Plot complex inequality in Julia

Question

I have the mathematical expression |z - (-1)| < 1 |z - (-1)| < 1 , with z element of Complexes, which is equivalent of a circle of radius 1 centered in (x,y)=(-1,0) .

1. How can I plot this expression,
2. preserving the structure of the mathematical expression it was derived from, as much as possible?
3. It should be an area.

What I tried so far:

using ImplicitEquations, Plots

f(a,b) = abs.(a+im*b - (-1))
plot(f<1)

The error I got:

ERROR: MethodError: no method matching isless(::typeof(f), ::Int64)
Closest candidates are:
  isless(::Union{StatsBase.PValue, StatsBase.TestStat}, ::Real) at /home/buddhilw/.julia/packages/StatsBase/PGTj8/src/statmodels.jl:514
  isless(::AbstractGray{T} where T, ::Real) at /home/buddhilw/.julia/packages/ColorTypes/6m8P7/src/operations.jl:31
  isless(::ForwardDiff.Dual{Tx, V, N} where {V, N}, ::Integer) where
Tx at /home/buddhilw/.julia/packages/ForwardDiff/UDrkY/src/dual.jl:144
  ...
Stacktrace:
 [1] <(x::Function, y::Int64)
   @ Base ./operators.jl:279
 [2] top-level scope
   @ REPL[62]:1

Answer 1

There's not a lot of documentation for ImplicitEquations, but something stands out: you're not using the right operators. The package relies on unusual operators to represent math expressions with Julia functions: ≪ (\\ll[tab]), ≦ (\\leqq[tab]), ⩵ (\\Equal[tab]), ≶ (\\lessgtr[tab]) or ≷ (\\gtrless[tab]), ≧ (\\geqq[tab]), ≫ (\\leqq[tab]).

So that fix would look like:

using ImplicitEquations, Plots

f(a,b) = abs(a + im*b - (-1))
plot(f ≪ 1)

Update:

The above causes a method ambiguity error, even though you had already refactored a complex function to a 2D Cartesian one. f(a, b) = hypot(a+1, b) , which is more or less what abs does, also causes the error. This does work:

f(a, b) = sqrt((a+1)^2 + b^2)

, but we're already way past preserving your original expression's structure.

It looks like the issue is that at some point in hypot , OInterval(x::Ointerval) is called, but dispatch could not pick between (::Type{T})(x::T) where T<:Number in boot.jl or OInterval(a) in intervals.jl . Just redefining OInterval(a::Ointerval) = a won't work either because you run into another MethodError for decompose(::OInterval) , which is a method intended for processing floats. Looking at the comments in intervals.jl , this seems like a work in progress.