diff --git a/buglist b/buglist index 47f9b97..d6b065b 100644 --- a/buglist +++ b/buglist @@ -1,6 +1,6 @@ ========================================================================= -bug 7249: +bug 7250: todo 336: wish 1011: meh 5: @@ -12,6 +12,85 @@ dup 50006: nonextend 60077: ========================================================================= +bug 7249: radicalSolve(z^7=1, z) + +Obviously, all the roots of the equation z^7 = 1 can be expressed in +radicals, and Mathematica can easily produce the explicit expressions +in terms of radicals. + +Solve[z^7 == 1, z] + +{{z -> 1}, {z -> -(-1)^(1/7)}, {z -> (-1)^(2/7)}, {z -> -(-1)^(3/7)}, +{{z -> {z -> (-1)^(4/7)}, {z -> -(-1)^(5/7)}, {z -> (-1)^(6/7)}} + +To save the space, below the only example is given. + +FunctionExpand[ComplexExpand[-(-1)^(1/7)]] + +(1/2)*((1/3)*((1/2)*(-1 + I*Sqrt[7]) + ((-1 + I*Sqrt[3])*((1/2)*(-1 + +I*Sqrt[7]) + (1/2)*(-1 - I*Sqrt[7])*((1/2)*(-1 + I*Sqrt[3]) + +(1/4)*(-1 + I*Sqrt[3])^2)))/(2*(6 + (3/4)*(-1 + I*Sqrt[3])*(-1 + +I*Sqrt[7]) + (1/2)*(-1 - I*Sqrt[7])*(1 + (3/4)*(-1 + +I*Sqrt[3])^2))^(1/3)) + (1/4)*(-1 + I*Sqrt[3])^2*(6 + (3/4)*(-1 + +I*Sqrt[3])*(-1 + I*Sqrt[7]) + (1/2)*(-1 - I*Sqrt[7])*(1 + (3/4)*(-1 + +I*Sqrt[3])^2))^(1/3)) +(1/3)*((1/2)*(1 + I*Sqrt[7]) - ((-1 + +I*Sqrt[3])^2*((1/2)*(-1 -I*Sqrt[7]) + (1/2)*(-1 + +I*Sqrt[7])*((1/2)*(-1 + I*Sqrt[3]) + (1/4)*(-1 + I*Sqrt[3])^2)))/(4*(6 ++ (3/4)*(-1 + I*Sqrt[3])*(-1 - I*Sqrt[7]) + (1/2)*(-1 + I*Sqrt[7])*(1 ++ (3/4)*(-1 + I*Sqrt[3])^2))^(1/3)) -(1/2)*(-1 + I*Sqrt[3])*(6 + +(3/4)*(-1 + I*Sqrt[3])*(-1 - I*Sqrt[7]) + (1/2)*(-1 + I*Sqrt[7])*(1 + +(3/4)*(-1 + I*Sqrt[3])^2))^(1/3))) + (1/2)*((1/3)*((1/2)*(-1 + +I*Sqrt[7]) + ((-1 + I*Sqrt[3])*((1/2)*(-1 + I*Sqrt[7]) + (1/2)*(-1 - +I*Sqrt[7])*((1/2)*(-1 + I*Sqrt[3]) + (1/4)*(-1 + I*Sqrt[3])^2)))/(2*(6 ++ (3/4)*(-1 + I*Sqrt[3])*(-1 + I*Sqrt[7]) + (1/2)*(-1 - I*Sqrt[7])*(1 ++ (3/4)*(-1 + I*Sqrt[3])^2))^(1/3)) +(1/4)*(-1 + I*Sqrt[3])^2*(6 + +(3/4)*(-1 + I*Sqrt[3])*(-1 + I*Sqrt[7]) + (1/2)*(-1 - I*Sqrt[7])*(1 + +(3/4)*(-1 + I*Sqrt[3])^2))^(1/3)) + (1/3)*((1/2)*(-1 - I*Sqrt[7]) ++((-1 + I*Sqrt[3])^2*((1/2)*(-1 - I*Sqrt[7]) + (1/2)*(-1 + +I*Sqrt[7])*((1/2)*(-1 + I*Sqrt[3]) + (1/4)*(-1 + I*Sqrt[3])^2)))/(4*(6 ++ (3/4)*(-1 + I*Sqrt[3])*(-1 - I*Sqrt[7]) + (1/2)*(-1 + I*Sqrt[7])*(1 ++ (3/4)*(-1 + I*Sqrt[3])^2))^(1/3)) +(1/2)*(-1 + I*Sqrt[3])*(6 + +(3/4)*(-1 + I*Sqrt[3])*(-1 - I*Sqrt[7]) + (1/2)*(-1 + I*Sqrt[7])*(1 + +(3/4)*(-1 + I*Sqrt[3])^2))^(1/3))) + + +According to the AXIOM Book + +AXIOM Book> Use radicalSolve if you want your solutions expressed in +AXIOM Book> terms of radicals. + +However, already for z^7 = 1 this is not so, + +-> radicalSolve(z^7=1, z) + + [z= 1] + +and the problem exists for 11, 13, 14, 15, 17, 19 etc + +-> for i in 1..20 repeat print([i,#radicalSolve(z^i=1,z)]) + + [1,1] + [2,2] + [3,3] + [4,4] + [5,5] + [6,6] + [7,1] <-- not good + [8,8] + [9,9] + [10,10] + [11,1] <-- not good + [12,12] + [13,1] <-- not good + [14,2] <-- not good + [15,7] <-- not good + [16,16] + [17,1] <-- not good + [18,18] + [19,1] <-- not good + [20,20] + +========================================================================= bug 7248: differentiation bug in D(1,z) (94) -> D(sin(z)^2+cos(z)^2, z) diff --git a/changelog b/changelog index cdfd7ae..1b47789 100644 --- a/changelog +++ b/changelog @@ -1,5 +1,7 @@ +20140613 tpd src/axiom-website/patches.html 20140613.09.tpd.patch +20140613 tpd buglist bug 7249: radicalSolve(z^7=1, z) 20140613 tpd src/axiom-website/patches.html 20140613.08.tpd.patch -20140613 tpd bug 7248: differentiation bug in D(1,z) +20140613 tpd buglist bug 7248: differentiation bug in D(1,z) 20140613 tpd src/axiom-website/patches.html 20140613.07.tpd.patch 20140613 tpd buglist 7142 exp^log(x) Cannot take first of an empty list update 20140613 tpd src/axiom-website/patches.html 20140613.06.tpd.patch diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 5d7997f..8de5390 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -4435,7 +4435,9 @@ buglist 7142 exp^log(x) Cannot take first of an empty list update 20140613.07.tpd.patch buglist 7142 exp^log(x) Cannot take first of an empty list update 20140613.08.tpd.patch -bug 7248: differentiation bug in D(1,z) +buglist bug 7248: differentiation bug in D(1,z) +20140613.09.tpd.patch +buglist bug 7249: radicalSolve(z^7=1, z)