Macsyma elliptic integrals
- Subject: Macsyma elliptic integrals
- From: Richard Fateman
- Date: Fri, 19 Oct 2007 17:08:32 -0700
Macsyma does do some elliptic integrals.
Just not the ones you said..
Here's stuff from their demo..
(c2) INTANALYSIS : FALSE$
(c3) INTEGRATE(1/(SQRT(1-T^2)*SQRT(2-T^2)),T,0,1);
1
elliptic_kc(-)
2
(d3) --------------
sqrt(2)
(c4) SFLOAT(%);
(d4) 1.31103
(c5) INTEGRATE(1/(SQRT(1-T)*SQRT(1+T)*SQRT(2-T^2)),T,0,1);
1
elliptic_kc(-)
2
(d5) --------------
sqrt(2)
(c6) INTEGRATE(1/(SQRT(4-T^2)*SQRT(9-T^2)),T,1,2);
4 %pi 4
elliptic_kc(-) elliptic_f(---, -)
9 6 9
(d6) -------------- - ------------------
3 3
(c7) SFLOAT(%);
(d7) 0.42515
(c8) INTEGRATE(SQRT(2-T^2)/SQRT(1-T^2),T,0,1);
1
(d8) sqrt(2) elliptic_ec(-)
2
(c9) SFLOAT(%);
(d9) 1.9101
(c10) INTEGRATE(SQRT(9-T^2)/SQRT(4-T^2),T,1,2);
4 %pi 4
(d10) 3 elliptic_ec(-) - 3 elliptic_e(---, -)
9 6 9
(c11) INTEGRATE(SQRT(3-T^2)/SQRT(2-T^2),T,0,SQRT(2));
2
(d11) sqrt(3) elliptic_ec(-)
3
(c12) INTEGRATE(SQRT(5-2*T^2)/SQRT(3-SQRT(2)*T^2),T,0,1);
1/4
2 6
sqrt(5) elliptic_e(asin(-------), ---------)
sqrt(3) 5 sqrt(2)
(d12) --------------------------------------------
1/4
2
(c13) INTEGRATE(SQRT(1-T^2)/SQRT(2-T^2),T,0,1);
1
elliptic_kc(-)
1 2
(d13) sqrt(2) elliptic_ec(-) - --------------
2 sqrt(2)
(c14) INTEGRATE(SQRT(4-T^2)/SQRT(9-T^2),T,1,2);
%pi 4 4
5 elliptic_f(---, -) 5 elliptic_kc(-)
6 9 %pi 4 9
(d14) -------------------- - 3 elliptic_e(---, -) - ----------------
3 6 9 3
4
+ 3 elliptic_ec(-)
9
(c15) INTEGRATE(SQRT(1-T^2)*SQRT(2-T^2),T,0,1);
1
sqrt(2) elliptic_kc(-)
1 2
(d15) sqrt(2) elliptic_ec(-) - ----------------------
2 3
(c16) INTEGRATE(SQRT(4-T^2)*SQRT(9-T^2),T,1,2);
%pi 4 %pi 4 4
(d16) 5 elliptic_f(---, -) - 13 elliptic_e(---, -) - 5 elliptic_kc(-)
6 9 6 9 9
4 2 sqrt(2) sqrt(3)
+ 13 elliptic_ec(-) - -----------------
9 3
(c17) INTEGRATE(1/(SQRT(T^2-2)*SQRT(T^2-4)),T,2,INF);
1
elliptic_kc(-)
2
(d17) --------------
2
(c18) INTEGRATE(1/(SQRT(T^2-2)*SQRT(T^2-4)),T,2,4);
1 %pi 1
elliptic_kc(-) elliptic_f(---, -)
2 6 2
(d18) -------------- - ------------------
2 2
(c19) INTEGRATE(1/(SQRT(T^2+1)*SQRT(T^2+2)),T,0,INF);
1
elliptic_kc(-)
2
(d19) --------------
sqrt(2)
(c20) INTEGRATE(1/(SQRT(T^2+1)*SQRT(T^2+2)),T,0,1);
%pi 1
elliptic_f(---, -)
4 2
(d20) ------------------
sqrt(2)
(c21) INTEGRATE(1/(SQRT(T^2+1)*SQRT(T^2+2)),T,1,2);
1 %pi 1
elliptic_f(atan(2), -) elliptic_f(---, -)
2 4 2
(d21) ---------------------- - ------------------
sqrt(2) sqrt(2)
(c22) INTEGRATE(1/(SQRT(T^2+1)*SQRT(T^2+2)),T,2,INF);
sqrt(2) 1
elliptic_f(atan(-------), -)
2 2
(d22) ----------------------------
sqrt(2)
(c23) 'INTEGRATE(1/SQRT((T-3)*(T-2)*(T-1)),T,4,6);
6
/
[ 1
(d23) I ----------------------------- dt
] sqrt((t - 3) (t - 2) (t - 1))
/
4
(c24) (ASSUME(U > 0),CHANGEVAR(%,U = SQRT(T-3),U,T));
sqrt(3)
/
[ 1
(d24) 2 I ------------------------- du
] 2 2
/ sqrt(u + 1) sqrt(u + 2)
1
(c25) (FORGET(U > 0),APPLY_NOUNS(%));
%pi 1 %pi 1
elliptic_f(---, -) elliptic_f(---, -)
3 2 4 2
(d25) 2 (------------------ - ------------------)
sqrt(2) sqrt(2)
(c26) [ELLIPTIC_F(%PI/2,M),ELLIPTIC_E(%PI/2,M)];
(d26) [elliptic_kc(m), elliptic_ec(m)]
(c27) [ELLIPTIC_KC(0),ELLIPTIC_EC(0),ELLIPTIC_EC(1)];
%pi %pi
(d27) [---, ---, 1]
2 2
(c28) LIMIT(ELLIPTIC_KC(M),M,1);
(d28) inf
(c29)
[ELLIPTIC_F(PHI,0),ELLIPTIC_E(PHI,0),ELLIPTIC_F(PHI,1),ELLIPTIC_E(PHI,1)];
(d29) [phi, phi, atanh(sin(phi)), sin(phi)]
(c30) ELLIPTIC_KC(1/2) = MAKEGAMMA(ELLIPTIC_KC(1/2));
2 1
gamma (-)
1 4
(d30) elliptic_kc(-) = -----------
2 4 sqrt(%pi)
(c31) ELLIPTIC_EC(1/2) = MAKEGAMMA(ELLIPTIC_EC(1/2));
2 3 2 1
gamma (-) gamma (-)
4 4
2 (--------- + ---------)
1 4 16
(d31) elliptic_ec(-) = -------------------------
2 sqrt(%pi)
(c32) ELLIPTIC_KC(1/2-SQRT(3)/4) = MAKEGAMMA(ELLIPTIC_KC(1/2-SQRT(3)/4));
1/4 3 1
3 gamma (-)
1 sqrt(3) 3
(d32) elliptic_kc(- - -------) = --------------
2 4 1/3
4 2 %pi
(c33) SFLOAT(%);
(d33) 1.59814 = 1.59814
(c34) [ELLIPTIC_F(-PHI,M),ELLIPTIC_E(-PHI,M)];
(d34) [- elliptic_f(phi, m), - elliptic_e(phi, m)]
(c35) [DIFF(ELLIPTIC_KC(M),M),DIFF(ELLIPTIC_EC(M),M)];
elliptic_ec(m) elliptic_kc(m)
(d35) [-------------- - --------------,
2 (1 - m) m 2 m
elliptic_ec(m) elliptic_kc(m)
-------------- - --------------]
2 m 2 m
(c36) [DIFF(ELLIPTIC_F(PHI,M),PHI),DIFF(ELLIPTIC_E(PHI,M),PHI)];
1 2
(d36) [---------------------, sqrt(1 - m sin (phi))]
2
sqrt(1 - m sin (phi))
(c37) [DIFF(ELLIPTIC_F(PHI,M),M),DIFF(ELLIPTIC_E(PHI,M),M)];
elliptic_f(phi, m) elliptic_e(phi, m)
(d37) [- ------------------ + ------------------
2 m 2 (1 - m) m
cos(phi) sin(phi)
- -------------------------------,
2
2 (1 - m) sqrt(1 - m sin (phi))
elliptic_e(phi, m) elliptic_f(phi, m)
------------------ - ------------------]
2 m 2 m
(c38) LR :
ELLIPTIC_EC(M)*ELLIPTIC_KC(1-M)+ELLIPTIC_EC(1-M)*ELLIPTIC_KC(M)-ELLIPTIC_KC(
M)*ELLIPTIC_KC(1-M);
(d38) - elliptic_kc(1 - m) elliptic_kc(m)
+ elliptic_ec(1 - m) elliptic_kc(m)
+ elliptic_kc(1 - m) elliptic_ec(m)
(c39) [SUBST(0.4,M,LR),SFLOAT(%PI/2)];
(d39) [1.57079, 1.5708]
(c40) RATSIMP(DIFF(LR,M));
(d40) 0
(c41) [POWERSERIES(ELLIPTIC_KC(M),M,0),POWERSERIES(ELLIPTIC_EC(M),M,0)];
inf 2 2 i1 - 1 i1
==== %pi genfact (2 i1 - 1, --------, 2) m
\ 2
(d41) [ > ---------------------------------------,
/ 2 i1 2
==== 2 2 i1!
i1 = 0
inf 2 2 i2 - 1 i2
==== %pi genfact (2 i2 - 1, --------, 2) m
\ 2
> (- ---------------------------------------)]
/ 2 i2 2
==== 2 2 (2 i2 - 1) i2!
i2 = 0
(c42) [TAYLOR(ELLIPTIC_KC(M),M,0,4),TAYLOR(ELLIPTIC_EC(M),M,0,4)];
2 3 4
%pi %pi m (9 %pi) m (25 %pi) m (1225 %pi) m
(d42)/T/ [--- + ----- + ---------- + ----------- + -------------
2 8 128 512 32768
2 3 4
%pi %pi m (3 %pi) m (5 %pi) m (175 %pi) m
+ . . ., --- - ----- - ---------- - ---------- - ------------
2 8 128 512 32768
+ . . .]
(c43) LIMIT((ELLIPTIC_KC(M)-ELLIPTIC_EC(M))/M,M,0);
%pi
(d43) ---
4
(c44) TLIMIT((ELLIPTIC_KC(M)-ELLIPTIC_EC(M))/M,M,0);
%pi
(d44) ---
4
(c45) [ELLIPTK(M),ELLIPTK1(M),ELLIPTE(M),ELLIPTE1(M)];
(d45) [elliptic_kc(m), elliptic_kc(1 - m), elliptic_ec(m),
elliptic_ec(1 - m)]
(c46) (REMVALUE(LR),RETSET(INTANALYSIS))$
(