Previous: Functions and Variables for Elliptic Functions [Contents][Index]
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.
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.
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.
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)
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.
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.
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))
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)
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)
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)
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