acb_elliptic.h

Sets k to the complete elliptic integral of the first kind $K(m) = \int_0^{\pi/2} (1 - m \sin^2 t)^{-1/2} \, dt$, computed as $K(m) = \pi / (2 M(\sqrt{1-m}))$ where $M(z) = \operatorname{agm}(1, z)$ is computed by acb_agm1. The branch cut is $m \in [1, \infty)$.

Sets w to the Taylor coefficients of $K(m + x)$ at $x = 0$, truncated to length len. The first two coefficients come from $(M, M')$ evaluated by acb_agm1_cpx, the others from the linear recurrence implied by the hypergeometric differential equation of $K$.

Set res to $K(m(x))$ truncated to length len, for a power series $m(x)$, by composing the output of acb_elliptic_k_jet with the nonconstant part of $m$. The underscore version requires len $\ge 1$ and reads $\min(\text{mlen}, \text{len})$ coefficients of m.

Sets res to the complete elliptic integral of the second kind $E(m) = \int_0^{\pi/2} (1 - m \sin^2 t)^{1/2} \, dt$, computed from $K(m)$ and $K'(m)$ by $E(m) = (1-m)(2m K'(m) + K(m))$, where $K'$ is the derivative (not the complementary integral).

Sets r to the complete elliptic integral of the third kind $\Pi(n, m) = \int_0^{\pi/2} (1 - n \sin^2 t)^{-1} (1 - m \sin^2 t)^{-1/2} \, dt$. For $n = 0$ this is $K(m)$ and for $m = 0$ it is $\frac{\pi}{2}(1-n)^{-1/2}$; otherwise it is computed as $R_F(0, 1-m, 1) + \frac{n}{3} R_J(0, 1-m, 1, 1-n)$, the same reduction as for the incomplete integral, which is slower than the AGM methods for $K$ and $E$.

Sets res to the incomplete elliptic integral of the first kind $F(\phi, m) = \int_0^\phi (1 - m \sin^2 t)^{-1/2} \, dt$, extended by $F(\phi + d\pi, m) = 2 d K(m) + F(\phi, m)$. With times_pi nonzero, $\phi$ is replaced by $\pi \phi$. In particular $F(\pi/2, m) = K(m)$.

Sets res to the incomplete elliptic integral of the second kind $E(\phi, m) = \int_0^\phi (1 - m \sin^2 t)^{1/2} \, dt$, extended by $E(\phi + d\pi, m) = 2 d E(m) + E(\phi, m)$. With times_pi nonzero, $\phi$ is replaced by $\pi \phi$. In particular $E(\pi/2, m) = E(m)$.

Sets res to the incomplete elliptic integral of the third kind $\Pi(n, \phi, m) = \int_0^\phi (1 - n \sin^2 t)^{-1} (1 - m \sin^2 t)^{-1/2} \, dt$, extended by $\Pi(n, \phi + d\pi, m) = 2 d \Pi(n, m) + \Pi(n, \phi, m)$. With times_pi nonzero, $\phi$ is replaced by $\pi \phi$. In particular $\Pi(n, \pi/2, m) = \Pi(n, m)$ and $\Pi(0, \phi, m) = F(\phi, m)$.

Sets res to the Carlson symmetric integral of the first kind $R_F(x, y, z) = \frac12 \int_0^\infty ((t+x)(t+y)(t+z))^{-1/2} \, dt$, by duplication steps followed by a Taylor series in the elementary symmetric polynomials of the deviations. The result is indeterminate if more than one argument contains zero.

Sets res to the Carlson symmetric integral of the second kind $R_G$, computed from $R_F$ and $R_D$. There are no restrictions on the arguments.

Set res to the Carlson symmetric integral of the third kind $R_J(x, y, z, p) = \frac32 \int_0^\infty (t+p)^{-1} ((t+x)(t+y)(t+z))^{-1/2} \, dt$. The carlson version uses the duplication algorithm, which can give wrong values for some complex arguments; the integration version moves the arguments to the right half-plane by numerical integration; the default version uses the duplication algorithm when it is known to be correct (including the case $p = z$, which computes $R_D(x, y, z)$) and integration otherwise.

Sets res to $R_C(1, 1+x) = \operatorname{atan}(\sqrt{x})/\sqrt{x}$, a helper for $R_J$, using a short Taylor series for small $|x|$.

Sets r to the Weierstrass elliptic function $\wp(z, \tau) = z^{-2} + \sum_{\omega \in \Lambda \setminus 0} ((z + \omega)^{-2} - \omega^{-2})$, computed by

Sets r to the derivative $\wp'(z, \tau)$ with respect to $z$, computed by $\wp'(z,\tau) = (-2\pi \eta(\tau)^2/\theta_1(z,\tau))^3 \, \theta_2(z,\tau) \theta_3(z,\tau) \theta_4(z,\tau)$, where $\eta$ is the Dedekind eta function.

Sets r to the Taylor coefficients of $\wp(z + x, \tau)$ at $x = 0$, truncated to length len, from the theta function jets of acb_modular_theta_jet. With len = 2 this gives $(\wp(z, \tau), \wp'(z, \tau))$, which generate the field of elliptic functions with periods 1 and $\tau$.

Set res to $\wp(z(x), \tau)$ truncated to length len, for a power series $z(x)$. The underscore version requires len $\ge 1$.

Sets g2 and g3 to the lattice invariants $g_2 = 60 G_4(\tau)$ and $g_3 = 140 G_6(\tau)$, where $G_{2k}$ are the Eisenstein series of acb_modular_eisenstein. They are the coefficients of the differential equation $\wp'^2 = 4 \wp^3 - g_2 \wp - g_3$.

Sets e1, e2, e3 to the lattice roots, the roots of $4x^3 - g_2 x - g_3$, which are $e_1 = \wp(1/2)$, $e_2 = \wp((1 + \tau)/2)$ and $e_3 = \wp(\tau/2)$, computed from the fourth powers of the theta constants [NIST2012, 23.6.2–23.6.4]. The outputs are indeterminate if $\operatorname{Im}(\tau)$ is not certainly positive.

Sets res to the inverse Weierstrass function, a solution $w$ of $\wp(w, \tau) = z$, given by the elliptic integral

Sets res to the Weierstrass zeta function $\zeta(z, \tau) = z^{-1} + \sum_{\omega \ne 0} ((z - \omega)^{-1} + \omega^{-1} + z \omega^{-2})$, which satisfies $\zeta' = -\wp$ and is quasi-periodic: $\zeta(z + 1) = \zeta(z) + 2\eta_1$, $\zeta(z + \tau) = \zeta(z) + 2\eta_3$ with $\eta_1 = \zeta(1/2)$ and $\eta_3 = \zeta(\tau/2)$. It is computed as $\zeta(z) = 2 \eta_1 z + \theta_1'(z, \tau)/\theta_1(z, \tau)$, where $\eta_1 = -\theta_1'''(0, \tau) / (6 \theta_1'(0, \tau))$.

Sets res to the Weierstrass sigma function $\sigma(z, \tau) = z \prod_{\omega \ne 0} (1 - z/\omega) \exp(z/\omega + z^2/(2\omega^2))$, which satisfies $\sigma'/\sigma = \zeta$, computed as $\sigma(z) = e^{\eta_1 z^2} \theta_1(z, \tau) / \theta_1'(0, \tau)$.