3 * Introduction to Elliptic Functions and Integrals::
4 * Functions and Variables for Elliptic Functions::
5 * Functions and Variables for Elliptic Integrals::
10 @node Introduction to Elliptic Functions and Integrals, Functions and Variables for Elliptic Functions, , Top
11 @comment node-name, next, previous, up
13 @section Introduction to Elliptic Functions and Integrals
15 Maxima includes support for Jacobian elliptic functions and for
16 complete and incomplete elliptic integrals. This includes symbolic
17 manipulation of these functions and numerical evaluation as well.
18 Definitions of these functions and many of their properties can by
19 found in Abramowitz and Stegun, Chapter 16--17. As much as possible,
20 we use the definitions and relationships given there.
22 In particular, all elliptic functions and integrals use the parameter
23 @math{m} instead of the modulus @math{k} or the modular angle
24 @math{\alpha}. This is one area where we differ from Abramowitz and
25 Stegun who use the modular angle for the elliptic functions. The
26 following relationships are true:
63 The elliptic functions and integrals are primarily intended to support
64 symbolic computation. Therefore, most of derivatives of the functions
65 and integrals are known. However, if floating-point values are given,
66 a floating-point result is returned.
68 Support for most of the other properties of elliptic functions and
69 integrals other than derivatives has not yet been written.
71 Some examples of elliptic functions:
76 @c diff (jacobi_sn (u, m), u);
77 @c diff (jacobi_sn (u, m), m);
80 (%i1) jacobi_sn (u, m);
82 (%i2) jacobi_sn (u, 1);
84 (%i3) jacobi_sn (u, 0);
86 (%i4) diff (jacobi_sn (u, m), u);
87 (%o4) jacobi_cn(u, m) jacobi_dn(u, m)
88 (%i5) diff (jacobi_sn (u, m), m);
89 (%o5) jacobi_cn(u, m) jacobi_dn(u, m)
91 elliptic_e(asin(jacobi_sn(u, m)), m)
92 (u - ------------------------------------)/(2 m)
96 jacobi_cn (u, m) jacobi_sn(u, m)
97 + --------------------------------
101 Some examples of elliptic integrals:
103 @c elliptic_f (phi, m);
104 @c elliptic_f (phi, 0);
105 @c elliptic_f (phi, 1);
106 @c elliptic_e (phi, 1);
107 @c elliptic_e (phi, 0);
108 @c elliptic_kc (1/2);
110 @c diff (elliptic_f (phi, m), phi);
111 @c diff (elliptic_f (phi, m), m);
114 (%i1) elliptic_f (phi, m);
115 (%o1) elliptic_f(phi, m)
116 (%i2) elliptic_f (phi, 0);
118 (%i3) elliptic_f (phi, 1);
120 (%o3) log(tan(--- + ---))
122 (%i4) elliptic_e (phi, 1);
124 (%i5) elliptic_e (phi, 0);
126 (%i6) elliptic_kc (1/2);
136 (%i8) diff (elliptic_f (phi, m), phi);
138 (%o8) ---------------------
140 sqrt(1 - m sin (phi))
141 (%i9) diff (elliptic_f (phi, m), m);
142 elliptic_e(phi, m) - (1 - m) elliptic_f(phi, m)
143 (%o9) (-----------------------------------------------
147 - ---------------------)/(2 (1 - m))
149 sqrt(1 - m sin (phi))
152 Support for elliptic functions and integrals was written by Raymond
153 Toy. It is placed under the terms of the General Public License (GPL)
154 that governs the distribution of Maxima.
156 @opencatbox{Categories:}
157 @category{Elliptic functions}
160 @node Functions and Variables for Elliptic Functions, Functions and Variables for Elliptic Integrals, Introduction to Elliptic Functions and Integrals, Top
161 @comment node-name, next, previous, up
163 @section Functions and Variables for Elliptic Functions
166 @deffn {Function} jacobi_sn (@var{u}, @var{m})
167 The Jacobian elliptic function @math{sn(u,m)}.
169 @opencatbox{Categories:}
170 @category{Elliptic functions}
175 @deffn {Function} jacobi_cn (@var{u}, @var{m})
176 The Jacobian elliptic function @math{cn(u,m)}.
178 @opencatbox{Categories:}
179 @category{Elliptic functions}
184 @deffn {Function} jacobi_dn (@var{u}, @var{m})
185 The Jacobian elliptic function @math{dn(u,m)}.
187 @opencatbox{Categories:}
188 @category{Elliptic functions}
193 @deffn {Function} jacobi_ns (@var{u}, @var{m})
194 The Jacobian elliptic function @math{ns(u,m) = 1/sn(u,m)}.
196 @opencatbox{Categories:}
197 @category{Elliptic functions}
202 @deffn {Function} jacobi_sc (@var{u}, @var{m})
203 The Jacobian elliptic function @math{sc(u,m) = sn(u,m)/cn(u,m)}.
205 @opencatbox{Categories:}
206 @category{Elliptic functions}
211 @deffn {Function} jacobi_sd (@var{u}, @var{m})
212 The Jacobian elliptic function @math{sd(u,m) = sn(u,m)/dn(u,m)}.
214 @opencatbox{Categories:}
215 @category{Elliptic functions}
220 @deffn {Function} jacobi_nc (@var{u}, @var{m})
221 The Jacobian elliptic function @math{nc(u,m) = 1/cn(u,m)}.
223 @opencatbox{Categories:}
224 @category{Elliptic functions}
229 @deffn {Function} jacobi_cs (@var{u}, @var{m})
230 The Jacobian elliptic function @math{cs(u,m) = cn(u,m)/sn(u,m)}.
232 @opencatbox{Categories:}
233 @category{Elliptic functions}
238 @deffn {Function} jacobi_cd (@var{u}, @var{m})
239 The Jacobian elliptic function @math{cd(u,m) = cn(u,m)/dn(u,m)}.
241 @opencatbox{Categories:}
242 @category{Elliptic functions}
247 @deffn {Function} jacobi_nd (@var{u}, @var{m})
248 The Jacobian elliptic function @math{nd(u,m) = 1/dn(u,m)}.
250 @opencatbox{Categories:}
251 @category{Elliptic functions}
256 @deffn {Function} jacobi_ds (@var{u}, @var{m})
257 The Jacobian elliptic function @math{ds(u,m) = dn(u,m)/sn(u,m)}.
259 @opencatbox{Categories:}
260 @category{Elliptic functions}
265 @deffn {Function} jacobi_dc (@var{u}, @var{m})
266 The Jacobian elliptic function @math{dc(u,m) = dn(u,m)/cn(u,m)}.
268 @opencatbox{Categories:}
269 @category{Elliptic functions}
273 @anchor{inverse_jacobi_sn}
274 @deffn {Function} inverse_jacobi_sn (@var{u}, @var{m})
275 The inverse of the Jacobian elliptic function @math{sn(u,m)}.
277 @opencatbox{Categories:}
278 @category{Elliptic functions}
282 @anchor{inverse_jacobi_cn}
283 @deffn {Function} inverse_jacobi_cn (@var{u}, @var{m})
284 The inverse of the Jacobian elliptic function @math{cn(u,m)}.
286 @opencatbox{Categories:}
287 @category{Elliptic functions}
291 @anchor{inverse_jacobi_dn}
292 @deffn {Function} inverse_jacobi_dn (@var{u}, @var{m})
293 The inverse of the Jacobian elliptic function @math{dn(u,m)}.
295 @opencatbox{Categories:}
296 @category{Elliptic functions}
300 @anchor{inverse_jacobi_ns}
301 @deffn {Function} inverse_jacobi_ns (@var{u}, @var{m})
302 The inverse of the Jacobian elliptic function @math{ns(u,m)}.
304 @opencatbox{Categories:}
305 @category{Elliptic functions}
309 @anchor{inverse_jacobi_sc}
310 @deffn {Function} inverse_jacobi_sc (@var{u}, @var{m})
311 The inverse of the Jacobian elliptic function @math{sc(u,m)}.
313 @opencatbox{Categories:}
314 @category{Elliptic functions}
318 @anchor{inverse_jacobi_sd}
319 @deffn {Function} inverse_jacobi_sd (@var{u}, @var{m})
320 The inverse of the Jacobian elliptic function @math{sd(u,m)}.
322 @opencatbox{Categories:}
323 @category{Elliptic functions}
327 @anchor{inverse_jacobi_nc}
328 @deffn {Function} inverse_jacobi_nc (@var{u}, @var{m})
329 The inverse of the Jacobian elliptic function @math{nc(u,m)}.
331 @opencatbox{Categories:}
332 @category{Elliptic functions}
336 @anchor{inverse_jacobi_cs}
337 @deffn {Function} inverse_jacobi_cs (@var{u}, @var{m})
338 The inverse of the Jacobian elliptic function @math{cs(u,m)}.
340 @opencatbox{Categories:}
341 @category{Elliptic functions}
345 @anchor{inverse_jacobi_cd}
346 @deffn {Function} inverse_jacobi_cd (@var{u}, @var{m})
347 The inverse of the Jacobian elliptic function @math{cd(u,m)}.
349 @opencatbox{Categories:}
350 @category{Elliptic functions}
354 @anchor{inverse_jacobi_nd}
355 @deffn {Function} inverse_jacobi_nd (@var{u}, @var{m})
356 The inverse of the Jacobian elliptic function @math{nd(u,m)}.
358 @opencatbox{Categories:}
359 @category{Elliptic functions}
363 @anchor{inverse_jacobi_ds}
364 @deffn {Function} inverse_jacobi_ds (@var{u}, @var{m})
365 The inverse of the Jacobian elliptic function @math{ds(u,m)}.
367 @opencatbox{Categories:}
368 @category{Elliptic functions}
372 @anchor{inverse_jacobi_dc}
373 @deffn {Function} inverse_jacobi_dc (@var{u}, @var{m})
374 The inverse of the Jacobian elliptic function @math{dc(u,m)}.
376 @opencatbox{Categories:}
377 @category{Elliptic functions}
382 @node Functions and Variables for Elliptic Integrals, , Functions and Variables for Elliptic Functions, Top
383 @comment node-name, next, previous, up
385 @section Functions and Variables for Elliptic Integrals
388 @deffn {Function} elliptic_f (@var{phi}, @var{m})
389 The incomplete elliptic integral of the first kind, defined as
393 @math{ $$ \int_0^{\phi} {\frac{d\theta}{\sqrt{1-m\sin^2\theta}}} $$ }
396 @math{integrate(1/sqrt(1 - m*sin(x)^2), x, 0, phi)}
400 @math{integrate(1/sqrt(1 - m*sin(x)^2), x, 0, phi)}
404 $$ \int_0^{\phi} {\frac{d\theta}{\sqrt{1-m\sin^2\theta}}} $$
407 See also @ref{elliptic_e} and @ref{elliptic_kc}.
409 @opencatbox{Categories:}
410 @category{Elliptic integrals}
415 @deffn {Function} elliptic_e (@var{phi}, @var{m})
416 The incomplete elliptic integral of the second kind, defined as
420 @math{ $$ \int_0^\phi {\sqrt{1 - m\sin^2\theta}} d\theta $$ }
423 @math{elliptic_e(phi, m) = integrate(sqrt(1 - m*sin(x)^2), x, 0, phi)}
427 @math{elliptic_e(phi, m) = integrate(sqrt(1 - m*sin(x)^2), x, 0, phi)}
431 $$ \int_0^\phi \sqrt{1 - m\sin^2\theta} d\theta $$
434 See also @ref{elliptic_f} and @ref{elliptic_ec}.
436 @opencatbox{Categories:}
437 @category{Elliptic integrals}
442 @deffn {Function} elliptic_eu (@var{u}, @var{m})
443 The incomplete elliptic integral of the second kind, defined as
447 @math{ $$ \int_0^u {\rm dn}(v, m) dv = \int_0^\tau \sqrt{\frac{1-m t^2}{1-t^2}} dt $$ }
449 @math{ $$ \tau = sn(u,m) $$ }
452 @math{integrate(dn(v,m)^2,v,0,u) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2), t, 0, tau)}
453 where @math{tau = sn(u,m)}.
457 @math{integrate(dn(v,m)^2,v,0,u) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2), t, 0, tau)}
459 where @math{tau = sn(u,m)}.
462 $$ \int_0^u {\rm dn}(v, m) dv = \int_0^\tau \sqrt{\frac{1-m t^2}{1-t^2}} dt $$
464 where $\tau = {\rm sn}(u, m)$.
467 This is related to @code{elliptic_e} by
471 @math{ $$ E(u,m) = E(\phi, m) $$ }
473 @math{ $$ \phi = \sin^{-1} {\rm sn}(u, m) $$ }
476 @math{elliptic_eu(u, m) = elliptic_e(asin(sn(u,m)),m)}
480 @math{elliptic_eu(u, m) = elliptic_e(asin(sn(u,m)),m)}
484 $$E(u,m) = E(\phi, m)$$
486 where $\phi = \sin^{-1} {\rm sn}(u, m)$.
489 See also @ref{elliptic_e}.
490 @opencatbox{Categories:}
491 @category{Elliptic integrals}
496 @deffn {Function} elliptic_pi (@var{n}, @var{phi}, @var{m})
497 The incomplete elliptic integral of the third kind, defined as
501 @math{ $$ \int_0^\phi {{d\theta}\over{(1-n\sin^2 \theta)\sqrt{1 - m\sin^2\theta}}} $$ }
504 @math{integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, 0, phi)}
508 @math{integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, 0, phi)}
511 $$\int_0^\phi {{d\theta}\over{(1-n\sin^2 \theta)\sqrt{1 - m\sin^2\theta}}}$$
514 @opencatbox{Categories:}
515 @category{Elliptic integrals}
520 @deffn {Function} elliptic_kc (@var{m})
521 The complete elliptic integral of the first kind, defined as
525 @math{ $$ \int_0^{\frac{\pi}{2}} {{d\theta}\over{\sqrt{1 - m\sin^2\theta}}} $$ }
528 @math{integrate(1/sqrt(1 - m*sin(x)^2), x, 0, pi/2)}
532 @math{integrate(1/sqrt(1 - m*sin(x)^2), x, 0, pi/2)}
536 $$\int_0^{\frac{\pi}{2}} {{d\theta}\over{\sqrt{1 - m\sin^2\theta}}}$$
539 For certain values of @math{m}, the value of the integral is known in
540 terms of @math{Gamma} functions. Use @mref{makegamma} to evaluate them.
542 @opencatbox{Categories:}
543 @category{Elliptic integrals}
548 @deffn {Function} elliptic_ec (@var{m})
549 The complete elliptic integral of the second kind, defined as
553 @math{ $$ \int_0^{\frac{\pi}{2}} \sqrt{1 - m\sin^2\theta} d\theta $$ }
556 @math{integrate(sqrt(1 - m*sin(x)^2), x, 0, pi/2)}
560 @math{integrate(sqrt(1 - m*sin(x)^2), x, 0, pi/2)}
564 $$\int_0^{\frac{\pi}{2}} \sqrt{1 - m\sin^2\theta} d\theta$$
567 For certain values of @math{m}, the value of the integral is known in
568 terms of @math{Gamma} functions. Use @mref{makegamma} to evaluate them.
570 @opencatbox{Categories:}
571 @category{Elliptic integrals}
576 @deffn {Function} carlson_rc (@var{x}, @var{y})
577 Carlson's RC integral is defined by
581 @math{ $$ R_C(x, y) = \frac{1}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}(t+y)}\, dt $$ }
584 @math{integrate(1/2*(t+x)^(-1/2)/(t+y), t, 0, inf)}
588 @math{integrate(1/2*(t+x)^(-1/2)/(t+y), t, 0, inf)}
592 $$ R_C(x, y) = \frac{1}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}(t+y)}\, dt $$
595 This integral is related to many elementary functions in the following
603 \log x &= (x-1) R_C\left(\left({\frac{1+x}{2}}\right)^2, x\right), x > 0 \cr
604 \sin^{-1} x &= x R_C(1-x^2, 1), |x| \le 1 \cr
605 \cos^{-1} x &= \sqrt{1-x^2} R_C(x^2,1), 0 \le x \le 1 \cr
606 \tan^{-1} x &= x R_C(1,1+x^2) \cr
607 \sinh^{-1} x &= x R_C(1+x^2,1) \cr
608 \cosh^{-1} x &= \sqrt{x^2-1} R_C(x^2,1), x \ge 1 \cr
609 \tanh^{-1}(x) &= x R_C(1,1-x^2), |x| \le 1
615 @math{log(x) = (x-1)*rc(((1+x)/2)^2, x), x > 0}
617 @math{asin(x) = x * rc(1-x^2, 1), |x|<= 1}
619 @math{acos(x) = sqrt(1-x^2)*rc(x^2,1), 0 <= x <=1}
621 @math{atan(x) = x * rc(1,1+x^2)}
623 @math{asinh(x) = x * rc(1+x^2,1)}
625 @math{acosh(x) = sqrt(x^2-1) * rc(x^2,1), x >= 1}
627 @math{atanh(x) = x * rc(1,1-x^2), |x|<=1}
631 @math{log(x) = (x-1)*rc(((1+x)/2)^2, x), x > 0}
633 @math{asin(x) = x * rc(1-x^2, 1), |x|<= 1}
635 @math{acos(x) = sqrt(1-x^2)*rc(x^2,1), 0 <= x <=1}
637 @math{atan(x) = x * rc(1,1+x^2)}
639 @math{asinh(x) = x * rc(1+x^2,1)}
641 @math{acosh(x) = sqrt(x^2-1) * rc(x^2,1), x >= 1}
643 @math{atanh(x) = x * rc(1,1-x^2), |x|<=1}
649 \log x &= (x-1) R_C\left(\left({\frac{1+x}{2}}\right)^2, x\right), x > 0 \cr
650 \sin^{-1} x &= x R_C(1-x^2, 1), |x| \le 1 \cr
651 \cos^{-1} x &= \sqrt{1-x^2} R_C(x^2,1), 0 \le x \le 1 \cr
652 \tan^{-1} x &= x R_C(1,1+x^2) \cr
653 \sinh^{-1} x &= x R_C(1+x^2,1) \cr
654 \cosh^{-1} x &= \sqrt{x^2-1} R_C(x^2,1), x \ge 1 \cr
655 \tanh^{-1}(x) &= x R_C(1,1-x^2), |x| \le 1
660 Also, we have the relationship
666 R_C(x,y) = R_F(x,y,y)
671 @math{R_C(x,y) = R_F(x,y,y)}
676 @math{R_C(x,y) = R_F(x,y,y)}
681 R_C(x,y) = R_F(x,y,y)
690 \eqalign{R_C(0, 1) &= \frac{\pi}{2} \cr
691 R_C(0, 1/4) &= \pi \cr
692 R_C(2,1) &= \log(\sqrt{2} + 1) \cr
693 R_C(i,i+1) &= \frac{\pi}{4} + \frac{i}{2} \log(\sqrt{2}-1) \cr
694 R_C(0,i) &= (1-i)\frac{\pi}{2\sqrt{2}} \cr
700 @math{R_C(0,1) = pi/2}
702 @math{R_C(0,1/4) = pi}
704 @math{R_C(2,1) = log(sqrt(2)+1)}
706 @math{R_C(i, i+1) = pi/4 + i/2*log(sqrt(2)+1)}
708 @math{R_C(0, i) = (1-i)*pi/(2*sqrt(2))}
713 @math{R_C(0,1) = pi/2}
715 @math{R_C(0,1/4) = pi}
717 @math{R_C(2,1) = log(sqrt(2)+1)}
719 @math{R_C(i, i+1) = pi/4 + i/2*log(sqrt(2)+1)}
721 @math{R_C(0, i) = (1-i)*pi/(2*sqrt(2))}
726 \eqalign{R_C(0, 1) &= \frac{\pi}{2} \cr
727 R_C(0, 1/4) &= \pi \cr
728 R_C(2,1) &= \log(\sqrt{2} + 1) \cr
729 R_C(i,i+1) &= \frac{\pi}{4} + \frac{i}{2} \log(\sqrt{2}-1) \cr
730 R_C(0,i) &= (1-i)\frac{\pi}{2\sqrt{2}} \cr
736 @opencatbox{Categories:}
737 @category{Elliptic integrals}
742 @deffn {Function} carlson_rd (@var{x}, @var{y}, @var{z})
743 Carlson's RD integral is defined by
749 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
754 @math{R_D(x,y,z) = 3/2*integrate(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)*(t+z)), t, 0, inf)}
759 @math{R_D(x,y,z) = 3/2*integrate(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)*(t+z)), t, 0, inf)}
764 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
769 We also have the special values
776 R_D(x,x,x) &= x^{-\frac{3}{2}} \cr
777 R_D(0,y,y) &= \frac{3}{4} \pi y^{-\frac{3}{2}} \cr
778 R_D(0,2,1) &= 3 \sqrt{\pi} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}
784 @math{R_D(x,x,x) = x^(-3/2)}
786 @math{R_D(0,y,y) = 3/4*pi*y^(-3/2)}
788 @math{R_D(0,2,1) = 3 sqrt(pi) gamma(3/4)/gamma(1/4)}
793 @math{R_D(x,x,x) = x^(-3/2)}
795 @math{R_D(0,y,y) = 3/4*pi*y^(-3/2)}
797 @math{R_D(0,2,1) = 3 sqrt(pi) gamma(3/4)/gamma(1/4)}
803 R_D(x,x,x) &= x^{-\frac{3}{2}} \cr
804 R_D(0,y,y) &= \frac{3}{4} \pi y^{-\frac{3}{2}} \cr
805 R_D(0,2,1) &= 3 \sqrt{\pi} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}
812 It is also related to the complete elliptic E function as follows
818 E(m) = R_F(0, 1 - m, 1) - \frac{m}{3} R_D(0, 1 - m, 1)
823 @math{E(m) = R_F(0, 1 - m, 1) - (m/3)* R_D(0, 1 - m, 1)}
828 @math{E(m) = R_F(0, 1 - m, 1) - (m/3)* R_D(0, 1 - m, 1)}
833 E(m) = R_F(0, 1 - m, 1) - \frac{m}{3} R_D(0, 1 - m, 1)
838 @opencatbox{Categories:}
839 @category{Elliptic integrals}
844 @deffn {Function} carlson_rf (@var{x}, @var{y}, @var{z})
845 Carlson's RF integral is defined by
851 R_F(x,y,z) = \frac{1}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}}\, dt
856 @math{R_F(x,y,z) = 1/2*integrate(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)), t, 0, inf)}
861 @math{R_F(x,y,z) = 1/2*integrate(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)), t, 0, inf)}
866 R_F(x,y,z) = \frac{1}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}}\, dt
871 We also have the special values
878 R_F(0,1,2) &= \frac{\Gamma({\frac{1}{4}})^2}{4\sqrt{2\pi}} \cr
879 R_F(i,-i,0) &= \frac{\Gamma({\frac{1}{4}})^2}{4\sqrt{\pi}}
885 @math{R_F(0,1,2) = gamma(1/4)^2/(4*sqrt(2*pi))}
887 @math{R_F(i,-i,0) = gamma(1/4)^2/(4*sqrt(pi))}
892 @math{R_F(0,1,2) = gamma(1/4)^2/(4*sqrt(2*pi))}
894 @math{R_F(i,-i,0) = gamma(1/4)^2/(4*sqrt(pi))}
900 R_F(0,1,2) &= \frac{\Gamma({\frac{1}{4}})^2}{4\sqrt{2\pi}} \cr
901 R_F(i,-i,0) &= \frac{\Gamma({\frac{1}{4}})^2}{4\sqrt{\pi}}
908 It is also related to the complete elliptic E function as follows
914 E(m) = R_F(0, 1 - m, 1) - \frac{m}{3} R_D(0, 1 - m, 1)
919 @math{E(m) = R_F(0, 1 - m, 1) - (m/3)* R_D(0, 1 - m, 1)}
924 @math{E(m) = R_F(0, 1 - m, 1) - (m/3)* R_D(0, 1 - m, 1)}
929 E(m) = R_F(0, 1 - m, 1) - \frac{m}{3} R_D(0, 1 - m, 1)
935 @opencatbox{Categories:}
936 @category{Elliptic integrals}
941 @deffn {Function} carlson_rj (@var{x}, @var{y}, @var{z}, @var{p})
942 Carlson's RJ integral is defined by
948 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
953 @math{R_J(x,y,z) = 1/2*integrate(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)*(t+p)), t, 0, inf)}
958 @math{R_J(x,y,z) = 1/2*integrate(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)*(t+p)), t, 0, inf)}
963 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
968 @opencatbox{Categories:}
969 @category{Elliptic integrals}