What's wrong in FunctionPeriod?

Question

FunctionPeriod[Sin[16 Pi x/5 - Pi/8], x, Integers]

80

then I ListPlot the function:

ListPlot[Transpose[{Range[0, 90], Sin[16/5 π Range[0, 90] - π/8]}], Filling -> Axis]

then I test the number:

Union[Simplify[Sin[16/5 π Range[0, 90] - π/8]]]

{-Cos[π/40], Cos[(7 π)/40],Cos[(9 π)/40], -Sin[(3 π)/40], -Sin[π/8]}

so I wonder why Mathematica gives 80 instead of 5?

Answer 1

FunctionPeriod is certainly imperfect. There is already discussion of this in this question, though that predates FunctionPeriod and uses lower level tools buried in the Periodic context. Nonetheless, the same comments apply:

FunctionPeriod[Sin[3 t] Sin[5 t], t]
(* Incorrect result: 2Pi *)

FunctionPeriod[TrigFactor[Sin[3 t] Sin[5 t]], t]
(* Correct result: Pi *)

Similarly,

FunctionPeriod[Sin[16 Pi x/5 - Pi/8], x, Integers]
(* Incorrect result: 80 *)

FunctionPeriod[TrigFactor[Sin[16 Pi x/5 - Pi/8]], x, Integers]
(* Correct result: 5 *)

I don't know of any cases where FunctionPeriod returns a result that is too small so you might generally try applying several of the Trig* commands and grab the absolute value of the minimum.