(maxima.info)Functions and Variables for Elliptic Integrals

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16.3 Functions and Variables for Elliptic Integrals
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 -- 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 Note: elliptic_e and Note: elliptic_kc.

 -- 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 Note: elliptic_f and Note: elliptic_ec.

 -- 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 Note: elliptic_e.

 -- 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)

 -- 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.

 -- 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.