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1: 20.4 Values at z = 0
Jacobi’s Identity
2: 27.13 Functions
27.13.5 ( ϑ ( x ) ) 2 = 1 + n = 1 r 2 ( n ) x n .
One of Jacobi’s identities implies that … Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities. …
3: Bibliography K
  • A. Khare, A. Lakshminarayan, and U. Sukhatme (2003) Cyclic identities for Jacobi elliptic and related functions. J. Math. Phys. 44 (4), pp. 1822–1841.
  • A. Khare and U. Sukhatme (2002) Cyclic identities involving Jacobi elliptic functions. J. Math. Phys. 43 (7), pp. 3798–3806.
  • 4: Bibliography B
  • J. M. Borwein and P. B. Borwein (1991) A cubic counterpart of Jacobi’s identity and the AGM. Trans. Amer. Math. Soc. 323 (2), pp. 691–701.
  • 5: 20.3 Graphics
    See accompanying text
    Figure 20.3.18: θ 1 ( 0.1 | u + i v ) , - 1 u 1 , 0.005 v 0.5 . …1 of z is chosen arbitrarily since θ 1 vanishes identically when z = 0 . Magnify 3D Help
    6: 20.7 Identities
    20.7.15 A A ( τ ) = 1 / θ 4 ( 0 | 2 τ ) ,
    20.7.20 B B ( τ ) = 1 / ( θ 3 ( 0 | τ ) θ 4 ( 0 | τ ) θ 3 ( 1 4 π | τ ) ) ,
    7: 20.11 Generalizations and Analogs
    This is the discrete analog of the Poisson identity1.8(iv)). … This is Jacobi’s inversion problem of §20.9(ii). … Each provides an extension of Jacobi’s inversion problem. …
    8: 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 ,
    22.8.14 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 .
    For these and related identities see Copson (1935, pp. 415–416). …
    9: 22.17 Moduli Outside the Interval [0,1]
    22.17.1 p q ( z , k ) = p q ( z , - k ) ,
    22.17.2 sn ( z , 1 / k ) = k sn ( z / k , k ) ,
    22.17.3 cn ( z , 1 / k ) = dn ( z / k , k ) ,
    22.17.4 dn ( z , 1 / k ) = cn ( z / k , k ) .
    In terms of the coefficients of the power series of §22.10(i), the above equations are polynomial identities in k . …
    10: 22.9 Cyclic Identities
    §22.9 Cyclic Identities
    §22.9(ii) Typical Identities of Rank 2
    §22.9(iii) Typical Identities of Rank 3