Should integrate detect orthogonality of functions in the integrand?

Question

What would be the correct way to tell Mathematica that

 Integrate[Sin[3 x] Sin[n x], {x, 0, Pi}]

Should not be zero when n is an integer for all n in the above? Since when n=3 the answer should be Pi/2 and zero for all other n values.

But Mathematica does not seem to detect this, even when told that n is integer:

ClearAll[x,n]
Integrate[Sin[3 x] Sin[n x], {x, 0, Pi},Assumptions -> Element[n, Integers] && n > 0]

 % /. n -> 3

And if

Assuming[Element[n,Integers]&&n>0,Integrate[Sin[3 x] Sin[n x],{x,0,Pi}]]
(*0*)

While

n=3;
Integrate[Sin[3 x] Sin[n x],{x,0,Pi}]
(* Pi/2*)

The question is, is it the programmer responsibility to tell Mathematica that sin(3 x)*sin(n x) is special case when n=3? May be make a function around it, something like

foo[n_Integer]:=
   If[n==3,
      Integrate[Sin[3 x]^2,{x,0,Pi}],
      Integrate[Sin[3 x] Sin[n x],{x,0,Pi}]] (*0*)
   ];

But I do not think the above is a good way to do in in general.

I think a special rule is needed to tell Mathematica about orthogonality of trig functions, and then ask Integrate to use this rule? But how to do this?

The question is: Should Mathematica have been able to automatically handle the special case of n=3 above? If not, what would be the best way to add a rule to tell Integrate to do this?

Answer 1

One way to get the answer:

Integrate[Sin[3 x] Sin[n x], {x, 0, Pi}, Assumptions -> Element[n, Integers] && n > 0]
Normal@Series[%, {n, 3, 0}]  (* evaluate at n -> 3 *)
(* Pi/2 *)

I suppose Mathematica should get the right answer, but there's way in which this shows the answer returned by Integrate contains the correct answer at n == 3. Algebraic notation is sometimes deficient, and for an analytic object like this result, the power series is probably the best mathematical representation of the object. The undefined points n == ±3 of the expression returned by Integrate are not even singularities of the series. (It is to address such a deficiency that the need for the Sinc function was felt.) Thus I think Series (also Limit) is a mathematically correct way to evaluate the result.

Answer 2

For this particular example, you could try using FourierSinCoefficient:

FourierSinCoefficient[Sin[3 x], x, n, FourierParameters->{-1,1}]

1/2 π DiscreteDelta[-3 + n]

Answer 3

I believe it's a bug!

I0=Simplify[Integrate[Sin[3 x] Sin[n x], {x, 0, Pi} ] ](*-((3 Sin[n \[Pi]])/(-9 + n^2))*)
I1=Simplify[Integrate[Sin[3 x] Sin[n x], {x, 0, Pi} ],Assumptions ->Element[n, Integers]](*0*)
I2=Integrate[Sin[3 x] Sin[n x], {x, 0, Pi}, Assumptions -> n != 3](*-((3 Sin[n \[Pi]])/(-9 + n^2))*)
I3=Integrate[Sin[3 x] Sin[n x], {x, 0, Pi}, Assumptions -> n ==  3](*\[Pi]/2*)

The integrals I0,I2,I3 are evaluated as expected whereras I1 gives a wrong result! It looks like the Limit/Singularity n->3 isn't recognized by the Simplify-Command!

Answer 4

f[n_] = Piecewise[{{Integrate[Sin[3 x]^2, {x, 0, Pi}], n == 3}},
  Integrate[Sin[3 x] Sin[n x], {x, 0, Pi}]]

Answer 5

EDIT: Didn't understand the question at first

First, note that to get 0, which is the answer when n≠3, you need to use Assuming instead of Assumptions:

ClearAll[x, n]
Assuming[
 Element[n, Integers] && n > 0,
 Integrate[Sin[3 x] Sin[n x], {x, 0, Pi}]
 ]

Output:

0

Now the question is why Mathematica fails to notice the special case of n=3. I speculate that Mathematica's ability to recognize the need of a delta function is lacking here. For instance, if I type an integral that is equivalent to the delta function, I just get an error:

Integrate[Exp[-I k x], {k, -\[Infinity], \[Infinity]}]

Output:

During evaluation of In[1845]:= Integrate::idiv: Integral of E^(-I k x) does not converge on {-\[Infinity],\[Infinity]}.

Your case may be another specific case of Integrate not identifying opportunities for the delta function. I would perhaps report it as a bug.