### 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.

  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.

  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.

  Categories: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)

  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.

  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.

  Categories:Elliptic integrals ·

Function: carlson_rc (x, y)

  Carlson’s RC integral is defined by

  integrate(1/2*(t+x)^(-1/2)/(t+y), t, 0, inf)

  This integral is related to many elementary functions in the following way:

  log(x) = (x-1)*rc(((1+x)/2)^2, x), x > 0

  asin(x) = x * rc(1-x^2, 1), |x|<= 1

  acos(x) = sqrt(1-x^2)*rc(x^2,1), 0 <= x <=1

  atan(x) = x * rc(1,1+x^2)

  asinh(x) = x * rc(1+x^2,1)

  acosh(x) = sqrt(x^2-1) * rc(x^2,1), x >= 1

  atanh(x) = x * rc(1,1-x^2), |x|<=1

  Also, we have the relationship

  R_C(x,y) = R_F(x,y,y)

  Some special values: R_C(0,1) = pi/2

  R_C(0,1/4) = pi

  R_C(2,1) = log(sqrt(2)+1)

  R_C(i, i+1) = pi/4 + i/2*log(sqrt(2)+1)

  R_C(0, i) = (1-i)*pi/(2*sqrt(2))

  Categories:Elliptic integrals ·

Function: carlson_rd (x, y, z)

  Carlson’s RD integral is defined by

  R_D(x,y,z) = 3/2*integrate(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)*(t+z)), t, 0, inf)

  We also have the special values

  R_D(x,x,x) = x^(-3/2)

  R_D(0,y,y) = 3/4*pi*y^(-3/2)

  R_D(0,2,1) = 3 sqrt(pi) gamma(3/4)/gamma(1/4)

  It is also related to the complete elliptic E function as follows

  E(m) = R_F(0, 1 - m, 1) - (m/3)* R_D(0, 1 - m, 1)

  Categories:Elliptic integrals ·

Function: carlson_rf (x, y, z)

  Carlson’s RF integral is defined by

  R_F(x,y,z) = 1/2*integrate(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)), t, 0, inf)

  We also have the special values

  R_F(0,1,2) = gamma(1/4)^2/(4*sqrt(2*pi))

  R_F(i,-i,0) = gamma(1/4)^2/(4*sqrt(pi))

  It is also related to the complete elliptic E function as follows

  E(m) = R_F(0, 1 - m, 1) - (m/3)* R_D(0, 1 - m, 1)

  Categories:Elliptic integrals ·

Function: carlson_rj (x, y, z, p)

  Carlson’s RJ integral is defined by

  R_J(x,y,z) = 1/2*integrate(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)*(t+p)), t, 0, inf)

  Categories:Elliptic integrals ·