# Jacobi imaginary transformation

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##### 2: 29.10 Lamé Functions with Imaginary Periods
###### §29.10 Lamé Functions with Imaginary Periods
29.10.2 $z^{\prime}=\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),$
transform (29.2.1) into
29.10.3 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z^{\prime}}^{2}}+(h^{\prime}-\nu(\nu+1){k^% {\prime}}^{2}{\operatorname{sn}}^{2}\left(z^{\prime},k^{\prime}\right))w=0.$
The first and the fourth functions have period $2\mathrm{i}{K^{\prime}}$; the second and the third have period $4\mathrm{i}{K^{\prime}}$. …
##### 3: 14.31 Other Applications
###### §14.31(ii) Conical Functions
The conical functions $\mathsf{P}^{m}_{-\frac{1}{2}+i\tau}\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). These functions are also used in the Mehler–Fock integral transform14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …
##### 4: 20.14 Methods of Computation
For instance, the first three terms of (20.2.1) give the value of $\theta_{1}\left(2-i\middle|i\right)$ ($=\theta_{1}\left(2-i,e^{-\pi}\right)$) to 12 decimal places. For values of $\left|q\right|$ near $1$ the transformations of §20.7(viii) can be used to replace $\tau$ with a value that has a larger imaginary part and hence a smaller value of $\left|q\right|$. For instance, to find $\theta_{3}\left(z,0.9\right)$ we use (20.7.32) with $q=0.9=e^{i\pi\tau}$, $\tau=-i\ln\left(0.9\right)/\pi$. …Hence the first term of the series (20.2.3) for $\theta_{3}\left(z\tau^{\prime}\middle|\tau^{\prime}\right)$ suffices for most purposes. In theory, starting from any value of $\tau$, a finite number of applications of the transformations $\tau\to\tau+1$ and $\tau\to-1/\tau$ will result in a value of $\tau$ with $\Im\tau\geq\sqrt{3}/2$; see §23.18. …
##### 5: 20.10 Integrals
20.10.1 $\int_{0}^{\infty}x^{s-1}\theta_{2}\left(0\middle|ix^{2}\right)\mathrm{d}x=2^{s% }(1-2^{-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$
20.10.2 $\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|ix^{2}\right)-1)\mathrm{d}x=% \pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$
20.10.3 $\int_{0}^{\infty}x^{s-1}(1-\theta_{4}\left(0\middle|ix^{2}\right))\mathrm{d}x=% (1-2^{1-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right).$
##### 6: 20.7 Identities
20.7.33 $(-i\tau)^{1/2}\theta_{4}\left(z\middle|\tau\right)=\exp\left(i\tau^{\prime}z^{% 2}/\pi\right)\theta_{2}\left(z\tau^{\prime}\middle|\tau^{\prime}\right).$
##### 7: 15.9 Relations to Other Functions
###### §15.9(ii) Jacobi Function
The Jacobi transform is defined as …with inverse … …
##### 8: 29.18 Mathematical Applications
when transformed to sphero-conal coordinates $r,\beta,\gamma$:
$x=kr\operatorname{sn}\left(\beta,k\right)\operatorname{sn}\left(\gamma,k\right),$
$y=\mathrm{i}\frac{k}{k^{\prime}}r\operatorname{cn}\left(\beta,k\right)% \operatorname{cn}\left(\gamma,k\right),$
The wave equation (29.18.1), when transformed to ellipsoidal coordinates $\alpha,\beta,\gamma$: …
$z=\frac{\mathrm{i}}{kk^{\prime}}\operatorname{dn}\left(\alpha,k\right)% \operatorname{dn}\left(\beta,k\right)\operatorname{dn}\left(\gamma,k\right),$
##### 9: 15.17 Mathematical Applications
The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. …
###### §15.17(iii) Group Representations
For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function (§15.9(ii)). …Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform1.14(ii)) or as a specialization of a group Fourier transform. … Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). …
##### 10: 23.15 Definitions
In §§23.1523.19, $k$ and $k^{\prime}$ $(\in\mathbb{C})$ denote the Jacobi modulus and complementary modulus, respectively, and $q=e^{i\pi\tau}$ ($\Im\tau>0$) denotes the nome; compare §§20.1 and 22.1. … Also $\mathcal{A}$ denotes a bilinear transformation on $\tau$, given by …The set of all bilinear transformations of this form is denoted by SL$(2,\mathbb{Z})$ (Serre (1973, p. 77)). … In (23.15.9) the branch of the cube root is chosen to agree with the second equality; in particular, when $\tau$ lies on the positive imaginary axis the cube root is real and positive. …