### 16.3 Functions and Variables for Elliptic Integrals

Function: elliptic_f (phi, m)

  The incomplete elliptic integral of the first kind, defined as

  integrate(1/sqrt(1 - m*sin(x)^2), x, 0, phi)

  See also elliptic_e and elliptic_kc.

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  Categories: Elliptic integrals

Function: elliptic_e (phi, m)

  The incomplete elliptic integral of the second kind, defined as

  elliptic_e(phi, m) = integrate(sqrt(1 - m*sin(x)^2), x, 0, phi)

  See also elliptic_f and elliptic_ec.

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  Categories: Elliptic integrals

Function: elliptic_eu (u, m)

  The incomplete elliptic integral of the second kind, defined as

  integrate(dn(v,m)^2,v,0,u) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2), t, 0, tau)

  where tau = sn(u,m).

  This is related to elliptic_e by

  elliptic_eu(u, m) = elliptic_e(asin(sn(u,m)),m)

  See also elliptic_e.

  ·Elliptic integrals

Function: elliptic_pi (n, phi, m)

  The incomplete elliptic integral of the third kind, defined as

  integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, 0, phi)

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  Categories: Elliptic integrals

Function: elliptic_kc (m)

  The complete elliptic integral of the first kind, defined as

  integrate(1/sqrt(1 - m*sin(x)^2), x, 0, %pi/2)

  For certain values of m, the value of the integral is known in terms of Gamma functions. Use `makegamma` to evaluate them.

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  Categories: Elliptic integrals

Function: elliptic_ec (m)

  The complete elliptic integral of the second kind, defined as

  integrate(sqrt(1 - m*sin(x)^2), x, 0, %pi/2)

  For certain values of m, the value of the integral is known in terms of Gamma functions. Use `makegamma` to evaluate them.

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  Categories: Elliptic integrals