acb_elliptic_k res m prec

Computes the complete elliptic integral of the first kind

\[`\] \[K(m) = \int_0^{\pi/2} \frac{dt}{\sqrt{1-m \sin^2 t}} = \int_0^1 \frac{dt}{\left(\sqrt{1-t^2}\right)\left(\sqrt{1-mt^2}\right)}\]

using the arithmetic-geometric mean: \(K(m) = \pi / (2 M(\sqrt{1-m}))\).

acb_elliptic_k_jet res m len prec

Sets the coefficients in the array res to the power series expansion of the complete elliptic integral of the first kind at the point m truncated to length len, i.e. \(K(m+x) \in \mathbb{C}[[x]]\).

acb_elliptic_k_series res m len prec

Sets res to the complete elliptic integral of the first kind of the power series m, truncated to length len.

acb_elliptic_e res m prec

Computes the complete elliptic integral of the second kind

\[`\] \[E(m) = \int_0^{\pi/2} \sqrt{1-m \sin^2 t} \, dt = \int_0^1 \frac{\sqrt{1-mt^2}}{\sqrt{1-t^2}} \, dt\]

using \(E(m) = (1-m)(2m K'(m) + K(m))\) (where the prime denotes a derivative, not a complementary integral).

acb_elliptic_pi res n m prec

Evaluates the complete elliptic integral of the third kind

\[`\] \[\Pi(n, m) = \int_0^{\pi/2} \frac{dt}{(1-n \sin^2 t) \sqrt{1-m \sin^2 t}} = \int_0^1 \frac{dt}{(1-nt^2) \sqrt{1-t^2} \sqrt{1-mt^2}}.\]

This implementation currently uses the same algorithm as the corresponding incomplete integral. It is therefore less efficient than the implementations of the first two complete elliptic integrals which use the AGM.

acb_elliptic_f res phi m pi prec

Evaluates the Legendre incomplete elliptic integral of the first kind, given by

\[`\] \[F(\phi,m) = \int_0^{\phi} \frac{dt}{\sqrt{1-m \sin^2 t}} = \int_0^{\sin \phi} \frac{dt}{\left(\sqrt{1-t^2}\right)\left(\sqrt{1-mt^2}\right)}\]

on the standard strip \(-\pi/2 \le \operatorname{Re}(\phi) \le \pi/2\). Outside this strip, the function extends quasiperiodically as

\[`\] \[F(\phi + n \pi, m) = 2 n K(m) + F(\phi,m), n \in \mathbb{Z}.\]

Inside the standard strip, the function is computed via the symmetric integral \(R_F\).

If the flag pi is set to 1, the variable \(\phi\) is replaced by \(\pi \phi\), changing the quasiperiod to 1.

The function reduces to a complete elliptic integral of the first kind when \(\phi = \frac{\pi}{2}\); that is, \(F\left(\frac{\pi}{2}, m\right) = K(m)\).

acb_elliptic_e_inc res phi m pi prec

Evaluates the Legendre incomplete elliptic integral of the second kind, given by

\[`\] \[E(\phi,m) = \int_0^{\phi} \sqrt{1-m \sin^2 t} \, dt = \int_0^{\sin \phi} \frac{\sqrt{1-mt^2}}{\sqrt{1-t^2}} \, dt\]

on the standard strip \(-\pi/2 \le \operatorname{Re}(\phi) \le \pi/2\). Outside this strip, the function extends quasiperiodically as

\[`\] \[E(\phi + n \pi, m) = 2 n E(m) + E(\phi,m), n \in \mathbb{Z}.\]

Inside the standard strip, the function is computed via the symmetric integrals \(R_F\) and \(R_D\).

If the flag pi is set to 1, the variable \(\phi\) is replaced by \(\pi \phi\), changing the quasiperiod to 1.

The function reduces to a complete elliptic integral of the second kind when \(\phi = \frac{\pi}{2}\); that is, \(E\left(\frac{\pi}{2}, m\right) = E(m)\).

acb_elliptic_pi_inc res n phi m pi prec

Evaluates the Legendre incomplete elliptic integral of the third kind, given by

\[`\] \[\Pi(n, \phi, m) = \int_0^{\phi} \frac{dt}{(1-n \sin^2 t) \sqrt{1-m \sin^2 t}} = \int_0^{\sin \phi} \frac{dt}{(1-nt^2) \sqrt{1-t^2} \sqrt{1-mt^2}}\]

on the standard strip \(-\pi/2 \le \operatorname{Re}(\phi) \le \pi/2\). Outside this strip, the function extends quasiperiodically as

\[`\] \[\Pi(n, \phi + k \pi, m) = 2 k \Pi(n,m) + \Pi(n,\phi,m), k \in \mathbb{Z}.\]

Inside the standard strip, the function is computed via the symmetric integrals \(R_F\) and \(R_J\).

If the flag pi is set to 1, the variable \(\phi\) is replaced by \(\pi \phi\), changing the quasiperiod to 1.

The function reduces to a complete elliptic integral of the third kind when \(\phi = \frac{\pi}{2}\); that is, \(\Pi\left(n, \frac{\pi}{2}, m\right) = \Pi(n, m)\).

acb_elliptic_rf res x y z flags prec

Evaluates the Carlson symmetric elliptic integral of the first kind

\[`\] \[R_F(x,y,z) = \frac{1}{2} \int_0^{\infty} \frac{dt}{\sqrt{(t+x)(t+y)(t+z)}}\]

where the square root extends continuously from positive infinity. The integral is well-defined for \(x,y,z \notin (-\infty,0)\), and with at most one of \(x,y,z\) being zero. When some parameters are negative real numbers, the function is still defined by analytic continuation.

In general, one or more duplication steps are applied until \(x,y,z\) are close enough to use a multivariate Taylor series.

The special case \(R_C(x, y) = R_F(x, y, y) = \frac{1}{2} \int_0^{\infty} (t+x)^{-1/2} (t+y)^{-1} dt\) may be computed by setting y and z to the same variable. (This case is not yet handled specially, but might be optimized in the future.)

The flags parameter is reserved for future use and currently does nothing. Passing 0 results in default behavior.

acb_elliptic_rg res x y z flags prec

Evaluates the Carlson symmetric elliptic integral of the second kind

\[`\] \[R_G(x,y,z) = \frac{1}{4} \int_0^{\infty} \frac{t}{\sqrt{(t+x)(t+y)(t+z)}} \left( \frac{x}{t+x} + \frac{y}{t+y} + \frac{z}{t+z}\right) dt\]

where the square root is taken continuously as in \(R_F\). The evaluation is done by expressing \(R_G\) in terms of \(R_F\) and \(R_D\). There are no restrictions on the variables.

acb_elliptic_rj_integration res x y z p flags prec

Evaluates the Carlson symmetric elliptic integral of the third kind

\[`\] \[R_J(x,y,z,p) = \frac{3}{2} \int_0^{\infty} \frac{dt}{(t+p)\sqrt{(t+x)(t+y)(t+z)}}\]

where the square root is taken continuously as in \(R_F\).

Three versions of this function are available: the carlson version applies one or more duplication steps until \(x,y,z,p\) are close enough to use a multivariate Taylor series.

The duplication algorithm is not correct for all possible combinations of complex variables, since the square roots taken during the computation can introduce spurious branch cuts. According to [Car1995], a sufficient (but not necessary) condition for correctness is that x, y, z have nonnegative real part and that p has positive real part.

In other cases, the algorithm might still be correct, but no attempt is made to check this; it is up to the user to verify that the duplication algorithm is appropriate for the given parameters before calling this function.

The integration algorithm uses explicit numerical integration to translate the parameters to the right half-plane. This is reliable but can be slow.

The default method uses the carlson algorithm when it is certain to be correct, and otherwise falls back to the slow integration algorithm.

The special case \(R_D(x, y, z) = R_J(x, y, z, z)\) may be computed by setting z and p to the same variable. This case is handled specially to avoid redundant arithmetic operations. In this case, the carlson algorithm is correct for all x, y and z.

The flags parameter is reserved for future use and currently does nothing. Passing 0 results in default behavior.

acb_elliptic_rc1 res x prec

This helper function computes the special case \(R_C(1, 1+x) = \operatorname{atan}(\sqrt{x})/\sqrt{x} = {}_2F_1(1,1/2,3/2,-x)\), which is needed in the evaluation of \(R_J\).

acb_elliptic_p res z tau prec

Computes Weierstrass's elliptic function

\[`\] \[\wp(z, \tau) = \frac{1}{z^2} + \sum_{n^2+m^2 \ne 0} \left[ \frac{1}{(z+m+n\tau)^2} - \frac{1}{(m+n\tau)^2} \right]\]

which satisfies \(\wp(z, \tau) = \wp(z + 1, \tau) = \wp(z + \tau, \tau)\). To evaluate the function efficiently, we use the formula

\[`\] \[\wp(z, \tau) = \pi^2 \theta_2^2(0,\tau) \theta_3^2(0,\tau) \frac{\theta_4^2(z,\tau)}{\theta_1^2(z,\tau)} - \frac{\pi^2}{3} \left[ \theta_2^4(0,\tau) + \theta_3^4(0,\tau)\right].\]

acb_elliptic_p_prime res z tau prec

Computes the derivative \(\wp'(z, \tau)\) of Weierstrass's elliptic function \(\wp(z, \tau)\).

acb_elliptic_p_jet res z tau len prec

Computes the formal power series \(\wp(z + x, \tau) \in \mathbb{C}[[x]]\), truncated to length len. In particular, with len = 2, simultaneously computes \(\wp(z, \tau), \wp'(z, \tau)\) which together generate the field of elliptic functions with periods 1 and \(\tau\).

acb_elliptic_p_series res z tau len prec

Sets res to the Weierstrass elliptic function of the power series z, with periods 1 and tau, truncated to length len.

acb_elliptic_invariants g2 g3 tau prec

Computes the lattice invariants \(g_2, g_3\). The Weierstrass elliptic function satisfies the differential equation \([\wp'(z, \tau)]^2 = 4 [\wp(z,\tau)]^3 - g_2 \wp(z,\tau) - g_3\). Up to constant factors, the lattice invariants are the first two Eisenstein series (see acb_modular_eisenstein).

acb_elliptic_roots e1 e2 e3 tau prec

Computes the lattice roots \(e_1, e_2, e_3\), which are the roots of the polynomial \(4z^3 - g_2 z - g_3\).

acb_elliptic_inv_p res z tau prec

Computes the inverse of the Weierstrass elliptic function, which satisfies \(\wp(\wp^{-1}(z, \tau), \tau) = z\). This function is given by the elliptic integral

\[`\] \[\wp^{-1}(z, \tau) = \frac{1}{2} \int_z^{\infty} \frac{dt}{\sqrt{(t-e_1)(t-e_2)(t-e_3)}} = R_F(z-e_1,z-e_2,z-e_3).\]

acb_elliptic_zeta res z tau prec

Computes the Weierstrass zeta function

\[`\] \[\zeta(z, \tau) = \frac{1}{z} + \sum_{n^2+m^2 \ne 0} \left[ \frac{1}{z-m-n\tau} + \frac{1}{m+n\tau} + \frac{z}{(m+n\tau)^2} \right]\]

which is quasiperiodic with \(\zeta(z + 1, \tau) = \zeta(z, \tau) + \zeta(1/2, \tau)\) and \(\zeta(z + \tau, \tau) = \zeta(z, \tau) + \zeta(\tau/2, \tau)\).

acb_elliptic_sigma res z tau prec

Computes the Weierstrass sigma function

\[`\] \[\sigma(z, \tau) = z \prod_{n^2+m^2 \ne 0} \left[ \left(1-\frac{z}{m+n\tau}\right) \exp\left(\frac{z}{m+n\tau} + \frac{z^2}{2(m+n\tau)^2} \right) \right]\]

which is quasiperiodic with \(\sigma(z + 1, \tau) = -e^{2 \zeta(1/2, \tau) (z+1/2)} \sigma(z, \tau)\) and \(\sigma(z + \tau, \tau) = -e^{2 \zeta(\tau/2, \tau) (z+\tau/2)} \sigma(z, \tau)\).