# acb_elliptic.h – elliptic integrals and functions of complex variables

This module supports computation of elliptic (doubly periodic) functions, and their inverses, elliptic integrals. See acb_modular.h for the closely related modular forms and Jacobi theta functions.

Warning: incomplete elliptic integrals have very complicated branch structure when extended to complex variables. For some functions in this module, branch cuts may be artifacts of the evaluation algorithm rather than having a natural mathematical justification. The user should, accordingly, watch out for edge cases where the functions implemented here may differ from other systems or literature. There may also exist points where a function should be well-defined but the implemented algorithm fails to produce a finite result due to artificial internal singularities.

## Complete elliptic integrals

void acb_elliptic_k(acb_t res, const acb_t m, slong 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: \(K(m) = \pi / (2 M(\sqrt{1-m}))\).

void acb_elliptic_k_jet(acb_ptr res, const acb_t m, slong len, slong 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. \(K(m+x) \in \mathbb{C}[[x]]\).

void _acb_elliptic_k_series(acb_ptr res, acb_srcptr m, slong mlen, slong len, slong prec)

void acb_elliptic_k_series(acb_poly_t res, const acb_poly_t m, slong len, slong prec)

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

void acb_elliptic_e(acb_t res, const acb_t m, slong 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).

void acb_elliptic_pi(acb_t res, const acb_t n, const acb_t m, slong 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.

## Legendre incomplete elliptic integrals

void acb_elliptic_f(acb_t res, const acb_t phi, const acb_t m, int pi, slong 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, \(F\left(\frac{\pi}{2}, m\right) = K(m)\).

void acb_elliptic_e_inc(acb_t res, const acb_t phi, const acb_t m, int pi, slong 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, \(E\left(\frac{\pi}{2}, m\right) = E(m)\).

void acb_elliptic_pi_inc(acb_t res, const acb_t n, const acb_t phi, const acb_t m, int pi, slong 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, \(\Pi\left(n, \frac{\pi}{2}, m\right) = \Pi(n, m)\).

## Carlson symmetric elliptic integrals

Carlson symmetric forms are the preferred form of incomplete elliptic integrals, due to their neat properties and relatively simple computation based on duplication theorems. There are five named functions: \(R_F, R_G, R_J\), and \(R_C\), \(R_D\) which are special cases of \(R_F\) and \(R_J\) respectively. We largely follow the definitions and algorithms in [Car1995] and chapter 19 in [NIST2012].

void acb_elliptic_rf(acb_t res, const acb_t x, const acb_t y, const acb_t z, int flags, slong 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.

void acb_elliptic_rg(acb_t res, const acb_t x, const acb_t y, const acb_t z, int flags, slong 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.

void acb_elliptic_rj(acb_t res, const acb_t x, const acb_t y, const acb_t z, const acb_t p, int flags, slong prec)

void acb_elliptic_rj_carlson(acb_t res, const acb_t x, const acb_t y, const acb_t z, const acb_t p, int flags, slong prec)

void acb_elliptic_rj_integration(acb_t res, const acb_t x, const acb_t y, const acb_t z, const acb_t p, int flags, slong 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.

void acb_elliptic_rc1(acb_t res, const acb_t x, slong 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 \(R_J\).

## Weierstrass elliptic functions

Elliptic functions may be defined on a general lattice \(\Lambda = \{m 2\omega_1 + n 2\omega_2\ : m, n \in \mathbb{Z}\}\) with half-periods \(\omega_1, \omega_2\). We simplify by setting \(2\omega_1 = 1, 2\omega_2 = \tau\) with \(\operatorname{im}(\tau) > 0\). To evaluate the functions on a general lattice, it is enough to make a linear change of variables. The main reference is chapter 23 in [NIST2012].

void acb_elliptic_p(acb_t res, const acb_t z, const acb_t tau, slong 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

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

void acb_elliptic_p_prime(acb_t res, const acb_t z, const acb_t tau, slong prec)

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

void acb_elliptic_p_jet(acb_ptr res, const acb_t z, const acb_t tau, slong len, slong 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 \(\tau\).

void _acb_elliptic_p_series(acb_ptr res, acb_srcptr z, slong zlen, const acb_t tau, slong len, slong prec)

void acb_elliptic_p_series(acb_poly_t res, const acb_poly_t z, const acb_t tau, slong len, slong prec)

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

void acb_elliptic_invariants(acb_t g2, acb_t g3, const acb_t tau, slong 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()`).

void acb_elliptic_roots(acb_t e1, acb_t e2, acb_t e3, const acb_t tau, slong prec)

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

void acb_elliptic_inv_p(acb_t res, const acb_t z, const acb_t tau, slong 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

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

void acb_elliptic_zeta(acb_t res, const acb_t z, const acb_t tau, slong 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 \(\zeta(z + \tau, \tau) = \zeta(z, \tau) + \zeta(\tau/2, \tau)\).

void acb_elliptic_sigma(acb_t res, const acb_t z, const acb_t tau, slong 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 \(\sigma(z + \tau, \tau) = -e^{2 \zeta(\tau/2, \tau) (z+\tau/2)} \sigma(z, \tau)\).