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implicitplot + transform...

vv 13805
December 19 2020
3 4 
restart;
F:=(x,y) -> x+y:
G:=(x,y) -> x^5+x^4*y+x^3*y+2*x^2*y^2+x*y^5+y^5-1:
p:=plots:-implicitplot(G, -5..5, -5..5):
plots:-display(plottools:-transform((x, y) -> [y, F(x,y)])(p), labels=[y,x]);

Or, you may want

plots:-display(plottools:-transform((x, y) -> [F(x,y),y])(p), labels=[x,y]);

[Edit] F being very simple, you could simply use

plots:-implicitplot(G(z-y,y), y=-5..5,z=-5..5)

 

special alias...

vv 13805
December 17 2020
1 0

I is a special alias for (-1)^(1/2), defined in the kernel as Complex(1). Complex is the constructor for complex numbers.

In very old Maple versions, I was defined via alias(I = (-1)^(1/2)).
complex(extended_numeric) is the basic type corresponding to numeric complex numbers (rational, floats or improper). 

 

Solutions...

vv 13805
December 17 2020
0 2

Here are the desired solutions found by Maple. There could be more, but then a more inolved search is needed.
Note that Im(x[k]) = +-Pi/2 <==> y[k] :: real

restart;
N := 4;
for k to N while k <> j do
    t[k] := sinh(x[k] + Gamma*I)^N*product(sinh(1/2*(x[k] - x[j] - 2*I*Gamma))^N, j = 1 .. N)/(sinh(x[k] - Gamma*I)^N*
product(sinh(1/2*(x[k] - x[j] + 2*I*Gamma))^N, j = 1 .. N));
end do;
sys := [t[1] = -1, t[2] = -1, t[3] = -1, t[4] = -1]:

numer~(`~`[lhs - rhs](simplify(eval(convert(sys, exp), [seq(x[k] = ln(I*y[k]), k = 1 .. N)])))):
S := factor~(simplify(eval(%, Gamma = Pi/4))) /~ 2:
nops~(S);
degree~(S);

n:=0:
for e1 in S[1] do for e2 in S[2] do for e3 in S[3] do for e4 in S[4] do 
n:=n+1;
sol:={}:
to 20 do
  sol1:=fsolve([e1,e2,e3,e4], {y[1], y[2],y[3],y[4]}, avoid=sol);
  if type(eval(sol1,1),function) then break fi;
  sol:=sol union {sol1};
od: SOL[n]:=sol; print(numsols[n]=nops(sol));
od:od:od:od:

'n'=n,numsols=add(nops(SOL[k]),k=1..n);
#                     n = 16, numsols = 145

X:=[seq(ln(I*y[k]),k=1..4)]:
crt:=1:
for i to n do for j to nops(SOL[i]) do
  print(Sol[crt++] = eval(X, SOL[i][j]))
od; od;

 

Sol[1] = [0.08627411361 - 1.570796327 I,
  0.8036414952 - 1.570796327 I, 0.08627411361 - 1.570796327 I,
  -1.822565644 - 1.570796327 I]

Sol[2] = [-0.04525896113 - 1.570796327 I,
  -1.548018529 + 1.570796327 I, -0.04525896113 - 1.570796327 I,
  -1.548018529 + 1.570796327 I]

 Sol[3] = [-0.4406867935 - 1.570796327 I,
   0.4406867935 - 1.570796327 I, 0.4406867935 - 1.570796327 I,
   -0.4406867935 + 1.570796327 I]

Sol[4] = [-0.5677693409 - 1.570796327 I,
  -0.5677693409 - 1.570796327 I, 0.1451722256 + 1.570796327 I,
  -1.942314249 - 1.570796327 I]

Sol[5] = [-1.942314249 + 1.570796327 I,
  -0.5677693409 + 1.570796327 I, -0.5677693409 + 1.570796327 I,
  0.1451722256 - 1.570796327 I]

Sol[6] = [-0.4406867935 + 1.570796327 I,
  0.4406867935 + 1.570796327 I, -0.4406867935 - 1.570796327 I,
  0.4406867935 + 1.570796327 I]

Sol[7] = [0.1716413778 - 1.570796327 I,
  -0.5630685995 + 1.570796327 I, 0.4614678655 + 1.570796327 I,
  1.131186650 + 1.570796327 I]

  Sol[8] = [0.1420880436 - 1.570796327 I,
    1.051187144 - 1.570796327 I, 1.051187144 - 1.570796327 I,
    -1.135161452 + 1.570796327 I]

 Sol[9] = [0.03257136864 - 1.570796327 I,
   0.4189103168 + 1.570796327 I, 0.4189103168 + 1.570796327 I,
   -0.001277707721 + 1.570796327 I]

 Sol[10] = [-0.5630685995 - 1.570796327 I,
   0.4614678655 - 1.570796327 I, 0.1716413778 + 1.570796327 I,
   1.131186650 - 1.570796327 I]

Sol[11] = [-0.6209595732 - 1.570796327 I,
  -0.02700782511 - 1.570796327 I, -0.6209595732 - 1.570796327 I,
  0.2611627098 - 1.570796327 I]

Sol[12] = [-1.068900508 + 1.570796327 I,
  0.01318893931 + 1.570796327 I, 0.01318893931 + 1.570796327 I,
  -0.1457687998 - 1.570796327 I]

 Sol[13] = [-0.1754775117 + 1.570796327 I,
   -1.487189168 + 1.570796327 I, -1.487189168 + 1.570796327 I,
   -1.329438275 - 1.570796327 I]

 Sol[14] = [0.03257136864 + 1.570796327 I,
   0.4189103168 - 1.570796327 I, 0.4189103168 - 1.570796327 I,
   -0.001277707721 - 1.570796327 I]

 Sol[15] = [0.6139924551 + 1.570796327 I,
   0.6139924551 + 1.570796327 I, 1.003955133 - 1.570796327 I,
   1.248786177 + 1.570796327 I]

 Sol[16] = [1.003955133 + 1.570796327 I,
   0.6139924551 - 1.570796327 I, 0.6139924551 - 1.570796327 I,
   1.248786177 - 1.570796327 I]

Sol[17] = [1.140069103 + 1.570796327 I,
  -0.2326672996 + 1.570796327 I, -0.2326672996 + 1.570796327 I,
  0.7167822839 + 1.570796327 I]

 Sol[18] = [1.003955133 - 1.570796327 I,
   0.6139924551 + 1.570796327 I, 1.248786177 + 1.570796327 I,
   0.6139924551 + 1.570796327 I]

 Sol[19] = [0.9981714534 - 1.570796327 I,
   0.9981714534 - 1.570796327 I, 0.4583235481 + 1.570796327 I,
   0.9981714534 - 1.570796327 I]

 Sol[20] = [0.3253917990 - 1.570796327 I,
   1.120005070 - 1.570796327 I, 0.6867968750 - 1.570796327 I,
   -1.502983465 - 1.570796327 I]

 Sol[21] = [0.1420880436 - 1.570796327 I,
   1.051187144 - 1.570796327 I, -1.135161452 + 1.570796327 I,
   1.051187144 - 1.570796327 I]

 Sol[22] = [-0.5630685995 - 1.570796327 I,
   0.4614678655 - 1.570796327 I, 1.131186650 - 1.570796327 I,
   0.1716413778 + 1.570796327 I]

Sol[23] = [-1.947380376 + 1.570796327 I,
  -0.4470338002 + 1.570796327 I, -1.130967546 + 1.570796327 I,
  -0.6364092887 - 1.570796327 I]

 Sol[24] = [-1.487189168 + 1.570796327 I,
   -1.487189168 + 1.570796327 I, -1.329438275 - 1.570796327 I,
   -0.1754775117 + 1.570796327 I]

Sol[25] = [0.03257136864 + 1.570796327 I,
  0.4189103168 - 1.570796327 I, -0.001277707721 - 1.570796327 I,
  0.4189103168 - 1.570796327 I]

 Sol[26] = [1.076626030 - 1.570796327 I,
   1.076626030 - 1.570796327 I, -0.2473913690 + 1.570796327 I,
   0.5583778254 - 1.570796327 I]

 Sol[27] = [0.5807479310 - 1.570796327 I,
   0.4615360925 + 1.570796327 I, 1.082572945 + 1.570796327 I,
   0.01036630967 + 1.570796327 I]

 Sol[28] = [-0.06620576889 - 1.570796327 I,
   0.6942090785 - 1.570796327 I, -1.681096685 + 1.570796327 I,
   0.2910093596 - 1.570796327 I]

      Sol[29] = [-0.4031997192 - 1.570796327 I,
        0.4031997194 + 1.570796327 I, 0. + 1.570796327 I,
        0. + 1.570796327 I]

Sol[30] = [-1.082572945 - 1.570796327 I,
  -0.01036630951 - 1.570796327 I, -0.5807479310 + 1.570796327 I,
  -0.4615360927 - 1.570796327 I]

Sol[31] = [-1.168790614 + 1.570796327 I,
  -0.04131220486 + 1.570796327 I, 0.4461003025 + 1.570796327 I,
  0.3618875142 - 1.570796327 I]

Sol[32] = [-0.4461003026 + 1.570796327 I,
  -0.3618875142 - 1.570796327 I, 0.04131220523 + 1.570796327 I,
  1.168790614 + 1.570796327 I]

      Sol[33] = [-0.4031997192 + 1.570796327 I,
        0.4031997194 - 1.570796327 I, 0. - 1.570796327 I,
        0. - 1.570796327 I]

Sol[34] = [0.5807479310 + 1.570796327 I,
  0.4615360925 - 1.570796327 I, 0.01036630967 - 1.570796327 I,
  1.082572945 - 1.570796327 I]

Sol[35] = [0.4189103168 - 1.570796327 I,
  -0.001277707721 - 1.570796327 I, 0.03257136864 + 1.570796327 I,
  0.4189103168 - 1.570796327 I]

  Sol[36] = [0.2369294769 - 1.570796327 I,
    1.386432315 - 1.570796327 I, 1.849698602 - 1.570796327 I,
    0.9747509077 - 1.570796327 I]

Sol[37] = [0.03257136864 - 1.570796327 I,
  -0.001277707721 + 1.570796327 I, 0.4189103168 + 1.570796327 I,
  0.4189103168 + 1.570796327 I]

Sol[38] = [-0.1056450350 - 1.570796327 I,
  0.1992691082 - 1.570796327 I, -0.1056450350 - 1.570796327 I,
  -2.082097231 - 1.570796327 I]

 Sol[39] = [-0.5630685995 - 1.570796327 I,
   1.131186650 - 1.570796327 I, 0.4614678655 - 1.570796327 I,
   0.1716413778 + 1.570796327 I]

Sol[40] = [-1.947380376 + 1.570796327 I,
  -1.130967546 + 1.570796327 I, -0.4470338002 + 1.570796327 I,
  -0.6364092887 - 1.570796327 I]

Sol[41] = [-1.068900508 + 1.570796327 I,
  -0.1457687998 - 1.570796327 I, 0.01318893931 + 1.570796327 I,
  0.01318893931 + 1.570796327 I]

Sol[42] = [-0.6364092887 + 1.570796327 I,
  -1.130967546 - 1.570796327 I, -0.4470338002 - 1.570796327 I,
  -1.947380376 - 1.570796327 I]

Sol[43] = [-0.5000525978 + 1.570796327 I,
  -0.06411040884 + 1.570796327 I, -0.8256973704 - 1.570796327 I,
  0.3287899986 + 1.570796327 I]

Sol[44] = [0.5807479310 - 1.570796327 I,
  0.01036630967 + 1.570796327 I, 0.4615360925 + 1.570796327 I,
  1.082572945 + 1.570796327 I]

 Sol[45] = [0.4031997194 - 1.570796327 I, 0. - 1.570796327 I,
   -0.4031997192 + 1.570796327 I, 0. - 1.570796327 I]

 Sol[46] = [-0.06620576889 - 1.570796327 I,
   0.2910093596 - 1.570796327 I, 0.6942090785 - 1.570796327 I,
   -1.681096685 + 1.570796327 I]

 Sol[47] = [-0.4031997192 - 1.570796327 I, 0. + 1.570796327 I,
   0.4031997194 + 1.570796327 I, 0. + 1.570796327 I]

Sol[48] = [-0.4406867935 - 1.570796327 I,
  0.4406867935 + 1.570796327 I, -0.4406867935 - 1.570796327 I,
  0.4406867935 + 1.570796327 I]

 Sol[49] = [-0.4461003026 - 1.570796327 I,
   1.168790614 - 1.570796327 I, -0.3618875142 + 1.570796327 I,
   0.04131220523 - 1.570796327 I]

Sol[50] = [-1.082572945 - 1.570796327 I,
  -0.4615360927 - 1.570796327 I, -0.01036630951 - 1.570796327 I,
  -0.5807479310 + 1.570796327 I]

Sol[51] = [-1.168790614 + 1.570796327 I,
  0.4461003025 + 1.570796327 I, -0.04131220486 + 1.570796327 I,
  0.3618875142 - 1.570796327 I]

Sol[52] = [0.4406867935 + 1.570796327 I,
  -0.4406867935 + 1.570796327 I, 0.4406867935 - 1.570796327 I,
  -0.4406867935 - 1.570796327 I]

 Sol[53] = [0.5807479310 + 1.570796327 I,
   1.082572945 - 1.570796327 I, 0.4615360925 - 1.570796327 I,
   0.01036630967 - 1.570796327 I]

Sol[54] = [0.6942090785 + 1.570796327 I,
  -1.681096685 - 1.570796327 I, -0.06620576889 + 1.570796327 I,
  0.2910093596 + 1.570796327 I]

Sol[55] = [1.076626030 - 1.570796327 I,
  0.5583778254 - 1.570796327 I, -0.2473913690 + 1.570796327 I,
  1.076626030 - 1.570796327 I]

 Sol[56] = [0.5807479310 - 1.570796327 I,
   0.01036630967 + 1.570796327 I, 1.082572945 + 1.570796327 I,
   0.4615360925 + 1.570796327 I]

 Sol[57] = [0.06526071873 - 1.570796327 I,
   0.3792213167 - 1.570796327 I, 0.3792213167 - 1.570796327 I,
   -1.832138209 - 1.570796327 I]

 Sol[58] = [-0.04131220486 - 1.570796327 I,
   0.4461003025 - 1.570796327 I, 0.3618875142 + 1.570796327 I,
   -1.168790614 - 1.570796327 I]

 Sol[59] = [-0.07966953883 - 1.570796327 I,
   0.8488969469 - 1.570796327 I, 0.8488969469 - 1.570796327 I,
   0.5134270623 - 1.570796327 I]

  Sol[60] = [-0.5873203142 - 1.570796327 I,
    1.287867298 - 1.570796327 I, 1.287867298 - 1.570796327 I,
    0.9525417822 + 1.570796327 I]

Sol[61] = [-1.795234421 - 1.570796327 I,
  -0.2301990663 + 1.570796327 I, -1.947977331 + 1.570796327 I,
  0.1782320712 + 1.570796327 I]

Sol[62] = [-2.167275854 - 1.570796327 I,
  -1.322814376 - 1.570796327 I, 0.06842942425 - 1.570796327 I,
  -0.1545705428 + 1.570796327 I]

Sol[63] = [-1.082572945 + 1.570796327 I,
  -0.4615360927 + 1.570796327 I, -0.5807479310 - 1.570796327 I,
  -0.01036630951 + 1.570796327 I]

 Sol[64] = [-0.4031997192 + 1.570796327 I, 0. - 1.570796327 I,
   0. - 1.570796327 I, 0.4031997194 - 1.570796327 I]

 Sol[65] = [0.4031997194 + 1.570796327 I, 0. + 1.570796327 I,
   0. + 1.570796327 I, -0.4031997192 - 1.570796327 I]

Sol[66] = [1.076626030 + 1.570796327 I,
  -0.2473913690 - 1.570796327 I, 0.5583778254 + 1.570796327 I,
  1.076626030 + 1.570796327 I]

  Sol[67] = [1.329438275 - 1.570796327 I,
    1.487189168 + 1.570796327 I, 1.487189168 + 1.570796327 I,
    0.1754775113 + 1.570796327 I]

Sol[68] = [0.1457687998 - 1.570796327 I,
  -0.01318893945 + 1.570796327 I, -0.01318893945 + 1.570796327 I,
  1.068900508 + 1.570796327 I]

Sol[69] = [0.001277707384 - 1.570796327 I,
  -0.4189103171 - 1.570796327 I, -0.03257136890 + 1.570796327 I,
  -0.4189103171 - 1.570796327 I]

 Sol[70] = [-0.2356095839 - 1.570796327 I,
   0.1053381512 - 1.570796327 I, 0.1053381512 - 1.570796327 I,
   -2.002834240 + 1.570796327 I]

 Sol[71] = [-0.2611627100 - 1.570796327 I,
   0.6209595735 - 1.570796327 I, 0.6209595735 - 1.570796327 I,
   0.02700782505 - 1.570796327 I]

Sol[72] = [-1.131186650 - 1.570796327 I,
  0.5630685995 - 1.570796327 I, -0.4614678653 - 1.570796327 I,
  -0.1716413780 + 1.570796327 I]

Sol[73] = [-1.248786177 + 1.570796327 I,
  -1.003955133 - 1.570796327 I, -0.6139924553 + 1.570796327 I,
  -0.6139924553 + 1.570796327 I]

Sol[74] = [-1.131186650 + 1.570796327 I,
  -0.4614678653 + 1.570796327 I, -0.1716413780 - 1.570796327 I,
  0.5630685995 + 1.570796327 I]

 Sol[75] = [-1.060866171 + 1.570796327 I,
   0.4849359742 + 1.570796327 I, 0.4849359742 + 1.570796327 I,
   0.8523057087 - 1.570796327 I]

Sol[76] = [0.1457687998 + 1.570796327 I,
  1.068900508 - 1.570796327 I, -0.01318893945 - 1.570796327 I,
  -0.01318893945 - 1.570796327 I]

 Sol[77] = [0.6867968750 - 1.570796327 I,
   1.120005070 - 1.570796327 I, 0.3253917990 - 1.570796327 I,
   -1.502983465 - 1.570796327 I]

Sol[78] = [0.6456099413 - 1.570796327 I,
  0.01956591987 - 1.570796327 I, -1.039649603 + 1.570796327 I,
  -1.039649603 + 1.570796327 I]

Sol[79] = [-0.06411040884 - 1.570796327 I,
  0.3287899986 - 1.570796327 I, -0.8256973704 + 1.570796327 I,
  -0.5000525978 - 1.570796327 I]

Sol[80] = [-0.1457687998 - 1.570796327 I,
  0.01318893931 + 1.570796327 I, 0.01318893931 + 1.570796327 I,
  -1.068900508 + 1.570796327 I]

Sol[81] = [-2.024517769 + 1.570796327 I,
  -0.2310059600 - 1.570796327 I, 0.4939779113 - 1.570796327 I,
  -0.2310059600 - 1.570796327 I]

Sol[82] = [-0.1457687998 + 1.570796327 I,
  0.01318893931 - 1.570796327 I, 0.01318893931 - 1.570796327 I,
  -1.068900508 - 1.570796327 I]

 Sol[83] = [-0.001277707721 + 1.570796327 I,
   0.4189103168 + 1.570796327 I, 0.4189103168 + 1.570796327 I,
   0.03257136864 - 1.570796327 I]

 Sol[84] = [0.4583235481 + 1.570796327 I,
   0.9981714534 - 1.570796327 I, 0.9981714534 - 1.570796327 I,
   0.9981714534 - 1.570796327 I]

Sol[85] = [0.7167822839 + 1.570796327 I,
  -0.2326672996 + 1.570796327 I, -0.2326672996 + 1.570796327 I,
  1.140069103 + 1.570796327 I]

 Sol[86] = [1.082572945 - 1.570796327 I,
   0.4615360925 - 1.570796327 I, 0.5807479310 + 1.570796327 I,
   0.01036630967 - 1.570796327 I]

  Sol[87] = [0.5583778254 - 1.570796327 I,
    1.076626030 - 1.570796327 I, 1.076626030 - 1.570796327 I,
    -0.2473913690 + 1.570796327 I]

Sol[88] = [0.3618875142 - 1.570796327 I,
  -1.168790614 + 1.570796327 I, -0.04131220486 + 1.570796327 I,
  0.4461003025 + 1.570796327 I]

Sol[89] = [0.2910093596 - 1.570796327 I,
  0.6942090785 - 1.570796327 I, -0.06620576889 - 1.570796327 I,
  -1.681096685 + 1.570796327 I]

 Sol[90] = [0. - 1.570796327 I, 0.4031997194 - 1.570796327 I,
   -0.4031997192 + 1.570796327 I, 0. - 1.570796327 I]

 Sol[91] = [-1.336148998 - 1.570796327 I,
   1.336148997 - 1.570796327 I, 0.2787419186 + 1.570796327 I,
   -0.2787419187 + 1.570796327 I]

 Sol[92] = [-0.4031997192 + 1.570796327 I, 0. + 1.570796327 I,
   0. + 1.570796327 I, 0.4031997194 - 1.570796327 I]

Sol[93] = [0.3618875142 + 1.570796327 I,
  -1.168790614 - 1.570796327 I, -0.04131220486 - 1.570796327 I,
  0.4461003025 - 1.570796327 I]

Sol[94] = [0.3792213167 + 1.570796327 I,
  -1.832138209 + 1.570796327 I, 0.06526071873 + 1.570796327 I,
  0.3792213167 + 1.570796327 I]

 Sol[95] = [0.5583778254 - 1.570796327 I,
   1.076626030 - 1.570796327 I, -0.2473913690 + 1.570796327 I,
   1.076626030 - 1.570796327 I]

 Sol[96] = [0.3618875142 - 1.570796327 I,
   -1.168790614 + 1.570796327 I, 0.4461003025 + 1.570796327 I,
   -0.04131220486 + 1.570796327 I]

 Sol[97] = [0.2910093596 - 1.570796327 I,
   0.6942090785 - 1.570796327 I, -1.681096685 + 1.570796327 I,
   -0.06620576889 - 1.570796327 I]

Sol[98] = [-0.5807479310 - 1.570796327 I,
  -1.082572945 + 1.570796327 I, -0.4615360927 + 1.570796327 I,
  -0.01036630951 + 1.570796327 I]

Sol[99] = [-1.947977331 - 1.570796327 I,
  0.1782320712 - 1.570796327 I, -0.2301990663 - 1.570796327 I,
  -1.795234421 + 1.570796327 I]

 Sol[100] = [0.01036630967 + 1.570796327 I,
   0.4615360925 + 1.570796327 I, 1.082572945 + 1.570796327 I,
   0.5807479310 - 1.570796327 I]

Sol[101] = [0.4286147013 + 1.570796327 I,
  -1.844405988 + 1.570796327 I, -0.2295150718 + 1.570796327 I,
  0.8318144201 + 1.570796327 I]

 Sol[102] = [0.5583778254 + 1.570796327 I,
   1.076626030 + 1.570796327 I, -0.2473913690 - 1.570796327 I,
   1.076626030 + 1.570796327 I]

 Sol[103] = [0.4876998405 - 1.570796327 I,
   0.9305671649 - 1.570796327 I, -1.202724258 + 1.570796327 I,
   0.4876998405 - 1.570796327 I]

Sol[104] = [-0.1716413780 - 1.570796327 I,
  -1.131186650 + 1.570796327 I, -0.4614678653 + 1.570796327 I,
  0.5630685995 + 1.570796327 I]

Sol[105] = [-2.002834240 + 1.570796327 I,
  -0.2356095839 - 1.570796327 I, 0.1053381512 - 1.570796327 I,
  0.1053381512 - 1.570796327 I]

Sol[106] = [-1.273995236 + 1.570796327 I,
  -0.7420312580 + 1.570796327 I, -0.3249455390 + 1.570796327 I,
  -2.872375031 - 1.570796327 I]

Sol[107] = [-0.3287899987 + 1.570796327 I,
  0.06411040857 + 1.570796327 I, 0.5000525977 + 1.570796327 I,
  0.8256973702 - 1.570796327 I]

 Sol[108] = [0.4470338004 + 1.570796327 I,
   1.130967546 + 1.570796327 I, 0.6364092886 - 1.570796327 I,
   1.947380376 + 1.570796327 I]

 Sol[109] = [0.6279598010 + 1.570796327 I,
   -1.528406919 + 1.570796327 I, 0.6279598010 + 1.570796327 I,
   0.6279598010 + 1.570796327 I]

Sol[110] = [0.4406867935 - 1.570796327 I,
  0.4406867935 - 1.570796327 I, -0.4406867935 + 1.570796327 I,
  -0.4406867935 + 1.570796327 I]

Sol[111] = [0.3792213167 - 1.570796327 I,
  0.3792213167 - 1.570796327 I, 0.06526071873 - 1.570796327 I,
  -1.832138209 - 1.570796327 I]

 Sol[112] = [0.04131220523 - 1.570796327 I,
   1.168790614 - 1.570796327 I, -0.3618875142 + 1.570796327 I,
   -0.4461003026 - 1.570796327 I]

 Sol[113] = [0. - 1.570796327 I, 0. - 1.570796327 I,
   -0.4031997192 + 1.570796327 I, 0.4031997194 - 1.570796327 I]

 Sol[114] = [-1.681096685 + 1.570796327 I,
   0.2910093596 - 1.570796327 I, 0.6942090785 - 1.570796327 I,
   -0.06620576889 - 1.570796327 I]

Sol[115] = [-1.322814376 + 1.570796327 I,
  0.06842942425 + 1.570796327 I, -2.167275854 + 1.570796327 I,
  -0.1545705428 - 1.570796327 I]

 Sol[116] = [-0.2787419187 + 1.570796327 I,
   -1.336148998 - 1.570796327 I, 1.336148997 - 1.570796327 I,
   0.2787419186 + 1.570796327 I]

 Sol[117] = [0. + 1.570796327 I, 0. + 1.570796327 I,
   -0.4031997192 - 1.570796327 I, 0.4031997194 + 1.570796327 I]

Sol[118] = [0.2910093596 + 1.570796327 I,
  -1.681096685 - 1.570796327 I, -0.06620576889 + 1.570796327 I,
  0.6942090785 + 1.570796327 I]

Sol[119] = [0.3618875142 + 1.570796327 I,
  0.4461003025 - 1.570796327 I, -0.04131220486 - 1.570796327 I,
  -1.168790614 - 1.570796327 I]

Sol[120] = [0.4406867935 + 1.570796327 I,
  0.4406867935 + 1.570796327 I, -0.4406867935 - 1.570796327 I,
  -0.4406867935 - 1.570796327 I]

Sol[121] = [0.6209595735 - 1.570796327 I,
  0.6209595735 - 1.570796327 I, -0.2611627100 - 1.570796327 I,
  0.02700782505 - 1.570796327 I]

 Sol[122] = [-0.01318893945 - 1.570796327 I,
   1.068900508 - 1.570796327 I, 0.1457687998 + 1.570796327 I,
   -0.01318893945 - 1.570796327 I]

Sol[123] = [-0.03257136890 - 1.570796327 I,
  -0.4189103171 + 1.570796327 I, 0.001277707384 + 1.570796327 I,
  -0.4189103171 + 1.570796327 I]

Sol[124] = [-0.07147320562 - 1.570796327 I,
  -0.07147320562 - 1.570796327 I, -0.7639142748 + 1.570796327 I,
  -0.07147320562 - 1.570796327 I]

Sol[125] = [-0.4189103171 - 1.570796327 I,
  -0.03257136890 + 1.570796327 I, 0.001277707384 - 1.570796327 I,
  -0.4189103171 - 1.570796327 I]

 Sol[126] = [-1.202724258 - 1.570796327 I,
   0.4876998405 + 1.570796327 I, 0.9305671649 + 1.570796327 I,
   0.4876998405 + 1.570796327 I]

 Sol[127] = [-1.202724258 + 1.570796327 I,
   0.4876998405 - 1.570796327 I, 0.9305671649 - 1.570796327 I,
   0.4876998405 - 1.570796327 I]

Sol[128] = [-0.3287899987 + 1.570796327 I,
  0.5000525977 + 1.570796327 I, 0.06411040857 + 1.570796327 I,
  0.8256973702 - 1.570796327 I]

Sol[129] = [-0.01318893945 + 1.570796327 I,
  -0.01318893945 + 1.570796327 I, 0.1457687998 - 1.570796327 I,
  1.068900508 + 1.570796327 I]

Sol[130] = [0.1053381512 + 1.570796327 I,
  -2.002834240 - 1.570796327 I, -0.2356095839 + 1.570796327 I,
  0.1053381512 + 1.570796327 I]

Sol[131] = [0.8256973702 + 1.570796327 I,
  0.5000525977 - 1.570796327 I, 0.06411040857 - 1.570796327 I,
  -0.3287899987 - 1.570796327 I]

Sol[132] = [1.039649603 - 1.570796327 I,
  1.039649603 - 1.570796327 I, -0.01956592001 + 1.570796327 I,
  -0.6456099410 + 1.570796327 I]

Sol[133] = [-0.4189103171 - 1.570796327 I,
  -0.4189103171 - 1.570796327 I, -0.03257136890 + 1.570796327 I,
  0.001277707384 - 1.570796327 I]

Sol[134] = [-0.4614678653 - 1.570796327 I,
  0.5630685995 - 1.570796327 I, -0.1716413780 + 1.570796327 I,
  -1.131186650 - 1.570796327 I]

 Sol[135] = [-1.812967751 - 1.570796327 I,
   0.02257676210 - 1.570796327 I, 1.152462801 - 1.570796327 I,
   -1.494949285 + 1.570796327 I]

Sol[136] = [-2.210985117 - 1.570796327 I,
  -0.4592776410 - 1.570796327 I, -0.4592776410 - 1.570796327 I,
  -2.136610451 + 1.570796327 I]

 Sol[137] = [-2.002834240 + 1.570796327 I,
   0.1053381512 - 1.570796327 I, 0.1053381512 - 1.570796327 I,
   -0.2356095839 - 1.570796327 I]

Sol[138] = [-0.4189103171 + 1.570796327 I,
  -0.03257136890 - 1.570796327 I, -0.4189103171 + 1.570796327 I,
  0.001277707384 + 1.570796327 I]

 Sol[139] = [0.8523057087 + 1.570796327 I,
   0.4849359742 - 1.570796327 I, 0.4849359742 - 1.570796327 I,
   -1.060866171 - 1.570796327 I]

Sol[140] = [1.039649603 + 1.570796327 I,
  1.039649603 + 1.570796327 I, -0.01956592001 - 1.570796327 I,
  -0.6456099410 - 1.570796327 I]

 Sol[141] = [0.2613891677 - 1.570796327 I,
   0.2613891677 - 1.570796327 I, 0.2613891677 - 1.570796327 I,
   -1.680493002 + 1.570796327 I]

Sol[142] = [-0.4406867935 - 1.570796327 I,
  -0.4406867935 + 1.570796327 I, -0.4406867935 + 1.570796327 I,
  -0.4406867935 - 1.570796327 I]

Sol[143] = [-0.5437908128 - 1.570796327 I,
  0.06432366650 - 1.570796327 I, -1.320973167 - 1.570796327 I,
  -0.5437908128 - 1.570796327 I]

Sol[144] = [-0.4406867935 + 1.570796327 I,
  -0.4406867935 - 1.570796327 I, -0.4406867935 - 1.570796327 I,
  -0.4406867935 + 1.570796327 I]

 Sol[145] = [0.2613891677 + 1.570796327 I,
   -1.680493002 - 1.570796327 I, 0.2613891677 + 1.570796327 I,
   0.2613891677 + 1.570796327 I]
 

 

Answer (reduction);...

vv 13805
December 15 2020
1 1 
# Remove Gamma=Pi/4; then:
numer~(`~`[lhs - rhs](simplify(eval(convert(sys, exp), [seq(x[k] = ln(y[k]), k = 1 .. N)])))):
S := factor~(simplify(eval(%, Gamma = Pi/4))) /~ 2:
nops~([S[]]);
degree~([S[]]);
                          [2, 2, 2, 2]
                        [20, 20, 20, 20]
S1:=map2(op,1, [S[]]):
map(degree,S1);
#                        [10, 10, 10, 10]

So, the problem reduces to 16 polynomial systems, each polynomial having the total degree 10.

No wonder that it cannot be solved!

One of the many solutions is (approx):
[x[1] = 1.07654607639368 - 3.14159265358979*I, x[2] = -0.263712436689289 + 3.14159265358979*I, x[3] = 0.263712436689284 - (4.03269792266791*10^(-31))*I, x[4] = -1.07654607639368 + (1.45550972708563*10^(-31))*I]

subsindets...

vv 13805
December 15 2020
2 3 
ex1:=1/cosh(x)+   (cosh(x)^2+cosh(x)*sinh(x)+x)/cosh(x)*_C1 + x*exp(_C3+x+_C2)*C;
ex2:=_C1/cosh(x)+ (cosh(x)^2+cosh(x)*sinh(x)+x)/cosh(x);
ex3:=_C1/cosh(x)+ (cosh(x)^2+cosh(x)*sinh(x)+x)/cosh(x)*_C2;

Cc:=proc(u)
  local Cc1 := C -> c[parse(convert(C,string)[3..])];
  subsindets(u, `*`,
    proc(u)
      local k,r;
      k,r:=selectremove(type, [op(u)], suffixed(_C,integer));
      if nops(k)<>1 then return u fi;
      `*`(Cc1(k[]),r[])
    end):
  subsindets(%, suffixed(_C,integer), Cc1)
end:

Cc(ex1); Cc(ex2); Cc(ex3);

fast convergence...

vv 13805
December 13 2020
2 1

Most probably the constant has not a symbolic representation, but is this important? The series has a fast convergence (just like Sum(1/j^5, j)), so Maple computes quickly a few hundred decimals.

evalf[500](Sum(fd(n)*ln(1-1/n^2), n=2..infinity));

-0.13193448210193902101168456260113057372954207877638029494362536124924096687277453818310396774949030858278425655631749947528907468010666626334748919975517016978153062099871194024801794507410911378951971698504881151995858660891084236232931902589675514220351056452721912565324355327946777370232965534555511025840021602746613401257447542954468015595944834472807189884301164078566662687373782315983259649170312137042952056677012051441069664477950335128241355390929177317762043273134991270883122867315911249

thru 3 points...

vv 13805
December 13 2020
0 0

> 

restart;
> 

conic:=(P,Q,R,a,b) -> a*Q*R+b*P*R+P*Q:
> 

line:=(x1,y1,x2,y2)->(x-x1)*(y2-y1)-(y-y1)*(x2-x1):
> 

Delta:=qq -> diff(qq,x,y)^2-diff(qq,x,x)*diff(qq,y,y):
> 

########
> 

pts:=[[1,2],[-2,2],[2,3]]:
> 

PQR:=line(pts[1][],pts[2][]), line(pts[2][],pts[3][]), line(pts[3][],pts[1][]):
> 

co:=conic(PQR, 2, b):
> 

solve(Delta(co)>0); evalf([%]);

RealRange(-infinity, Open(3-2*2^(1/2))), RealRange(Open(3+2*2^(1/2)), infinity)

  

[RealRange(-infinity, Open(.171572876)), RealRange(Open(5.828427124), infinity)]

 (1)
> 

co1:=eval(co,b=2): co2:=eval(co,b=-8):
> 

plots:-implicitplot( [PQR, co1, co2], x=-4..4, y=0..8, color=[red$3,blue,green], scaling=constrained);

 

 

Note. You may add a parabola for b=3+2*sqrt(2).

subs...

vv 13805
December 12 2020
2 4

Use subs instead of alias

S:=[seq(_C||k=c[k], k=0..10)]:
sol:=dsolve(diff(y(x),x) = x+y(x),y(x));
sort(subs(S,sol));    #sort~(subs(S,[sol]))[]

 

Cartesian...

vv 13805
December 12 2020
0 2

The homogeneous coordinates are very useful to avoid degenerate cases.
A similar construction can be obtained with Cartesian coordinates.

restart;
r:=rand(-12..12):
P,Q,R := 'r()*x+r()*y+r()'$3:
q :=(a,b) -> a*Q*R+b*P*R+P*Q:
plots:-implicitplot( [P, Q, R, q(6,-6)], x=-5..5, y=-5..5, color=[red$3,blue]);

answer...

vv 13805
December 10 2020
0 0

This seems to be about twice faster.

SP:= proc(A::list(set), B::list(set))
local i,j, M:=Array(1..nops(A),1..nops(B),datatype=integer[4]);
for i to nops(A) do for j to nops(B) do
  if A[i] subset B[j] then M[i,j]:=j fi od od;
[seq(subs(0=NULL,[entries(M[i],nolist)]),i=1..nops(A))]
end:

 

Envelope...

vv 13805
December 10 2020
2 0

Maybe you should always check the envelope.
In this case,  [y=sin(x+c), cos(x+c)=0] ==> y = +-1.

Note. For the IV problem

dsolve({diff(y(x),x)=sqrt(1-y(x)^2), y(0)=-1});

Maple gives  y(x)=-1, but it has infinitely many solutions in any nbd of 0, e.g.
S:=a -> piecewise(x<a, -1, -cos(x-a));    #  0 <= a,   x<a+Pi
which probably Maple will never (?) find. 

answer...

vv 13805
December 08 2020
2 1

> 

f:=product(x+i, i=1..n):
> 

answer:=simplify(residue(f/x^3,x=0)) assuming n::posint;

-(1/12)*(Pi^2-6*gamma^2-12*gamma*Psi(n+1)-6*Psi(n+1)^2-6*Psi(1, n+1))*GAMMA(n+1)

 (1)
> 

# check
simplify(eval(answer,n=10)) = coeff(expand(eval(f,n=10)),x^2);

12753576 = 12753576

 (2)

 

Same result with:

simplify(coeff(series(f,x), x^2));

 

Physics...

vv 13805
December 08 2020
2 1

 

> 

restart;
> 

with(Physics):
> 

Setup(noncommutativeprefix = {A,B,C})

[noncommutativeprefix = {A, B, C}]

 (1)
> 

(2*A+B*C)^*;

2*Physics:-Dagger(A)+Physics:-`*`(Physics:-Dagger(C), Physics:-Dagger(B))

 (2)
> 

lprint(%);

2*Physics:-Dagger(A)+Physics:-`*`(Physics:-Dagger(C),Physics:-Dagger(B))

  
> 

(2*A * B^(-1) - C)^*;

2*Physics:-`*`(Physics:-`^`(Physics:-Dagger(B), -1), Physics:-Dagger(A))-Physics:-Dagger(C)

 (3)
> 

lprint(%);

2*Physics:-`*`(Physics:-`^`(Physics:-Dagger(B),-1),Physics:-Dagger(A))-Physics
:-Dagger(C)

  
> 

 

 

Answer...

vv 13805
December 08 2020
0 0 
ex:=sqrt(4*n^2+5*n-7)/n:
sqrt(expand(ex^2));

(assuming n>0).

You should be aware that in Maple it could be difficult to obtain the answer in a preferred form.

polynomials...

vv 13805
December 06 2020
2 3

Maple can solve systems of polynomial inequalities. SemiAlgebraic is more flexible than solve.

sys := r-c>0, 0<=r^2 - 2*r*c -r +2*c, r^2 - 2*r*c -r +2*c  < (r-c)^2, 0<=c, c<=1 , 0<r, r<1, r<2*c:
SolveTools:-SemiAlgebraic([sys], [r], parameters=[c]);

       

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