diff --git a/changelog b/changelog index 8cfbeaa..c8c65b6 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,5 @@ +20130124 tpd src/axiom-website/patches.html 20130124.03.tpd.patch +20120124 tpd src/input/trigtests.input test new hyperbolic identities 20130124 tpd src/axiom-website/patches.html 20130124.02.tpd.patch 20130124 tpd books/bookvol10.4 expand hyperbolic identities 20130124 tpd src/axiom-website/patches.html 20130124.01.tpd.patch diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index ff185ed..43eaa45 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -3959,5 +3959,7 @@ src/Makefile clean up src/algebra properly books/bookvol10.4 document htrigs function in TRMANIP 20130124.02.tpd.patch books/bookvol10.4 expand hyperbolic identities +20130124.03.tpd.patch +src/input/trigtests.input test new hyperbolic identities diff --git a/src/input/trigtests.input.pamphlet b/src/input/trigtests.input.pamphlet index da3cc78..e47e386 100644 --- a/src/input/trigtests.input.pamphlet +++ b/src/input/trigtests.input.pamphlet @@ -22,7 +22,7 @@ ------------------------------------- -- Hyperbolic sine of x, p26, 8.1 ---S 1 of 89 +--S 1 of 118 t01:= sinh(x) - (exp(x)-exp(-x))/2 --R --R @@ -33,1037 +33,1270 @@ t01:= sinh(x) - (exp(x)-exp(-x))/2 --R Type: Expression(Integer) --E 1 +--S 2 of 118 +htrigs t01 +--R +--R +--R (2) 0 +--R Type: Expression(Integer) +--E 2 + -- Hyperbolic cosine of x, p26, 8.2 ---S 2 of 89 +--S 3 of 118 t02:= cosh(x) - (exp(x)+exp(-x))/2 --R --R --R x - x --R - %e - %e + 2cosh(x) ---R (2) ------------------------ +--R (3) ------------------------ --R 2 --R Type: Expression(Integer) ---E 2 +--E 3 + +--S 4 of 118 +htrigs t02 +--R +--R +--R (4) 0 +--R Type: Expression(Integer) +--E 4 + -- Hyperbolic tangent of x, p26, 8.3 ---S 3 of 89 +--S 5 of 118 t03:= tanh(x) - (exp(x)-exp(-x))/(exp(x)+exp(-x)) --R --R --R x - x x - x --R (%e + %e )tanh(x) - %e + %e ---R (3) ---------------------------------- +--R (5) ---------------------------------- --R x - x --R %e + %e --R Type: Expression(Integer) ---E 3 +--E 5 + +--S 6 of 118 +simplify htrigs t03 +--R +--R +--R (6) 0 +--R Type: Expression(Integer) +--E 6 -- Hyperbolic cotangent of x, p26, 8.4 ---S 4 of 89 +--S 7 of 118 t04:= coth(x) - (exp(x)+exp(-x))/(exp(x)-exp(-x)) --R --R --R x - x --R (coth(x) - 1)%e + (- coth(x) - 1)%e ---R (4) --------------------------------------- +--R (7) --------------------------------------- --R x - x --R %e - %e --R Type: Expression(Integer) ---E 4 +--E 7 + +--S 8 of 118 +simplify htrigs t04 +--R +--R +--R (8) 0 +--R Type: Expression(Integer) +--E 8 -- Hyperbolic secant of x, p26, 8.5 ---S 5 of 89 +--S 9 of 118 t05:= sech(x) - 2/(exp(x)+exp(-x)) --R --R --R x - x --R (%e + %e )sech(x) - 2 ---R (5) ------------------------ +--R (9) ------------------------ --R x - x --R %e + %e --R Type: Expression(Integer) ---E 5 +--E 9 + +--S 10 of 118 +simplify htrigs t05 +--R +--R +--R (10) 0 +--R Type: Expression(Integer) +--E 10 -- Hyperbolic cosecant of x, p26, 8.6 ---S 6 of 89 +--S 11 of 118 t06:= csch(x) - 2/(exp(x)-exp(-x)) --R --R ---R x - x ---R csch(x)%e - csch(x)%e - 2 ---R (6) ----------------------------- ---R x - x ---R %e - %e +--R x - x +--R csch(x)%e - csch(x)%e - 2 +--R (11) ----------------------------- +--R x - x +--R %e - %e --R Type: Expression(Integer) ---E 6 +--E 11 + +--S 12 of 118 +simplify htrigs t06 +--R +--R +--R (12) 0 +--R Type: Expression(Integer) +--E 12 ------------------------------------------- -- Relationships among Hyperbolic Functions ------------------------------------------- -- Hyperbolic tangent of x, p26, 8.7 ---S 7 of 89 +--S 13 of 118 t07:= tanh(x) - sinh(x)/cosh(x) --R --R ---R cosh(x)tanh(x) - sinh(x) ---R (7) ------------------------ ---R cosh(x) +--R cosh(x)tanh(x) - sinh(x) +--R (13) ------------------------ +--R cosh(x) --R Type: Expression(Integer) ---E 7 +--E 13 + +--S 14 of 118 +simplify t07 +--R +--R +--R (14) 0 +--R Type: Expression(Integer) +--E 14 -- Hyperbolic cotangent of x, p26, 8.8 ---S 8 of 89 +--S 15 of 118 t08:= coth(x) - 1/tanh(x) --R --R ---R coth(x)tanh(x) - 1 ---R (8) ------------------ ---R tanh(x) +--R coth(x)tanh(x) - 1 +--R (15) ------------------ +--R tanh(x) --R Type: Expression(Integer) ---E 8 +--E 15 + +--S 16 of 118 +simplify t08 +--R +--R +--R (16) 0 +--R Type: Expression(Integer) +--E 16 -- Hyperbolic cotangent of x, p26, 8.8 ---S 9 of 89 +--S 17 of 118 t09:= coth(x) - cosh(x)/sinh(x) --R --R ---R coth(x)sinh(x) - cosh(x) ---R (9) ------------------------ ---R sinh(x) +--R coth(x)sinh(x) - cosh(x) +--R (17) ------------------------ +--R sinh(x) --R Type: Expression(Integer) ---E 9 +--E 17 + +--S 18 of 118 +simplify t09 +--R +--R +--R (18) 0 +--R Type: Expression(Integer) +--E 18 -- Hyperbolic secant of x, p26, 8.9 ---S 10 of 89 +--S 19 of 118 t10:= sech(x) - 1/cosh(x) --R --R --R cosh(x)sech(x) - 1 ---R (10) ------------------ +--R (19) ------------------ --R cosh(x) --R Type: Expression(Integer) ---E 10 +--E 19 + +--S 20 of 118 +simplify t10 +--R +--R +--R (20) 0 +--R Type: Expression(Integer) +--E 20 -- Hyperbolic cosecant of x, p26, 8.10 ---S 11 of 89 +--S 21 of 118 t11:= csch(x) - 1/sinh(x) --R --R --R csch(x)sinh(x) - 1 ---R (11) ------------------ +--R (21) ------------------ --R sinh(x) --R Type: Expression(Integer) ---E 11 +--E 21 + +--S 22 of 118 +simplify t11 +--R +--R +--R (22) 0 +--R Type: Expression(Integer) +--E 22 -- p26, 8.11 ---S 12 of 89 +--S 23 of 118 t12:= cosh(x)^2-sinh(x)^2 - 1 --R --R --R 2 2 ---R (12) - sinh(x) + cosh(x) - 1 +--R (23) - sinh(x) + cosh(x) - 1 --R Type: Expression(Integer) ---E 12 +--E 23 + +--S 24 of 118 +simplify t12 +--R +--R +--R (24) 0 +--R Type: Expression(Integer) +--E 24 -- p26, 8.12 ---S 13 of 89 +--S 25 of 118 t13:= sech(x)^2+tanh(x)^2 - 1 --R --R --R 2 2 ---R (13) tanh(x) + sech(x) - 1 +--R (25) tanh(x) + sech(x) - 1 --R Type: Expression(Integer) ---E 13 +--E 25 + +--S 26 of 118 +simplify t13 +--R +--R +--R (26) 0 +--R Type: Expression(Integer) +--E 26 -- p26, 8.13 ---S 14 of 89 +--S 27 of 118 t14:= coth(x)^2-csch(x)^2 - 1 --R --R --R 2 2 ---R (14) - csch(x) + coth(x) - 1 +--R (27) - csch(x) + coth(x) - 1 --R Type: Expression(Integer) ---E 14 +--E 27 + +--S 28 of 118 +simplify t14 +--R +--R +--R (28) 0 +--R Type: Expression(Integer) +--E 28 ---------------------------------- -- Functions of Negative Arguments ---------------------------------- -- p26, 8.14 ---S 15 of 89 +--S 29 of 118 t15:= sinh(-x) - (-sinh(x)) --R --R ---R (15) 0 +--R (29) 0 --R Type: Expression(Integer) ---E 15 +--E 29 -- p26, 8.15 ---S 16 of 89 +--S 30 of 118 t16:= cosh(-x) - cosh(x) --R --R ---R (16) 0 +--R (30) 0 --R Type: Expression(Integer) ---E 16 +--E 30 -- p26, 8.16 ---S 17 of 89 +--S 31 of 118 t17:= tanh(-x) - (-tanh(x)) --R --R ---R (17) 0 +--R (31) 0 --R Type: Expression(Integer) ---E 17 +--E 31 -- p26, 8.17 ---S 18 of 89 +--S 32 of 118 t18:= csch(-x) - (-csch(x)) --R --R ---R (18) 0 +--R (32) 0 --R Type: Expression(Integer) ---E 18 +--E 32 -- p26, 8.18 ---S 19 of 89 +--S 33 of 118 t19:= sech(-x) - sech(x) --R --R ---R (19) 0 +--R (33) 0 --R Type: Expression(Integer) ---E 19 +--E 33 -- p26, 8.19 ---S 20 of 89 +--S 34 of 118 t20:= coth(-x) - (-coth(x)) --R --R ---R (20) 0 +--R (34) 0 --R Type: Expression(Integer) ---E 20 +--E 34 -------------------- -- Addition Formulas -------------------- -- p27, 8.20 ---S 21 of 89 -t21:= sinh(x+y) - sinh(x)*cosh(y)+cosh(x)*sinh(y) +--S 35 of 118 +t21:= sinh(x+y) - (sinh(x)*cosh(y)+cosh(x)*sinh(y)) --R --R ---R (21) sinh(y + x) + cosh(x)sinh(y) - cosh(y)sinh(x) +--R (35) sinh(y + x) - cosh(x)sinh(y) - cosh(y)sinh(x) --R Type: Expression(Integer) ---E 21 +--E 35 ---S 22 of 89 -t22:= sinh(x-y) - sinh(x)*cosh(y)-cosh(x)*sinh(y) +--S 36 of 118 +expandTrigProducts t21 --R --R ---R (22) - cosh(x)sinh(y) - sinh(y - x) - cosh(y)sinh(x) +--R (36) 0 --R Type: Expression(Integer) ---E 22 +--E 36 + +--S 37 of 118 +t22:= sinh(x-y) - (sinh(x)*cosh(y)-cosh(x)*sinh(y)) +--R +--R +--R (37) cosh(x)sinh(y) - sinh(y - x) - cosh(y)sinh(x) +--R Type: Expression(Integer) +--E 37 + +--S 38 of 118 +expandTrigProducts t22 +--R +--R +--R (38) 0 +--R Type: Expression(Integer) +--E 38 -- p27, 8.21 ---S 23 of 89 -t23:= cosh(x+y) - cosh(x)*cosh(y)+sinh(x)*sinh(y) +--S 39 of 118 +t23:= cosh(x+y) - (cosh(x)*cosh(y)+sinh(x)*sinh(y)) --R --R ---R (23) sinh(x)sinh(y) + cosh(y + x) - cosh(x)cosh(y) +--R (39) - sinh(x)sinh(y) + cosh(y + x) - cosh(x)cosh(y) --R Type: Expression(Integer) ---E 23 +--E 39 ---S 24 of 89 -t24:= cosh(x-y) - cosh(x)*cosh(y)-sinh(x)*sinh(y) +--S 40 of 118 +expandTrigProducts t23 --R --R ---R (24) - sinh(x)sinh(y) - cosh(x)cosh(y) + cosh(y - x) +--R (40) 0 --R Type: Expression(Integer) ---E 24 +--E 40 + +--S 41 of 118 +t24:= cosh(x-y) - (cosh(x)*cosh(y)-sinh(x)*sinh(y)) +--R +--R +--R (41) sinh(x)sinh(y) - cosh(x)cosh(y) + cosh(y - x) +--R Type: Expression(Integer) +--E 41 + +--S 42 of 118 +expandTrigProducts t24 +--R +--R +--R (42) 0 +--R Type: Expression(Integer) +--E 42 -- p27, 8.22 ---S 25 of 89 +--S 43 of 118 t25:= tanh(x+y) - (tanh(x)+tanh(y))/(1+tanh(x)*tanh(y)) --R --R --R (tanh(x)tanh(y) + 1)tanh(y + x) - tanh(y) - tanh(x) ---R (25) --------------------------------------------------- +--R (43) --------------------------------------------------- --R tanh(x)tanh(y) + 1 --R Type: Expression(Integer) ---E 25 +--E 43 ---S 26 of 89 +--S 44 of 118 +expandTrigProducts t25 +--R +--R +--R (44) 0 +--R Type: Expression(Integer) +--E 44 + +--S 45 of 118 t26:= tanh(x-y) - (tanh(x)-tanh(y))/(1-tanh(x)*tanh(y)) --R --R --R (- tanh(x)tanh(y - x) - 1)tanh(y) + tanh(y - x) + tanh(x) ---R (26) --------------------------------------------------------- +--R (45) --------------------------------------------------------- --R tanh(x)tanh(y) - 1 --R Type: Expression(Integer) ---E 26 +--E 45 + +--S 46 of 118 +expandTrigProducts t26 +--R +--R +--R (46) 0 +--R Type: Expression(Integer) +--E 46 -- p27, 8.23 ---S 27 of 89 +--S 47 of 118 t27:= coth(x+y) - (coth(x)*coth(y)+1)/(coth(y)+coth(x)) --R --R --R (coth(y) + coth(x))coth(y + x) - coth(x)coth(y) - 1 ---R (27) --------------------------------------------------- +--R (47) --------------------------------------------------- --R coth(y) + coth(x) --R Type: Expression(Integer) ---E 27 +--E 47 ---S 28 of 89 +--S 48 of 118 +expandTrigProducts t27 +--R +--R +--R (48) 0 +--R Type: Expression(Integer) +--E 48 + +--S 49 of 118 t28:= coth(x-y) - (coth(x)*coth(y)-1)/(coth(y)-coth(x)) --R --R --R (- coth(y - x) - coth(x))coth(y) + coth(x)coth(y - x) + 1 ---R (28) --------------------------------------------------------- +--R (49) --------------------------------------------------------- --R coth(y) - coth(x) --R Type: Expression(Integer) ---E 28 +--E 49 + +--S 50 of 118 +expandTrigProducts t28 +--R +--R +--R (50) 0 +--R Type: Expression(Integer) +--E 50 ------------------------ -- Double Angle Formulas ------------------------ -- p27, 8.24 ---S 29 of 89 +--S 51 of 118 t29:= sinh(2*x) - 2*sinh(x)*cosh(x) --R --R ---R (29) sinh(2x) - 2cosh(x)sinh(x) +--R (51) sinh(2x) - 2cosh(x)sinh(x) --R Type: Expression(Integer) ---E 29 +--E 51 -- p27, 8.25 ---S 30 of 89 +--S 52 of 118 t30:= cosh(2*x) - cosh(x)^2+sinh(x)^2 --R --R --R 2 2 ---R (30) sinh(x) + cosh(2x) - cosh(x) +--R (52) sinh(x) + cosh(2x) - cosh(x) --R Type: Expression(Integer) ---E 30 +--E 52 ---S 31 of 89 +--S 53 of 118 t31:= cosh(2*x) - 2*cosh(x)^2-1 --R --R --R 2 ---R (31) cosh(2x) - 2cosh(x) - 1 +--R (53) cosh(2x) - 2cosh(x) - 1 --R Type: Expression(Integer) ---E 31 +--E 53 ---S 32 of 89 +--S 54 of 118 t32:= cosh(2*x) - 1+2*sinh(x)^2 --R --R --R 2 ---R (32) 2sinh(x) + cosh(2x) - 1 +--R (54) 2sinh(x) + cosh(2x) - 1 --R Type: Expression(Integer) ---E 32 +--E 54 -- p27, 8.26 ---S 33 of 89 +--S 55 of 118 t33:= tanh(2*x) - (2*tanh(x))/(1+tanh(x)^2) --R --R --R 2 --R (tanh(x) + 1)tanh(2x) - 2tanh(x) ---R (33) --------------------------------- +--R (55) --------------------------------- --R 2 --R tanh(x) + 1 --R Type: Expression(Integer) ---E 33 +--E 55 ---------------------- -- Half Angle Formulas ---------------------- -- p27, 8.27, assume x > 0 ---S 34 of 89 +--S 56 of 118 t34:= sinh(x/2) - sqrt((cosh(x)-1)/2) --R --R --R +-----------+ +-+ x --R - \|cosh(x) - 1 + \|2 sinh(-) --R 2 ---R (34) ------------------------------ +--R (56) ------------------------------ --R +-+ --R \|2 --R Type: Expression(Integer) ---E 34 +--E 56 -- p27, 8.27, assume x < 0 ---S 35 of 89 +--S 57 of 118 t35:= sinh(x/2) - (-sqrt((cosh(x)-1)/2)) --R --R --R +-----------+ +-+ x --R \|cosh(x) - 1 + \|2 sinh(-) --R 2 ---R (35) ---------------------------- +--R (57) ---------------------------- --R +-+ --R \|2 --R Type: Expression(Integer) ---E 35 +--E 57 -- p27, 8.28 ---S 36 of 89 +--S 58 of 118 t36:= cosh(x/2) - sqrt((cosh(x)+1)/2) --R --R --R +-----------+ +-+ x --R - \|cosh(x) + 1 + \|2 cosh(-) --R 2 ---R (36) ------------------------------ +--R (58) ------------------------------ --R +-+ --R \|2 --R Type: Expression(Integer) ---E 36 +--E 58 -- p27, 8.29, assume x > 0 ---S 37 of 89 +--S 59 of 118 t37:= tanh(x/2) - sqrt((cosh(x)-1)/(cosh(x)+1)) --R --R --R +-----------+ --R |cosh(x) - 1 x ---R (37) - |----------- + tanh(-) +--R (59) - |----------- + tanh(-) --R \|cosh(x) + 1 2 --R Type: Expression(Integer) ---E 37 +--E 59 -- p27, 8.29, assume x < 0 ---S 38 of 89 +--S 60 of 118 t38:= tanh(x/2) - (-sqrt((cosh(x)-1)/(cosh(x)+1))) --R --R --R +-----------+ --R |cosh(x) - 1 x ---R (38) |----------- + tanh(-) +--R (60) |----------- + tanh(-) --R \|cosh(x) + 1 2 --R Type: Expression(Integer) ---E 38 +--E 60 -- p27, 8.29 ---S 39 of 89 +--S 61 of 118 t39:= tanh(x/2) - sinh(x)/(cosh(x)+1) --R --R --R x --R (cosh(x) + 1)tanh(-) - sinh(x) --R 2 ---R (39) ------------------------------ +--R (61) ------------------------------ --R cosh(x) + 1 --R Type: Expression(Integer) ---E 39 +--E 61 -- p27, 8.29 ---S 40 of 89 +--S 62 of 118 t40:= tanh(x/2) - (cosh(x)-1)/sinh(x) --R --R --R x --R sinh(x)tanh(-) - cosh(x) + 1 --R 2 ---R (40) ---------------------------- +--R (62) ---------------------------- --R sinh(x) --R Type: Expression(Integer) ---E 40 +--E 62 -------------------------- -- Multiple Angle Formulas -------------------------- -- p27, 8.30 ---S 41 of 89 +--S 63 of 118 t41:= sinh(3*x) - 3*sinh(x)+4*sinh(x)^3 --R --R --R 3 ---R (41) sinh(3x) + 4sinh(x) - 3sinh(x) +--R (63) sinh(3x) + 4sinh(x) - 3sinh(x) --R Type: Expression(Integer) ---E 41 +--E 63 -- p27, 8.31 ---S 42 of 89 +--S 64 of 118 t42:= cosh(3*x) - 4*cosh(x)^3-3*cosh(x) --R --R --R 3 ---R (42) cosh(3x) - 4cosh(x) - 3cosh(x) +--R (64) cosh(3x) - 4cosh(x) - 3cosh(x) --R Type: Expression(Integer) ---E 42 +--E 64 -- p27, 8.32 ---S 43 of 89 +--S 65 of 118 t43:= tanh(3*x) - (3*tanh(x)+tanh(x)^3)/(1+3*tanh(x)^2) --R --R --R 2 3 --R (3tanh(x) + 1)tanh(3x) - tanh(x) - 3tanh(x) ---R (43) --------------------------------------------- +--R (65) --------------------------------------------- --R 2 --R 3tanh(x) + 1 --R Type: Expression(Integer) ---E 43 +--E 65 -- p27, 8.33 ---S 44 of 89 +--S 66 of 118 t44:= sinh(4*x) - 8*sinh(x)^3*cosh(x)+4*sinh(x)*cosh(x) --R --R --R 3 ---R (44) sinh(4x) - 8cosh(x)sinh(x) + 4cosh(x)sinh(x) +--R (66) sinh(4x) - 8cosh(x)sinh(x) + 4cosh(x)sinh(x) --R Type: Expression(Integer) ---E 44 +--E 66 -- p27, 8.34 ---S 45 of 89 +--S 67 of 118 t45:= cosh(4*x) - 8*cosh(x)^4-8*cosh(x)^2+1 --R --R --R 4 2 ---R (45) cosh(4x) - 8cosh(x) - 8cosh(x) + 1 +--R (67) cosh(4x) - 8cosh(x) - 8cosh(x) + 1 --R Type: Expression(Integer) ---E 45 +--E 67 -- p27, 8.35 ---S 46 of 89 +--S 68 of 118 t46:= tanh(4*x) - (4*tanh(x)+4*tanh(x)^3)/(1+6*tanh(x)^2+tanh(x)^4) --R --R --R 4 2 3 --R (tanh(x) + 6tanh(x) + 1)tanh(4x) - 4tanh(x) - 4tanh(x) ---R (46) --------------------------------------------------------- +--R (68) --------------------------------------------------------- --R 4 2 --R tanh(x) + 6tanh(x) + 1 --R Type: Expression(Integer) ---E 46 +--E 68 --------------------------------- -- Powers of Hyperbolic Functions --------------------------------- -- p28, 8.36 ---S 47 of 89 +--S 69 of 118 t47:= sinh(x)^2 - cosh(2*x)/2-1/2 --R --R --R 2 --R 2sinh(x) - cosh(2x) - 1 ---R (47) ------------------------ +--R (69) ------------------------ --R 2 --R Type: Expression(Integer) ---E 47 +--E 69 -- p28, 8.37 ---S 48 of 89 +--S 70 of 118 t48:= cosh(x)^2 - cosh(2*x)/2+1/2 --R --R --R 2 --R - cosh(2x) + 2cosh(x) + 1 ---R (48) -------------------------- +--R (70) -------------------------- --R 2 --R Type: Expression(Integer) ---E 48 +--E 70 -- p28, 8.38 ---S 49 of 89 +--S 71 of 118 t49:= sinh(x)^3 - sinh(3*x)/4-3*sinh(x)/4 --R --R --R 3 --R - sinh(3x) + 4sinh(x) - 3sinh(x) ---R (49) --------------------------------- +--R (71) --------------------------------- --R 4 --R Type: Expression(Integer) ---E 49 +--E 71 -- p28, 8.39 ---S 50 of 89 +--S 72 of 118 t50:= cosh(x)^3 - cosh(3*x)/4+3*cosh(x)/4 --R --R --R 3 --R - cosh(3x) + 4cosh(x) + 3cosh(x) ---R (50) --------------------------------- +--R (72) --------------------------------- --R 4 --R Type: Expression(Integer) ---E 50 +--E 72 -- p28, 8.40 ---S 51 of 89 +--S 73 of 118 t51:= sinh(x)^4 - 3/8-cosh(2*x)/2+cosh(4*x)/8 --R --R --R 4 --R 8sinh(x) + cosh(4x) - 4cosh(2x) - 3 ---R (51) ------------------------------------ +--R (73) ------------------------------------ --R 8 --R Type: Expression(Integer) ---E 51 +--E 73 -- p28, 8.41 ---S 52 of 89 +--S 74 of 118 t52:= cosh(x)^4 - 3/8+cosh(2*x)/2+cosh(4*x)/8 --R --R --R 4 --R cosh(4x) + 4cosh(2x) + 8cosh(x) - 3 ---R (52) ------------------------------------ +--R (74) ------------------------------------ --R 8 --R Type: Expression(Integer) ---E 52 +--E 74 ------------------------------------------------------- --- Sum, Difference and Product of Hyperbolic Functions ------------------------------------------------------- -- p28, 8.42 ---S 53 of 89 +--S 75 of 118 t53:= sinh(x)+sinh(y) - 2*sinh((x+y)/2)*cosh((x-y)/2) --R --R --R y - x y + x ---R (53) sinh(y) - 2cosh(-----)sinh(-----) + sinh(x) +--R (75) sinh(y) - 2cosh(-----)sinh(-----) + sinh(x) --R 2 2 --R Type: Expression(Integer) ---E 53 +--E 75 + +--S 76 of 118 +expandTrigProducts t53 +--R +--R +--R (76) 0 +--R Type: Expression(Integer) +--E 76 -- p28, 8.43 ---S 54 of 89 +--S 77 of 118 t54:= sinh(x)-sinh(y) - 2*cosh((x+y)/2)*sinh((x-y)/2) --R --R --R y + x y - x ---R (54) - sinh(y) + 2cosh(-----)sinh(-----) + sinh(x) +--R (77) - sinh(y) + 2cosh(-----)sinh(-----) + sinh(x) --R 2 2 --R Type: Expression(Integer) ---E 54 +--E 77 + +--S 78 of 118 +expandTrigProducts t54 +--R +--R +--R (78) 0 +--R Type: Expression(Integer) +--E 78 -- p28, 8.44 ---S 55 of 89 +--S 79 of 118 t55:= cosh(x)+cosh(y) - 2*cosh((x+y)/2)*cosh((x-y)/2) --R --R --R y - x y + x ---R (55) cosh(y) - 2cosh(-----)cosh(-----) + cosh(x) +--R (79) cosh(y) - 2cosh(-----)cosh(-----) + cosh(x) --R 2 2 --R Type: Expression(Integer) ---E 55 +--E 79 + +--S 80 of 118 +expandTrigProducts t55 +--R +--R +--R (80) 0 +--R Type: Expression(Integer) +--E 80 -- p28, 8.45 ---S 56 of 89 +--S 81 of 118 t56:= cosh(x)-cosh(y) - 2*sinh((x+y)/2)*sinh((x-y)/2) --R --R --R y - x y + x ---R (56) 2sinh(-----)sinh(-----) - cosh(y) + cosh(x) +--R (81) 2sinh(-----)sinh(-----) - cosh(y) + cosh(x) --R 2 2 --R Type: Expression(Integer) ---E 56 +--E 81 + +--S 82 of 118 +expandTrigProducts t56 +--R +--R +--R (82) 0 +--R Type: Expression(Integer) +--E 82 -- p28, 8.46 ---S 57 of 89 +--S 83 of 118 t57:= sinh(x)*sinh(y) - (cosh(x+y)-cosh(x-y))/2 --R --R --R 2sinh(x)sinh(y) - cosh(y + x) + cosh(y - x) ---R (57) ------------------------------------------- +--R (83) ------------------------------------------- --R 2 --R Type: Expression(Integer) ---E 57 +--E 83 + +--S 84 of 118 +expandTrigProducts t57 +--R +--R +--R (84) 0 +--R Type: Expression(Integer) +--E 84 -- p28, 8.47 ---S 58 of 89 +--S 85 of 118 t58:= cosh(x)*cosh(y) - (cosh(x+y)+cosh(x-y))/2 --R --R --R - cosh(y + x) + 2cosh(x)cosh(y) - cosh(y - x) ---R (58) --------------------------------------------- +--R (85) --------------------------------------------- --R 2 --R Type: Expression(Integer) ---E 58 +--E 85 + +--S 86 of 118 +expandTrigProducts t58 +--R +--R +--R (86) 0 +--R Type: Expression(Integer) +--E 86 -- p28, 8.48 ---S 59 of 89 +--S 87 of 118 t59:= sinh(x)*cosh(y) - (sinh(x+y)+sinh(x-y))/2 --R --R --R - sinh(y + x) + sinh(y - x) + 2cosh(y)sinh(x) ---R (59) --------------------------------------------- +--R (87) --------------------------------------------- --R 2 --R Type: Expression(Integer) ---E 59 +--E 87 + +--S 88 of 118 +expandTrigProducts t59 +--R +--R +--R (88) 0 +--R Type: Expression(Integer) +--E 88 -------------------------------------------------------- -- Expression of Hyperbolic Functions in terms of others -------------------------------------------------------- -- p28, table ---S 60 of 89 +--S 89 of 118 t60:= sinh(x) - sqrt(cosh(x)-1) --R --R --R +-----------+ ---R (60) - \|cosh(x) - 1 + sinh(x) +--R (89) - \|cosh(x) - 1 + sinh(x) --R Type: Expression(Integer) ---E 60 +--E 89 ---S 61 of 89 +--S 90 of 118 t61:= sinh(x) - tanh(x)/sqrt(1-tanh(x)^2) --R --R --R +--------------+ --R | 2 --R sinh(x)\|- tanh(x) + 1 - tanh(x) ---R (61) ---------------------------------- +--R (90) ---------------------------------- --R +--------------+ --R | 2 --R \|- tanh(x) + 1 --R Type: Expression(Integer) ---E 61 +--E 90 ---S 62 of 89 +--S 91 of 118 t62:= sinh(x) - 1/sqrt(coth(x)^2-1) --R --R --R +------------+ --R | 2 --R sinh(x)\|coth(x) - 1 - 1 ---R (62) -------------------------- +--R (91) -------------------------- --R +------------+ --R | 2 --R \|coth(x) - 1 --R Type: Expression(Integer) ---E 62 +--E 91 ---S 63 of 89 +--S 92 of 118 t63:= sinh(x) - sqrt(1-sech(x)^2)/sech(x) --R --R --R +--------------+ --R | 2 --R - \|- sech(x) + 1 + sech(x)sinh(x) ---R (63) ------------------------------------ +--R (92) ------------------------------------ --R sech(x) --R Type: Expression(Integer) ---E 63 +--E 92 ---S 64 of 89 +--S 93 of 118 t64:= sinh(x) - 1/csch(x) --R --R --R csch(x)sinh(x) - 1 ---R (64) ------------------ +--R (93) ------------------ --R csch(x) --R Type: Expression(Integer) ---E 64 +--E 93 ---S 65 of 89 +--S 94 of 118 t65:= cosh(x) - sqrt(1+sinh(x)^2) --R --R --R +------------+ --R | 2 ---R (65) - \|sinh(x) + 1 + cosh(x) +--R (94) - \|sinh(x) + 1 + cosh(x) --R Type: Expression(Integer) ---E 65 +--E 94 ---S 66 of 89 +--S 95 of 118 t66:= cosh(x) - 1/sqrt(1-tanh(x)^2) --R --R --R +--------------+ --R | 2 --R cosh(x)\|- tanh(x) + 1 - 1 ---R (66) ---------------------------- +--R (95) ---------------------------- --R +--------------+ --R | 2 --R \|- tanh(x) + 1 --R Type: Expression(Integer) ---E 66 +--E 95 ---S 67 of 89 +--S 96 of 118 t67:= cosh(x) - coth(x)/sqrt(coth(x)^2-1) --R --R --R +------------+ --R | 2 --R cosh(x)\|coth(x) - 1 - coth(x) ---R (67) -------------------------------- +--R (96) -------------------------------- --R +------------+ --R | 2 --R \|coth(x) - 1 --R Type: Expression(Integer) ---E 67 +--E 96 ---S 68 of 89 +--S 97 of 118 t68:= cosh(x) - 1/sech(x) --R --R --R cosh(x)sech(x) - 1 ---R (68) ------------------ +--R (97) ------------------ --R sech(x) --R Type: Expression(Integer) ---E 68 +--E 97 ---S 69 of 89 +--S 98 of 118 t69:= cosh(x) - sqrt(1+csch(x)^2)/csch(x) --R --R --R +------------+ --R | 2 --R - \|csch(x) + 1 + cosh(x)csch(x) ---R (69) ---------------------------------- +--R (98) ---------------------------------- --R csch(x) --R Type: Expression(Integer) ---E 69 +--E 98 ---S 70 of 89 +--S 99 of 118 t70:= tanh(x) - sinh(x)/sqrt(1+sinh(x)^2) --R --R --R +------------+ --R | 2 --R tanh(x)\|sinh(x) + 1 - sinh(x) ---R (70) -------------------------------- +--R (99) -------------------------------- --R +------------+ --R | 2 --R \|sinh(x) + 1 --R Type: Expression(Integer) ---E 70 +--E 99 ---S 71 of 89 +--S 100 of 118 t71:= tanh(x) - sqrt(cosh(x)^2-1)/cosh(x) --R --R ---R +------------+ ---R | 2 ---R - \|cosh(x) - 1 + cosh(x)tanh(x) ---R (71) ---------------------------------- ---R cosh(x) +--R +------------+ +--R | 2 +--R - \|cosh(x) - 1 + cosh(x)tanh(x) +--R (100) ---------------------------------- +--R cosh(x) --R Type: Expression(Integer) ---E 71 +--E 100 ---S 72 of 89 +--S 101 of 118 t72:= tanh(x) - 1/coth(x) --R --R ---R coth(x)tanh(x) - 1 ---R (72) ------------------ ---R coth(x) +--R coth(x)tanh(x) - 1 +--R (101) ------------------ +--R coth(x) --R Type: Expression(Integer) ---E 72 +--E 101 ---S 73 of 89 +--S 102 of 118 t73:= tanh(x) - sqrt(1-sech(x)^2) --R --R ---R +--------------+ ---R | 2 ---R (73) - \|- sech(x) + 1 + tanh(x) +--R +--------------+ +--R | 2 +--R (102) - \|- sech(x) + 1 + tanh(x) --R Type: Expression(Integer) ---E 73 +--E 102 ---S 74 of 89 +--S 103 of 118 t74:= tanh(x) - 1/sqrt(1+csch(x)^2) --R --R +--R +------------+ +--R | 2 +--R tanh(x)\|csch(x) + 1 - 1 +--R (103) -------------------------- --R +------------+ --R | 2 ---R tanh(x)\|csch(x) + 1 - 1 ---R (74) -------------------------- ---R +------------+ ---R | 2 ---R \|csch(x) + 1 +--R \|csch(x) + 1 --R Type: Expression(Integer) ---E 74 +--E 103 ---S 75 of 89 +--S 104 of 118 t75:= coth(x) - sqrt(sinh(x)^2+1)/sinh(x) --R --R ---R +------------+ ---R | 2 ---R - \|sinh(x) + 1 + coth(x)sinh(x) ---R (75) ---------------------------------- ---R sinh(x) +--R +------------+ +--R | 2 +--R - \|sinh(x) + 1 + coth(x)sinh(x) +--R (104) ---------------------------------- +--R sinh(x) --R Type: Expression(Integer) ---E 75 +--E 104 ---S 76 of 89 +--S 105 of 118 t76:= coth(x) - cosh(x)/sqrt(cosh(x)^2-1) --R --R ---R +------------+ ---R | 2 ---R coth(x)\|cosh(x) - 1 - cosh(x) ---R (76) -------------------------------- ---R +------------+ ---R | 2 ---R \|cosh(x) - 1 +--R +------------+ +--R | 2 +--R coth(x)\|cosh(x) - 1 - cosh(x) +--R (105) -------------------------------- +--R +------------+ +--R | 2 +--R \|cosh(x) - 1 --R Type: Expression(Integer) ---E 76 +--E 105 ---S 77 of 89 +--S 106 of 118 t77:= coth(x) - 1/tanh(x) --R --R ---R coth(x)tanh(x) - 1 ---R (77) ------------------ ---R tanh(x) +--R coth(x)tanh(x) - 1 +--R (106) ------------------ +--R tanh(x) --R Type: Expression(Integer) ---E 77 +--E 106 ---S 78 of 89 +--S 107 of 118 t78:= coth(x) - 1/sqrt(1-sech(x)^2) --R --R +--R +--------------+ +--R | 2 +--R coth(x)\|- sech(x) + 1 - 1 +--R (107) ---------------------------- --R +--------------+ --R | 2 ---R coth(x)\|- sech(x) + 1 - 1 ---R (78) ---------------------------- ---R +--------------+ ---R | 2 ---R \|- sech(x) + 1 +--R \|- sech(x) + 1 --R Type: Expression(Integer) ---E 78 +--E 107 ---S 79 of 89 +--S 108 of 118 t79:= coth(x) - sqrt(1+csch(x)^2) --R --R ---R +------------+ ---R | 2 ---R (79) - \|csch(x) + 1 + coth(x) +--R +------------+ +--R | 2 +--R (108) - \|csch(x) + 1 + coth(x) --R Type: Expression(Integer) ---E 79 +--E 108 ---S 80 of 89 +--S 109 of 118 t80:= sech(x) - 1/sqrt(1+sinh(x)^2) --R --R +--R +------------+ +--R | 2 +--R sech(x)\|sinh(x) + 1 - 1 +--R (109) -------------------------- --R +------------+ --R | 2 ---R sech(x)\|sinh(x) + 1 - 1 ---R (80) -------------------------- ---R +------------+ ---R | 2 ---R \|sinh(x) + 1 +--R \|sinh(x) + 1 --R Type: Expression(Integer) ---E 80 +--E 109 ---S 81 of 89 +--S 110 of 118 t81:= sech(x) - 1/cosh(x) --R --R ---R cosh(x)sech(x) - 1 ---R (81) ------------------ ---R cosh(x) +--R cosh(x)sech(x) - 1 +--R (110) ------------------ +--R cosh(x) --R Type: Expression(Integer) ---E 81 +--E 110 ---S 82 of 89 +--S 111 of 118 t82:= sech(x) - sqrt(1-tanh(x)^2) --R --R ---R +--------------+ ---R | 2 ---R (82) - \|- tanh(x) + 1 + sech(x) +--R +--------------+ +--R | 2 +--R (111) - \|- tanh(x) + 1 + sech(x) --R Type: Expression(Integer) ---E 82 +--E 111 ---S 83 of 89 +--S 112 of 118 t83:= sech(x) - sqrt(coth(x)^2-1)/coth(x) --R --R ---R +------------+ ---R | 2 ---R - \|coth(x) - 1 + coth(x)sech(x) ---R (83) ---------------------------------- ---R coth(x) +--R +------------+ +--R | 2 +--R - \|coth(x) - 1 + coth(x)sech(x) +--R (112) ---------------------------------- +--R coth(x) --R Type: Expression(Integer) ---E 83 +--E 112 ---S 84 of 89 +--S 113 of 118 t84:= sech(x) - csch(x)/sqrt(1+csch(x)^2) --R --R ---R +------------+ ---R | 2 ---R sech(x)\|csch(x) + 1 - csch(x) ---R (84) -------------------------------- ---R +------------+ ---R | 2 ---R \|csch(x) + 1 +--R +------------+ +--R | 2 +--R sech(x)\|csch(x) + 1 - csch(x) +--R (113) -------------------------------- +--R +------------+ +--R | 2 +--R \|csch(x) + 1 --R Type: Expression(Integer) ---E 84 +--E 113 ---S 85 of 89 +--S 114 of 118 t85:= csch(x) - 1/sinh(x) --R --R ---R csch(x)sinh(x) - 1 ---R (85) ------------------ ---R sinh(x) +--R csch(x)sinh(x) - 1 +--R (114) ------------------ +--R sinh(x) --R Type: Expression(Integer) ---E 85 +--E 114 ---S 86 of 89 +--S 115 of 118 t86:= csch(x) - 1/sqrt(cosh(x)^2-1) --R --R +--R +------------+ +--R | 2 +--R csch(x)\|cosh(x) - 1 - 1 +--R (115) -------------------------- --R +------------+ --R | 2 ---R csch(x)\|cosh(x) - 1 - 1 ---R (86) -------------------------- ---R +------------+ ---R | 2 ---R \|cosh(x) - 1 +--R \|cosh(x) - 1 --R Type: Expression(Integer) ---E 86 +--E 115 ---S 87 of 89 +--S 116 of 118 t87:= csch(x) - sqrt(1-tanh(x)^2)/tanh(x) --R --R ---R +--------------+ ---R | 2 ---R - \|- tanh(x) + 1 + csch(x)tanh(x) ---R (87) ------------------------------------ ---R tanh(x) +--R +--------------+ +--R | 2 +--R - \|- tanh(x) + 1 + csch(x)tanh(x) +--R (116) ------------------------------------ +--R tanh(x) --R Type: Expression(Integer) ---E 87 +--E 116 ---S 88 of 89 +--S 117 of 118 t88:= csch(x) - sqrt(coth(x)^2-1) --R --R ---R +------------+ ---R | 2 ---R (88) - \|coth(x) - 1 + csch(x) +--R +------------+ +--R | 2 +--R (117) - \|coth(x) - 1 + csch(x) --R Type: Expression(Integer) ---E 88 +--E 117 ---S 89 of 89 +--S 118 of 118 t89:= csch(x) - sech(x)/sqrt(1-sech(x)^2) --R --R ---R +--------------+ ---R | 2 ---R csch(x)\|- sech(x) + 1 - sech(x) ---R (89) ---------------------------------- ---R +--------------+ ---R | 2 ---R \|- sech(x) + 1 +--R +--------------+ +--R | 2 +--R csch(x)\|- sech(x) + 1 - sech(x) +--R (118) ---------------------------------- +--R +--------------+ +--R | 2 +--R \|- sech(x) + 1 --R Type: Expression(Integer) ---E 89 +--E 118 -- --