# Jacobi identity

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##### 2: 27.13 Functions
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.
##### 6: 20.7 Identities
20.7.15 $A\equiv A(\tau)=\ifrac{1}{\theta_{4}\left(0\middle|2\tau\right)},$
20.7.20 $B\equiv B(\tau)=\ifrac{1}{\left(\theta_{3}\left(0\middle|\tau\right)\theta_{4}% \left(0\middle|\tau\right)\theta_{3}\left(\tfrac{1}{4}\pi\middle|\tau\right)% \right)},$
##### 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
22.8.14 $\operatorname{sn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}u\operatorname% {dn}v+\operatorname{sn}v\operatorname{cn}v\operatorname{dn}u}{\operatorname{cn% }u\operatorname{cn}v+\operatorname{sn}u\operatorname{dn}u\operatorname{sn}v% \operatorname{dn}v},$
22.8.15 $\operatorname{cn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}u\operatorname% {dn}v-\operatorname{sn}v\operatorname{cn}v\operatorname{dn}u}{\operatorname{sn% }u\operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v\operatorname{cn}u% \operatorname{dn}u},$
22.8.17 $\operatorname{dn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}v\operatorname% {dn}u-\operatorname{sn}v\operatorname{cn}u\operatorname{dn}v}{\operatorname{sn% }u\operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v\operatorname{cn}u% \operatorname{dn}u},$
22.8.23 $\begin{vmatrix}\operatorname{sn}z_{1}\operatorname{cn}z_{1}&\operatorname{cn}z% _{1}\operatorname{dn}z_{1}&\operatorname{cn}z_{1}&\operatorname{dn}z_{1}\\ \operatorname{sn}z_{2}\operatorname{cn}z_{2}&\operatorname{cn}z_{2}% \operatorname{dn}z_{2}&\operatorname{cn}z_{2}&\operatorname{dn}z_{2}\\ \operatorname{sn}z_{3}\operatorname{cn}z_{3}&\operatorname{cn}z_{3}% \operatorname{dn}z_{3}&\operatorname{cn}z_{3}&\operatorname{dn}z_{3}\\ \operatorname{sn}z_{4}\operatorname{cn}z_{4}&\operatorname{cn}z_{4}% \operatorname{dn}z_{4}&\operatorname{cn}z_{4}&\operatorname{dn}z_{4}\end{% vmatrix}=0.$
For these and related identities see Copson (1935, pp. 415–416). …
##### 9: 22.17 Moduli Outside the Interval [0,1]
22.17.1 $\operatorname{pq}\left(z,k\right)=\operatorname{pq}\left(z,-k\right),$
22.17.2 $\operatorname{sn}\left(z,1/k\right)=k\operatorname{sn}\left(z/k,k\right),$
In terms of the coefficients of the power series of §22.10(i), the above equations are polynomial identities in $k$. …