Elliptic Functions

`carlson_rc` (x, y) — Function

Carlson’s RC integral is defined by

$$R_C(x, y) = \frac{1}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}(t+y)}, dt$$

$$R_C(x, y) = \frac{1}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}(t+y)}, dt $$

See https://arxiv.org/pdf/math/9409227Numerical Computation of Real or Complex Elliptic Integrals for more information.

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

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

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

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_rd` (x, y, z) — Function

Carlson’s RD integral is defined by

$$R_D(x,y,z) = \frac{3}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z},(t+z)}, dt$$

$$R_D(x,y,z) = \frac{3}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z},(t+z)}, dt $$

See https://arxiv.org/pdf/math/9409227Numerical Computation of Real or Complex Elliptic Integrals for more information.

We also have the special values

$$\eqalign{ R_D(x,x,x) &= x^{-\frac{3}{2}} \cr R_D(0,y,y) &= \frac{3}{4} \pi y^{-\frac{3}{2}} \cr R_D(0,2,1) &= 3 \sqrt{\pi} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})} }$$

$$\eqalign{ R_D(x,x,x) &= x^{-\frac{3}{2}} \cr R_D(0,y,y) &= \frac{3}{4} \pi y^{-\frac{3}{2}} \cr R_D(0,2,1) &= 3 \sqrt{\pi} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})} } $$

It is also related to the complete elliptic integral of the second kind, $E$, (elliptic_ec) as follows

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

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

`carlson_rf` (x, y, z) — Function

Carlson’s RF integral is defined by

$$R_F(x,y,z) = \frac{1}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}}, dt$$

$$R_F(x,y,z) = \frac{1}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}}, dt $$

See https://arxiv.org/pdf/math/9409227Numerical Computation of Real or Complex Elliptic Integrals for more information.

We also have the special values

$$\eqalign{ R_F(0,1,2) &= \frac{\Gamma({\frac{1}{4}})^2}{4\sqrt{2\pi}} \cr R_F(i,-i,0) &= \frac{\Gamma({\frac{1}{4}})^2}{4\sqrt{\pi}} }$$

$$\eqalign{ R_F(0,1,2) &= \frac{\Gamma({\frac{1}{4}})^2}{4\sqrt{2\pi}} \cr R_F(i,-i,0) &= \frac{\Gamma({\frac{1}{4}})^2}{4\sqrt{\pi}} } $$

It is also related to the complete elliptic integral of the second kind, $E$, (elliptic_ec) as follows

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

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

`carlson_rj` (x, y, z, p) — Function

Carlson’s RJ integral is defined by

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

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

See https://arxiv.org/pdf/math/9409227Numerical Computation of Real or Complex Elliptic Integrals for more information.

It is related to the elliptic integral of the third kind (elliptic_pi) by

$$\int_0^\phi {1\over \left(1+n\sin^2\theta\right) \sqrt{1-m\sin^2\theta}} , d\theta = R_F(c-1,c-m,c) - {n\over 3}R_j(c-1,c-m,c,c+n)$$

where $c = \csc\phi.$ Note that this differs in our definition of elliptic_pi by the sign of the parameter $n$.

`elliptic_e` (phi, m) — Function

The incomplete elliptic integral of the second kind, defined as

$$\int_0^\phi {\sqrt{1 - m\sin^2\theta}}, d\theta$$

$$\int_0^\phi {\sqrt{1 - m\sin^2\theta}}, d\theta $$

See also elliptic_005ff and elliptic_005fec.

`elliptic_ec` (m) — Function

The complete elliptic integral of the second kind, defined as

$$\int_0^{\frac{\pi}{2}} \sqrt{1 - m\sin^2\theta}, d\theta$$

$$\int_0^{\frac{\pi}{2}} \sqrt{1 - m\sin^2\theta}, d\theta $$

For certain values of $m$, the value of the integral is known in terms of gamma functions. Use makegamma to evaluate them.

`elliptic_eu` (u, m) — Function

The incomplete elliptic integral of the second kind, defined as

$$E(u, m) = \int_0^u {\rm dn}(v, m), dv = \int_0^\tau \sqrt{\frac{1-m t^2}{1-t^2}}, dt$$

$$E(u, m) = \int_0^u {\rm dn}(v, m), dv = \int_0^\tau \sqrt{\frac{1-m t^2}{1-t^2}}, dt $$

where $\tau = {\rm sn}(u,m) .$

This is related to elliptic_e by

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

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

`elliptic_f` (phi, m) — Function

The incomplete elliptic integral of the first kind, defined as

$$\int_0^{\phi} {\frac{d\theta}{\sqrt{1-m\sin^2\theta}}}$$

$$\int_0^{\phi} {\frac{d\theta}{\sqrt{1-m\sin^2\theta}}} $$

See also elliptic_005fe and elliptic_005fkc.

`elliptic_kc` (m) — Function

The complete elliptic integral of the first kind, defined as

$$\int_0^{\frac{\pi}{2}} {{d\theta}\over{\sqrt{1 - m\sin^2\theta}}}$$

$$\int_0^{\frac{\pi}{2}} {{d\theta}\over{\sqrt{1 - m\sin^2\theta}}} $$

For certain values of $m$, the value of the integral is known in terms of gamma functions. Use makegamma to evaluate them.

`elliptic_pi` (n, phi, m) — Function

The incomplete elliptic integral of the third kind, defined as

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

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

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