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1: 22.16 Related Functions
§22.16(i) Jacobi’s Amplitude ( am ) Function
§22.16(ii) Jacobi’s Epsilon Function
Integral Representations
§22.16(iii) Jacobi’s Zeta Function
Properties
2: 18.3 Definitions
§18.3 Definitions
Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … The Jacobi polynomials are in powers of x - 1 for n = 0 , 1 , , 6 . … However, most of these formulas can be obtained by specialization of formulas for Jacobi polynomials, via (18.7.4)–(18.7.6). …
3: 22.8 Addition Theorems
22.8.1 sn ( u + v ) = sn u cn v dn v + sn v cn u dn u 1 - k 2 sn 2 u sn 2 v ,
22.8.2 cn ( u + v ) = cn u cn v - sn u dn u sn v dn v 1 - k 2 sn 2 u sn 2 v ,
22.8.3 dn ( u + v ) = dn u dn v - k 2 sn u cn u sn v cn v 1 - k 2 sn 2 u sn 2 v .
22.8.14 sn ( u + v ) = sn u cn u dn v + sn v cn v dn u cn u cn v + sn u dn u sn v dn v ,
22.8.23 | sn z 1 cn z 1 cn z 1 dn z 1 cn z 1 dn z 1 sn z 2 cn z 2 cn z 2 dn z 2 cn z 2 dn z 2 sn z 3 cn z 3 cn z 3 dn z 3 cn z 3 dn z 3 sn z 4 cn z 4 cn z 4 dn z 4 cn z 4 dn z 4 | = 0 .
4: 22.6 Elementary Identities
22.6.2 1 + cs 2 ( z , k ) = k 2 + ds 2 ( z , k ) = ns 2 ( z , k ) ,
22.6.5 sn ( 2 z , k ) = 2 sn ( z , k ) cn ( z , k ) dn ( z , k ) 1 - k 2 sn 4 ( z , k ) ,
22.6.8 cd ( 2 z , k ) = cd 2 ( z , k ) - k 2 sd 2 ( z , k ) nd 2 ( z , k ) 1 + k 2 k 2 sd 4 ( z , k ) ,
§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)
Table 22.6.1: Jacobi’s imaginary transformation of Jacobian elliptic functions.
sn ( i z , k ) = i sc ( z , k ) dc ( i z , k ) = dn ( z , k )
5: 22.21 Tables
§22.21 Tables
Spenceley and Spenceley (1947) tabulates sn ( K x , k ) , cn ( K x , k ) , dn ( K x , k ) , am ( K x , k ) , ( K x , k ) for arcsin k = 1 ( 1 ) 89 and x = 0 ( 1 90 ) 1 to 12D, or 12 decimals of a radian in the case of am ( K x , k ) . … Lawden (1989, pp. 280–284 and 293–297) tabulates sn ( x , k ) , cn ( x , k ) , dn ( x , k ) , ( x , k ) , Z ( x | k ) to 5D for k = 0.1 ( .1 ) 0.9 , x = 0 ( .1 ) X , where X ranges from 1. … Zhang and Jin (1996, p. 678) tabulates sn ( K x , k ) , cn ( K x , k ) , dn ( K x , k ) for k = 1 4 , 1 2 and x = 0 ( .1 ) 4 to 7D. …
6: 22.4 Periods, Poles, and Zeros
For example, the poles of sn ( z , k ) , abbreviated as sn in the following tables, are at z = 2 m K + ( 2 n + 1 ) i K . … Then: (a) In any lattice unit cell p q ( z , k ) has a simple zero at z = p and a simple pole at z = q . (b) The difference between p and the nearest q is a half-period of p q ( z , k ) . This half-period will be plus or minus a member of the triple K , i K , K + i K ; the other two members of this triple are quarter periods of p q ( z , k ) . … For example, sn ( z + K , k ) = cd ( z , k ) . …
7: 20.4 Values at z = 0
20.4.1 θ 1 ( 0 , q ) = θ 2 ( 0 , q ) = θ 3 ( 0 , q ) = θ 4 ( 0 , q ) = 0 ,
Jacobi’s Identity
20.4.6 θ 1 ( 0 , q ) = θ 2 ( 0 , q ) θ 3 ( 0 , q ) θ 4 ( 0 , q ) .
20.4.7 θ 1 ′′ ( 0 , q ) = θ 2 ′′′ ( 0 , q ) = θ 3 ′′′ ( 0 , q ) = θ 4 ′′′ ( 0 , q ) = 0 .
20.4.12 θ 1 ′′′ ( 0 , q ) θ 1 ( 0 , q ) = θ 2 ′′ ( 0 , q ) θ 2 ( 0 , q ) + θ 3 ′′ ( 0 , q ) θ 3 ( 0 , q ) + θ 4 ′′ ( 0 , q ) θ 4 ( 0 , q ) .
8: 22.13 Derivatives and Differential Equations
22.13.1 ( d d z sn ( z , k ) ) 2 = ( 1 - sn 2 ( z , k ) ) ( 1 - k 2 sn 2 ( z , k ) ) ,
22.13.2 ( d d z cn ( z , k ) ) 2 = ( 1 - cn 2 ( z , k ) ) ( k 2 + k 2 cn 2 ( z , k ) ) ,
22.13.3 ( d d z dn ( z , k ) ) 2 = ( 1 - dn 2 ( z , k ) ) ( dn 2 ( z , k ) - k 2 ) .
22.13.4 ( d d z cd ( z , k ) ) 2 = ( 1 - cd 2 ( z , k ) ) ( 1 - k 2 cd 2 ( z , k ) ) ,
22.13.5 ( d d z sd ( z , k ) ) 2 = ( 1 - k 2 sd 2 ( z , k ) ) ( 1 + k 2 sd 2 ( z , k ) ) ,
9: 20.7 Identities
20.7.1 θ 3 2 ( 0 , q ) θ 3 2 ( z , q ) = θ 4 2 ( 0 , q ) θ 4 2 ( z , q ) + θ 2 2 ( 0 , q ) θ 2 2 ( z , q ) ,
20.7.6 θ 4 2 ( 0 , q ) θ 1 ( w + z , q ) θ 1 ( w - z , q ) = θ 3 2 ( w , q ) θ 2 2 ( z , q ) - θ 2 2 ( w , q ) θ 3 2 ( z , q ) ,
20.7.7 θ 4 2 ( 0 , q ) θ 2 ( w + z , q ) θ 2 ( w - z , q ) = θ 4 2 ( w , q ) θ 2 2 ( z , q ) - θ 1 2 ( w , q ) θ 3 2 ( z , q ) ,
20.7.8 θ 4 2 ( 0 , q ) θ 3 ( w + z , q ) θ 3 ( w - z , q ) = θ 4 2 ( w , q ) θ 3 2 ( z , q ) - θ 1 2 ( w , q ) θ 2 2 ( z , q ) ,
20.7.10 θ 1 ( 2 z , q ) = 2 θ 1 ( z , q ) θ 2 ( z , q ) θ 3 ( z , q ) θ 4 ( z , q ) θ 2 ( 0 , q ) θ 3 ( 0 , q ) θ 4 ( 0 , q ) .
10: 18.14 Inequalities
Jacobi
Jacobi
Jacobi
Szegő–Szász Inequality