Can I avoid expansion of complex `exp`?

[leostenzel]

Hi & thanks for developing this pretty cool tool!

I'm running into issues due to the expansion of exp(im * x) according to Euler's formula, i.e.,

using Symbolics
@variables x::Real
exp(im * x)
# cos(α) + im*sin(α)

exp(im * x) * exp(2im * x) |> expand |> simplify
# -sin(x)*sin(2x) + cos(x)*cos(2x) + im*(sin(x)*cos(2x) + cos(x)*sin(2x))

This probably makes sense, sometimes. But I'm multiplying a lot of exponentials, and thus get exponentially many terms.

As noted in the related issue #591, using @syms has the desired behavior, and keeps exp as is:

@syms y::Real
exp(im * y)
# exp((im)*y)

exp(im * y) * exp(2im * y) |> simplify
# exp((0 + 3im)*y)

PS: Unfortunately, just using @syms doesn't seem to work in my case, either, as some differentials don't get evaluated:

(Differential(y))(conj(exp(im * y))) |> expand_derivatives
# (0 + 1im)*exp((im)*y)*Differential(y)(conj(exp((im)*y)))

[ChrisRackauckas]

julia> using Symbolics

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

julia> exp(x)
exp(real(x))*cos(imag(x)) + im*exp(real(x))*sin(imag(x))

you probably want to use complex symbolic values, but indeed you will still get expression growth in some situations.

[leostenzel]

Thanks for your response, Chris!

I tried something along those lines, defining a @variables i::Complex and later substituting real(i) => 0.
That improves things in toy examples, but it's not perfect.

But I just realized sin & cos actually come from a call to base/complex.jl, so it's not even some Symbolics.jl rewrite…