Inverse transform cannot undo Fourier transform of a symfun

[cbm755]

octave:74> syms x w f(x) G(w)
octave:75> fourier(f)
ans = (sym)

                  ⎛          w ⎞
  FourierTransform⎜f(x), x, ───⎟
                  ⎝         2⋅π⎠

octave:76> syms x w f(x) G(w)
octave:77> F = fourier(f)
F = (sym)

                  ⎛          w ⎞
  FourierTransform⎜f(x), x, ───⎟
                  ⎝         2⋅π⎠

octave:78> g = ifourier(G)
g = (sym)

                         ⎛          x ⎞
  InverseFourierTransform⎜G(w), w, ───⎟
                         ⎝         2⋅π⎠
  ─────────────────────────────────────
                   2⋅π                 

octave:79> 

So far so good, but:

octave:79> fourier(g)
PYTHON: Error in cmd
Resetting the octsympy communication mechanism
Closing the Python pipe...
Traceback (most recent call last):
  File "<stdin>", line 10, in <module>
  File "<stdin>", line 7, in _fcn
  File "sympy/integrals/transforms.py", line 1378, in fourier_transform
    return FourierTransform(f, x, k).doit(**hints)
  File "sympy/integrals/transforms.py", line 117, in doit
    self.function_variable, self.transform_variable, **hints)
  File "sympy/integrals/transforms.py", line 1317, in _compute_transform
    self.__class__._name, **hints)
  File "sympy/integrals/transforms.py", line 195, in wrapper
    res = func(*args, **kwargs)
  File "sympy/integrals/transforms.py", line 1286, in _fourier_transform
    F = integrate(a*f*exp(b*I*x*k), (x, -oo, oo))
  File "sympy/utilities/decorator.py", line 35, in threaded_func
    return func(expr, *args, **kwargs)
  File "sympy/integrals/integrals.py", line 1251, in integrate
    integral = Integral(*args, **kwargs)
  File "sympy/integrals/integrals.py", line 74, in __new__
    obj = AddWithLimits.__new__(cls, function, *symbols, **assumptions)
  File "sympy/concrete/expr_with_limits.py", line 356, in __new__
    limits, orientation = _process_limits(*symbols)
  File "sympy/concrete/expr_with_limits.py", line 57, in _process_limits
    raise ValueError('Invalid limits given: %s' % str(symbols))
ValueError: Invalid limits given: ((x/(2*pi), -oo, oo),)

error: python_cmd: unexpected return
error: called from:
error:   /home/cbm/work/octsympy/octsympy.git/inst/python_cmd.m at line 143, column 5
error:   /home/cbm/work/octsympy/octsympy.git/inst/@sym/fourier.m at line 63, column 7

Looks like this might be because the way we now deal with the 2*pi factors. @maprieto maybe @AlexanderMisel had the right approach? Or possibly its an upstream problem: I haven't checked....

[cbm755]

Might be sympy's fault---along with many other transform bugs :(!

For example, even hacking around the limit error above:

octave:99> F = fourier(f, x, 2*sym(pi)*w)
F = (sym) FourierTransform(f(x), x, w)
octave:100> 2*sym(pi)*ifourier(F, w, x*2*sym(pi))
ans = (sym) InverseFourierTransform(FourierTransform(f(x), x, w), w, x)

note it does not simplify back to f(x).

[maprieto]

Confirmed: the direct and inverse application of the Fourier transform does not simplify the result adequately. However, to reproduce the SMT Fourier transform implementation (including the 2*pi factors) and to avoid the reported errors (arisen from a simplification step), as a provisional patch it should be added the sympy hint "simplify=False" to Fourier and inverse Fourier functions as follows:

import sympy as sp
from sympy.abc import x, w
# It does not simplify the direct and inverse transforms
f=sp.Function('f')
F=sp.fourier_transform(f(x),x,w)
g=sp.inverse_fourier_transform(F,w,x)
# SMT like implementation
F=sp.fourier_transform(f(x),x,w/(2*sp.pi),simplify=False)
g=sp.inverse_fourier_transform(F,w,x/(2*sp.pi),simplify=False)/(2*sp.pi)
# SMT like implementation
F=sp.Function('F')
f=sp.inverse_fourier_transform(F(w),w,x/(2*sp.pi),simplify=False)/(2*sp.pi)
G=sp.fourier_transform(f,x,w/(2*sp.pi),simplify=False)

So, it could be suggested to labelled also as "upstream"