions §22.18(iv)—§22.18(iv)
Jacobi–Anger expansions §10.12, §10.35
Jacobi’s amplitude function, see amplitude ( $\operatorname{am}$ ) 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’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

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)—§22.3(iv)
  - complex variable §22.3(iii)—§22.3(iii)
  - real variable §22.3(i)—§22.3(ii)
- half argument §22.6(iii)
- hyperbolic series for squares §22.11—§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 22.3.24, Figure 22.3.25, Figure 22.3.25, Figure 22.3.25, Figure 22.3.26, Figure 22.3.26, Figure 22.3.26, Figure 22.3.27, Figure 22.3.27, Figure 22.3.27, Figure 22.3.28, Figure 22.3.28, Figure 22.3.28, Figure 22.3.29, 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)—§22.4(i)
- poles §22.4(i)—§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)
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)

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