# Jacobi imaginary transformation

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## 11—20 of 23 matching pages

##### 11: 23.6 Relations to Other Functions
23.6.12 $\sigma\left(\omega_{3}\right)=2i\omega_{1}\frac{\exp\left(\tfrac{1}{2}\eta_{1}% \omega_{1}\tau^{2}\right)\theta_{4}\left(0,q\right)}{\pi q^{1/4}\theta_{1}'% \left(0,q\right)}.$
Also, $\mathbb{L}_{\mspace{1.0mu }1}$, $\mathbb{L}_{\mspace{1.0mu }2}$, $\mathbb{L}_{\mspace{1.0mu }3}$ are the lattices with generators $(4K,2\mathrm{i}{K^{\prime}})$, $(2K-2\mathrm{i}{K^{\prime}},2K+2\mathrm{i}{K^{\prime}})$, $(2K,4\mathrm{i}{K^{\prime}})$, respectively. …
23.6.29 $\zeta\left(z|\mathbb{L}_{\mspace{1.0mu }3}\right)-\zeta\left(z+2\mathrm{i}{K^{% \prime}}|\mathbb{L}_{\mspace{1.0mu }3}\right)-\zeta\left(2\mathrm{i}{K^{\prime% }}|\mathbb{L}_{\mspace{1.0mu }3}\right)=\operatorname{cs}\left(z,k\right).$
Similar results for the other nine Jacobi functions can be constructed with the aid of the transformations given by Table 22.4.3. For representations of the Jacobi functions $\operatorname{sn}$, $\operatorname{cn}$, and $\operatorname{dn}$ as quotients of $\sigma$-functions see Lawden (1989, §§6.2, 6.3). …
##### 12: 22.17 Moduli Outside the Interval [0,1]
###### §22.17(i) Real or Purely Imaginary Moduli
Jacobian elliptic functions with real moduli in the intervals $(-\infty,0)$ and $(1,\infty)$, or with purely imaginary moduli are related to functions with moduli in the interval $[0,1]$ by the following formulas. …
22.17.7 $\operatorname{cn}\left(z,ik\right)=\operatorname{cd}\left(z/k_{1}^{\prime},k_{% 1}\right),$
###### §22.17(ii) Complex Moduli
In particular, the Landen transformations in §§22.7(i) and 22.7(ii) are valid for all complex values of $k$, irrespective of which values of $\sqrt{k}$ and $k^{\prime}=\sqrt{1-k^{2}}$ are chosen—as long as they are used consistently. …
##### 14: 31.7 Relations to Other Functions
They are analogous to quadratic and cubic hypergeometric transformations (§§15.8(iii)15.8(v)). … Joyce (1994) gives a reduction in which the independent variable is transformed not polynomially or rationally, but algebraically. … With $z={\operatorname{sn}}^{2}\left(\zeta,k\right)$ and …The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities $\zeta=K$, $K+i{K^{\prime}}$, and $i{K^{\prime}}$, where $K$ and ${K^{\prime}}$ are related to $k$ as in §19.2(ii).
##### 15: Bibliography B
• W. N. Bailey (1929) Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.
• P. Baratella and L. Gatteschi (1988) The Bounds for the Error Term of an Asymptotic Approximation of Jacobi Polynomials. In Orthogonal Polynomials and Their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, pp. 203–221.
• S. Bielski (2013) Orthogonality relations for the associated Legendre functions of imaginary order. Integral Transforms Spec. Funct. 24 (4), pp. 331–337.
• 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.
• W. G. C. Boyd (1990b) Stieltjes transforms and the Stokes phenomenon. Proc. Roy. Soc. London Ser. A 429, pp. 227–246.
• ##### 16: 22.20 Methods of Computation
To compute $\operatorname{sn}$, $\operatorname{cn}$, $\operatorname{dn}$ to 10D when $x=0.8$, $k=0.65$. …
###### §22.20(iii) Landen Transformations
If either $\tau$ or $q=e^{i\pi\tau}$ is given, then we use $k={\theta_{2}}^{2}\left(0,q\right)/{\theta_{3}}^{2}\left(0,q\right)$, $k^{\prime}={\theta_{4}}^{2}\left(0,q\right)/{\theta_{3}}^{2}\left(0,q\right)$, $K=\frac{1}{2}\pi{\theta_{3}}^{2}\left(0,q\right)$, and $K^{\prime}=-i\tau K$, obtaining the values of the theta functions as in §20.14. …
##### 18: 20.11 Generalizations and Analogs
If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): … This is Jacobi’s inversion problem of §20.9(ii). … Each provides an extension of Jacobi’s inversion problem. … For $m=1,2,3,4$, $n=1,2,3,4$, and $m\neq n$, define twelve combined theta functions $\varphi_{m,n}\left(z,q\right)$ by …
##### 19: 22.18 Mathematical Applications
where $\mathcal{E}\left(u,k\right)$ is Jacobi’s epsilon function (§22.16(ii)). … By use of the functions $\operatorname{sn}$ and $\operatorname{cn}$, parametrizations of algebraic equations, such as …
###### §22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem
With the identification $x=\operatorname{sn}\left(z,k\right)$, $y=\ifrac{\mathrm{d}(\operatorname{sn}\left(z,k\right))}{\mathrm{d}z}$, the addition law (22.18.8) is transformed into the addition theorem (22.8.1); see Akhiezer (1990, pp. 42, 45, 73–74) and McKean and Moll (1999, §§2.14, 2.16). …
##### 20: 31.2 Differential Equations
###### $F$-Homotopic Transformations
By composing these three steps, there result $2^{3}=8$ possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1).