# Index J

• J-Matrix Theory of Scattering
• Jacobi fraction ($J$-fraction)
• Jacobi function
• applications
• definition
• relations to other functions
• associated Legendre functions
• conical functions
• hypergeometric function
• Jacobi matrix
• truncated
• applications
• Bieberbach conjecture
• associated
• asymptotic approximations
• Bateman-type sums
• computation
• definition
• derivatives
• differential equations
• expansions in series of
• Fourier transform
• generating functions
• graphs
• inequalities
• Bernstein-type
• positive sums
• Szegő–Szász
• Turán-type
• integral representations
• integrals
• fractional
• indefinite
• interrelations with other orthogonal polynomials
• Laplace transform
• limiting form
• as Bessel functions
• as Bessel polynomials
• limits to monomials
• local maxima and minima
• Mellin transform
• monic
• notation
• orthogonality properties
• parameter constraint
• recurrence relations
• relations to other functions
• generalized hypergeometric functions
• hypergeometric function
• orthogonal polynomials on the triangle
• Rodrigues formula
• shifted
• special values
• standardization
• symmetry
• tables of coefficients
• upper bounds
• weight function
• zeros
• asymptotic approximations
• Jacobi symbol
• number theory
• Jacobi transform
• inversion
• Jacobi-type polynomials
• Jacobian elliptic functions
• Jacobi–Anger expansions
• Jacobi’s amplitude function
• Jacobi’s epsilon function
• applications
• computation
• definition
• graphs
• integral representations
• quasi-periodicity
• relation to Legendre’s elliptic integrals
• relation to theta functions
• tables
• Jacobi’s identities
• number theory
• Jacobi’s imaginary transformation
• Jacobi’s inversion problem for elliptic functions
• Jacobi’s nome
• power-series expansion
• Jacobi’s theta functions, see theta functions.
• Jacobi’s triple product
• $q$-version
• Jacobi’s zeta function
• computation
• definition
• graphs
• tables
• Jacobian
• Jacobian elliptic functions
• analytic properties
• applications
• mathematical
• physical
• change of modulus
• computation
• congruent points
• coperiodic
• copolar
• cyclic identities
• notation
• points
• rank
• simultaneously permuted
• definitions
• derivatives
• differential equations
• first-order
• second-order
• double argument
• Eisenstein series
• elementary identities
• equianharmonic case
• expansions in doubly-infinite partial fractions
• Fourier series
• for squares
• fundamental unit cell
• Glaisher’s notation
• graphical interpretation via Glaisher’s notation
• graphics
• complex modulus
• complex variable
• real variable
• half argument
• hyperbolic series for squares
• integrals
• definite
• indefinite
• of squares
• interrelations
• inverse
• Jacobi’s imaginary transformation
• Landen transformations
• ascending
• descending
• generalized
• theta functions
• lattice
• computation
• lemniscatic case
• limiting forms as $k\to 0$ or $k\to 1$
• Maclaurin series
• in $k,k^{\prime}$
• in $z$
• modulus
• change of
• complex
• limiting values
• outside the interval $[0,1]$
• purely imaginary
• real
• nome
• notation
• periods
• poles
• poristic polygon constructions
• principal
• relations to other functions
• symmetric elliptic integrals
• theta functions
• Weierstrass elliptic functions
• rotation of argument
• special values of the variable
• subsidiary
• sums of squares
• tables
• translation of variable
• trigonometric series expansions
• zeros
• Jensen’s inequality for integrals
• Jonquière’s function, see polylogarithms.
• Jordan curve theorem
• Jordan’s function
• number theory
• Jordan’s inequality
• sine function
• Julia sets