Previous: Functions and Variables for Elliptic Functions, Up: Elliptic Functions [Contents][Index]
The incomplete elliptic integral of the first kind, defined as
$$ \int_0^{\phi} {\frac{d\theta}{\sqrt{1-m\sin^2\theta}}} $$See also elliptic_e and elliptic_kc.
The incomplete elliptic integral of the second kind, defined as
$$ \int_0^\phi {\sqrt{1 - m\sin^2\theta}}\, d\theta $$See also elliptic_f and elliptic_ec.
The incomplete elliptic integral of the second kind, defined as
$$ E(u, m) = \int_0^u {\rm dn}(v, m)\, dv = \int_0^\tau \sqrt{\frac{1-m t^2}{1-t^2}}\, dt $$where \(\tau = {\rm sn}(u,m) .\)
This is related to elliptic_e by
See also elliptic_e.
The incomplete elliptic integral of the third kind, defined as
$$ \int_0^\phi {{d\theta}\over{(1-n\sin^2 \theta)\sqrt{1 - m\sin^2\theta}}} $$The complete elliptic integral of the first kind, defined as
$$ \int_0^{\frac{\pi}{2}} {{d\theta}\over{\sqrt{1 - m\sin^2\theta}}} $$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
$$ \int_0^{\frac{\pi}{2}} \sqrt{1 - m\sin^2\theta}\, d\theta $$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
$$ R_C(x, y) = \frac{1}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}(t+y)}\, dt $$This integral is related to many elementary functions in the following way:
$$ \eqalign{ \log x &= (x-1) R_C\left(\left({\frac{1+x}{2}}\right)^2, x\right), \quad x > 0 \cr \sin^{-1} x &= x R_C(1-x^2, 1), \quad |x| \le 1 \cr \cos^{-1} x &= \sqrt{1-x^2} R_C(x^2,1), \quad 0 \le x \le 1 \cr \tan^{-1} x &= x R_C(1,1+x^2) \cr \sinh^{-1} x &= x R_C(1+x^2,1) \cr \cosh^{-1} x &= \sqrt{x^2-1} R_C(x^2,1), \quad x \ge 1 \cr \tanh^{-1}(x) &= x R_C(1,1-x^2), \quad |x| \le 1 } $$Also, we have the relationship
$$ R_C(x,y) = R_F(x,y,y) $$Some special values: $$ \eqalign{R_C(0, 1) &= \frac{\pi}{2} \cr R_C(0, 1/4) &= \pi \cr R_C(2,1) &= \log(\sqrt{2} + 1) \cr R_C(i,i+1) &= \frac{\pi}{4} + \frac{i}{2} \log(\sqrt{2}-1) \cr R_C(0,i) &= (1-i)\frac{\pi}{2\sqrt{2}} \cr } $$
Carlson’s RD integral is defined by
$$ R_D(x,y,z) = \frac{3}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}\,(t+z)}\, dt $$We also have the special values
$$ \eqalign{ R_D(x,x,x) &= x^{-\frac{3}{2}} \cr R_D(0,y,y) &= \frac{3}{4} \pi y^{-\frac{3}{2}} \cr R_D(0,2,1) &= 3 \sqrt{\pi} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})} } $$It is also related to the complete elliptic E function as follows
$$ E(m) = R_F(0, 1 - m, 1) - \frac{m}{3} R_D(0, 1 - m, 1) $$Carlson’s RF integral is defined by
$$ R_F(x,y,z) = \frac{1}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}}\, dt $$We also have the special values
$$ \eqalign{ R_F(0,1,2) &= \frac{\Gamma({\frac{1}{4}})^2}{4\sqrt{2\pi}} \cr R_F(i,-i,0) &= \frac{\Gamma({\frac{1}{4}})^2}{4\sqrt{\pi}} } $$It is also related to the complete elliptic E function as follows
$$ E(m) = R_F(0, 1 - m, 1) - \frac{m}{3} R_D(0, 1 - m, 1) $$Carlson’s RJ integral is defined by
$$ R_J(x,y,z) = \frac{1}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}\,(t+p)}\, dt $$Previous: Functions and Variables for Elliptic Functions, Up: Elliptic Functions [Contents][Index]