Real elliptic functions

• Conversions of parameters of elliptic functions
  • Elliptic nome \(q(k)\) (DAMath)
    • math53.elliptic_nome()
  • Elliptic modulus \(k(q)\) (DAMath)
    • math53.elliptic_modulus()
  • Jacobi amplitude, \(\mathrm{am}(x, k)\) (DAMath)
    • math53.jacobi_amplitude()
• Elliptic integrals
  • Legendre complete elliptic integral \(B(k)\)
    • math53.elliptic_b()
  • Legendre complete elliptic integral \(D(k)\)
    • math53.elliptic_d()
  • Legendre incomplete elliptic integral \(B(\phi, k)\)
    • math53.elliptic_b_inc()
  • Legendre incomplete elliptic integral \(D(\phi, k)\)
    • math53.elliptic_d_inc()
  • Heuman’s Lambda function, \(\Lambda_0(\phi,k)\)
    • math53.heuman_lambda()
  • Jacobi Zeta function, \(Z(\phi,k)\)
    • math53.jacobi_zeta()
• Bulirsch elliptic integrals
  • Complete elliptic integral of the 1st kind \(\mathrm{cel1}(k_c)\)
    • math53.cel1()
  • Complete elliptic integral of the 2nd kind \(\mathrm{cel2}(k_c, a, b)\)
    • math53.cel2()
  • General complete elliptic integral \(\mathrm{cel}(k_c, p, a, b)\)
    • math53.cel()
  • Incomplete elliptic integral of the 1st kind \(\mathrm{el1}(x, k_c)\)
    • math53.el1()
  • Incomplete elliptic integral of the 2nd kind \(\mathrm{el2}(x, k_c, a, b)\)
    • math53.el2()
  • Incomplete elliptic integral of the 3rd kind \(\mathrm{el3}(x, k_c, p)\)
    • math53.el3()
• Maple style elliptic integrals
  • Complete integral of the 1st kind, \(\mathrm{EllipticK}(k)\)
    • math53.EllipticK()
  • Complete integral of the 1st kind for imaginary modulus, \(\mathrm{EllipticKim}(k)\)
    • math53.EllipticKim()
  • Complementary complete integral of the 1st kind, \(\mathrm{EllipticCK}(k) = K'(k)\)
    • math53.EllipticCK()
  • Complete integral of the 2nd kind, \(\mathrm{EllipticEC}(k)\)
    • math53.EllipticEC()
  • Complete integral of the 2nd kind for imaginary modulus, \(\mathrm{EllipticECim}(k)\)
    • math53.EllipticECim()
  • Complementary complete integral of the 2nd kind, \(\mathrm{EllipticCE}(k)\)
    • math53.EllipticCE()
  • Complete integral of the 3rd kind, \(\mathrm{EllipticPiC}(\nu, k)\)
    • math53.EllipticPiC()
  • Complete integral of the 3rd kind for imaginary modulus, \(\mathrm{EllipticPiCim}(\nu, k)\)
    • math53.EllipticPiCim()
  • Complementary complete integral of the 3rd kind, \(\mathrm{EllipticCPi}(\nu, k)\)
    • math53.EllipticCPi()
  • Incomplete integral of the 1st kind, \(\mathrm{EllipticF}(z, k)\)
    • math53.EllipticF()
  • Incomplete integral of the 2nd kind, \(\mathrm{EllipticE}(z, k)\)
    • math53.EllipticE()
  • Incomplete integral of the 3rd kind, \(\mathrm{EllipticPi}(z, \nu, k)\)
    • math53.EllipticPi()
• Jacobi theta functions at \(x=0\) for \(0 \le q <1\)
  • Jacobi theta function \(\theta'_1(q)\)
    • math53.jacobi_theta1p()
  • Jacobi theta function \(\theta_2(q)\)
    • math53.jacobi_theta02()
  • Jacobi theta function \(\theta_3(q)\)
    • math53.jacobi_theta03()
  • Jacobi theta function \(\theta_4(q)\)
    • math53.jacobi_theta04()
• Inverse Jacobi elliptic functions
  • Inverse Jacobi elliptic function \(\mathrm{arcsn}(x, k)\)
    • math53.jacobi_arcsn()
  • Inverse Jacobi elliptic function \(\mathrm{arccn}(x, k)\)
    • math53.jacobi_arccn()
  • Inverse Jacobi elliptic function \(\mathrm{arcdn}(x, k)\)
    • math53.jacobi_arcdn()
  • Inverse Jacobi elliptic function \(\mathrm{arccd}(x, k)\)
    • math53.jacobi_arccd()
  • Inverse Jacobi elliptic function \(\mathrm{arcsd}(x, k)\)
    • math53.jacobi_arcsd()
  • Inverse Jacobi elliptic function \(\mathrm{arcnd}(x, k)\)
    • math53.jacobi_arcnd()
  • Inverse Jacobi elliptic function \(\mathrm{arcdc}(x, k)\)
    • math53.jacobi_arcdc()
  • Inverse Jacobi elliptic function \(\mathrm{arcnc}(x, k)\)
    • math53.jacobi_arcnc()
  • Inverse Jacobi elliptic function \(\mathrm{arcsc}(x, k)\)
    • math53.jacobi_arcsc()
  • Inverse Jacobi elliptic function \(\mathrm{arcns}(x, k)\)
    • math53.jacobi_arcns()
  • Inverse Jacobi elliptic function \(\mathrm{arcds}(x, k)\)
    • math53.jacobi_arcds()
  • Inverse Jacobi elliptic function \(\mathrm{arccs}(x, k)\)
    • math53.jacobi_arccs()
• Lemniscate functions
  • Lemniscate sine function, \(\mathrm{sinlemn}(x)\)
    • math53.sinlemn()
  • Lemniscate cosine function, \(\mathrm{coslemn}(x)\)
    • math53.coslemn()
  • Inverse lemniscate sine function, \(\mathrm{arcsl}(x)\)
    • math53.arcsl()
  • Inverse lemniscate cosine function, \(\mathrm{arccl}(x)\)
    • math53.arccl()