Complex missing Euler's formula

[AshtonSBradley]

Symbolics doesn't seem to be able to evaluate/define complex exponentials

@variables x 

exp(im*x)

MethodError: no method matching decompose(::Symbolics.Num)

Closest candidates are:

decompose(!Matched::Integer) at /Applications/Julia-1.7.app/Contents/Resources/julia/share/julia/base/float.jl:642

decompose(!Matched::Rational) at /Applications/Julia-1.7.app/Contents/Resources/julia/share/julia/base/rational.jl:515

decompose(!Matched::FixedPointNumbers.Normed) at ~/.julia/packages/FixedPointNumbers/HAGk2/src/normed.jl:296

...

isfinite(::Symbolics.Num)@float.jl:538
exp(::Complex{Symbolics.Num})@complex.jl:663
top-level scope@[Local: 1](http://localhost:1235/edit?id=a6a8d612-c9ba-11ec-173e-49370558ac76#)

Defining the function

t(x) = cos(x)+im*sin(x)

and trying to simplify gives the right result, but fails to express it as complex exponential:

simplify(t(x)^2)

im*sin(2x) + cos(2x)

Also, higher order tests are slow to generate correct but needlessly complicated expressions

simplify(t(x)^6)

im*((cos(2x)^2 - 4(cos(x)^2)*(sin(x)^2))*sin(2x) + (2//1)*(cos(2x)^2)*sin(2x)) + (cos(2x)^2 - 4(cos(x)^2)*(sin(x)^2))*cos(2x) - 8(cos(x)^2)*(sin(x)^2)*cos(2x)

[AshtonSBradley]

This seems to me to be required if complex numbers are to become first-class citizens of Symbolics

[AshtonSBradley]

#558

[shashi]

@YingboMa any thoughts on this?

[AshtonSBradley]

Using @syms gives quite different results. The exponential definition seems to work this time:

julia> @syms x
(x,)

julia> exp(im*x)
exp((im)*x)

but then the simplify step doesn't seem to work at all:

julia> t(x) = cos(x)+im*sin(x)
t (generic function with 1 method)

julia> simplify(t(x)^2)
((0 + 1im)*sin(x) + cos(x))^2

and yet

julia> simplify(exp(im*x)^2)
exp((0 + 2im)*x)

Not sure what the right way to handle this is - my knowledge of Symbolics is weak.

My intention is that I want to register some basic identities like exp(im*x) = cos(x)+im*sin(x) and the inverses for trig cos(x) = (exp(im*x)+exp(-im*x))/2, etc, and then I would expect to be able to derive identities by examining the real and imaginary parts of powers etc.

[oameye]

I think this is somewhat resolved.

julia> @variables a
1-element Vector{Num}:
 a

julia> exp(im*a)
cos(a) + im*sin(a)