issues with sinc function

[Djuleeo]

i have an issue with lambdification of the sinc function, due to the different definitions of the sinc

using SymPy
x = symbols("x", real=true)

fun      = SymPy.sinc(x/pi)
f        = lambdify(fun)
x_test   = 1.0
body     = convert(Expr, fun)                                           # correctly translated  :(x ^ -1 * sin(x))
f(x_test) - sin(x_test)/x_test                                          # is equal to zero, as expected
  #
#### BUT if i use the library to rewrite it :

fun2    = sinc(x/pi).rewrite(sinc)      
body2   = convert(Expr, fun2)               # :(sinc(x))
f2  = lambdify(fun2)                            # becomes sin(π x)/ πx   !!!!

f2(x_test) - sin(x_test)/x_test                             # different than 0 !

[jverzani]

So, I think this is a different in defintions:

In Julia, sinc is sin(pi x)/(pi x) and defined by

Base.sinc(x::Sym) = iszero(x) ? one(x) : sinpi(x)/(PI*x)

In Sympy (https://docs.sympy.org/latest/modules/functions/elementary.html#sympy.functions.elementary.trigonometric.sinc) it is sin(x)/x.

You are basically finding the difference between sympy.sinc (sinc(x)) and SymPy.sinc (sinpi(x)/(PI*x)). To smooth this out, we could modify the walk_expression function to handle the case of sinc (the first one above), though perhaps just documenting it better would be the best case forward.

[Djuleeo]

maybe it could be changed such that:

code                              symbolic
sinc(x)              ->    sinc( pi x ) (= sin(pi x) /(pi x))
SymPy.sinc(x)        ->    sinc(x) (= sin(x) /x)                              using the sympy.sinc definition
as currently  there is no difference between the two:
sinc(x)              ->    sinc( pi x ) 
SymPy.sinc(x)        ->    sinc( pi x)                  

provided that the difference is well documented.

in any case, in the code above the issue is that i get the wrong version of sinc when i lambdify:

code                                           symbolic
SymPy.sinc(x/pi)                       ->     sin(x)/x                    # uses the sinpi(x)/(pi x) definition
SymPy.sinc(x/pi).rewrite(sinc)         ->     sinc(x)                     # symbolic manipulation, sin(x)/x definition
everything is good so far, but if i lambdify
symbolic                                                  numeric
sinc(x) (=sin(x)/x)                      ->     sinc(x)  (=sin(pi x)/(pi x))  !!!!
instead it should be:
 sinc(x) (=sin(x)/x)                     ->     sinc(x/pi)  ( =sin(pi x/pi )/(pi x/pi) = sin(x)/x   )

in the current way, the net result is that using .rewrite(sinc) or a .simplify() that leads to a (symbolic) sinc and then lambdifying gives back a different function, when these methods are meant to change the form, but not modify the math.

[jverzani]

I think the issue is the distinction between sinc(x), SymPy.sinc(x), and sympy.sinc(x). The first two are the same, the latter different (it being the python libraries version sin(x)/x.) However, lambdify rewrites sympy.sinc(x) to Julia's sinc and so we get:

sympy.sinc(x)(1) --> sin(1)/1 lambdify(sympy.sinc(x))(1) --> sin(pi*1)/(pi*1)

It seems like the proper thing to do would be to modify lambdify to reexpress sinc(x) using the Python definition, not the Julia one. This would be the same as this:

SINC(x) = sinc(x/pi)
lambdify(sympy.sinc(x), fns=Dict("sinc"=>SINC))(1)