16.3 Functions and Variables for Elliptic Integrals

Function: elliptic_f (phi, m) ¶

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.

Categories: Elliptic integrals ·

Function: elliptic_e (phi, m) ¶

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.

Categories: Elliptic integrals ·

Function: elliptic_eu (u, m) ¶

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

\[E(u,m) = E(\sin^{-1} {\rm sn}(u, m), m) \]

See also elliptic_e.

Categories: Elliptic integrals ·

Function: elliptic_pi (n, phi, m) ¶

The incomplete elliptic integral of the third kind, defined as

\[\int_0^\phi {{d\theta}\over{(1-n\sin^2 \theta)\sqrt{1 - m\sin^2\theta}}} \]

Categories: Elliptic integrals ·

Function: elliptic_kc (m) ¶

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.

Categories: Elliptic integrals ·

Function: elliptic_ec (m) ¶

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.

Categories: Elliptic integrals ·

Function: carlson_rc (x, y) ¶

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:

\[\eqalign{R_C(0, 1) &= \frac{\pi}{2} \cr R_C(0, 1/4) &= \pi \cr R_C(2,1) &= \log(\sqrt{2} + 1) \cr R_C(i,i+1) &= \frac{\pi}{4} + \frac{i}{2} \log(\sqrt{2}-1) \cr R_C(0,i) &= (1-i)\frac{\pi}{2\sqrt{2}} \cr } \]

Categories: Elliptic integrals ·

Function: carlson_rd (x, y, z) ¶

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

\[E(m) = R_F(0, 1 - m, 1) - \frac{m}{3} R_D(0, 1 - m, 1) \]

Categories: Elliptic integrals ·

Function: carlson_rf (x, y, z) ¶

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

\[E(m) = R_F(0, 1 - m, 1) - \frac{m}{3} R_D(0, 1 - m, 1) \]

Categories: Elliptic integrals ·

Function: carlson_rj (x, y, z, p) ¶

Carlson’s RJ integral is defined by

\[R_J(x,y,z) = \frac{1}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}\,(t+p)}\, dt \]

Categories: Elliptic integrals ·