Skip to main content

list manipulation - Plotting complex solutions in Argand plane as arrows


I have:


Clear[z]
Solve[z^4 == 1 - Sqrt[3] I, z] // ComplexExpand

Which produces these solutions:


{{z -> -(1/(2 2^(1/4))) - Sqrt[3]/(2 2^(1/4)) + 
    I (-(1/(2 2^(1/4))) + Sqrt[3]/(2 2^(1/4)))}, {z -> 
   1/(2 2^(1/4)) - Sqrt[3]/(2 2^(1/4)) + 
    I (-(1/(2 2^(1/4))) - Sqrt[3]/(2 2^(1/4)))}, {z -> -(1/(

     2 2^(1/4))) + Sqrt[3]/(2 2^(1/4)) + 
    I (1/(2 2^(1/4)) + Sqrt[3]/(2 2^(1/4)))}, {z -> 
   1/(2 2^(1/4)) + Sqrt[3]/(2 2^(1/4)) + 
    I (1/(2 2^(1/4)) - Sqrt[3]/(2 2^(1/4)))}}

What would be the easiest way to convert these to a list of arrows to plot in the Argand plane. For example, the cube roots of -1:


Graphics[{
  Circle[],
  Blue, Thick,
  Arrow[{{0, 0}, {-1, 0}}],

  Arrow[{{0, 0}, {1/2, Sqrt[3]/2}}],
  Arrow[{{0, 0}, {1/2, -Sqrt[3]/2}}]
  }, Axes -> True, ImageSize -> Small]

Produces this image:


enter image description here


But I am looking for a cute way to convert all the data in the first problem, $z^4=1-\sqrt3 i$, into a list of arrows.



Answer



sol = z /. Solve[z^4 == 1 - Sqrt[3] I, z] // ComplexExpand


enter image description here


coords = N @ ReIm @ sol (* N not needed, I just wanted a compact output *)


{{-1.14869, 0.307789}, {-0.307789, -1.14869}, {0.307789, 1.14869}, {1.14869, -0.307789}}



arr = Arrow[{{0, 0}, #}] & /@ coords


{Arrow[{{0, 0}, {-1.14869, 0.307789}}], Arrow[{{0, 0}, {-0.307789, -1.14869}}], Arrow[{{0, 0}, {0.307789, 1.14869}}], Arrow[{{0, 0}, {1.14869, -0.307789}}]}




For the circle:


{r = Abs[1 - Sqrt[3] I]^(1/4), N@r}


{2^(1/4), 1.18921}



Graphics[{Circle[{0, 0}, r], Blue, Thick, arr}, Axes -> True, 
 AxesLabel -> {"Re z", "Im z"}]


enter image description here


Comments

Post a Comment

Popular posts from this blog

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
Post a Comment
Read more

mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...
Post a Comment
Read more