Orthogonality Sin(nx)Sin(mx)

Question

I want to prove that $$ \int_{0}^{\pi } \sin(n \, x) \, \sin(m \, x) \, \mathrm{d} x=0 $$ for m,n integers and $$ m\neq n$$ My try is:

$Assumptions = m ≠ n
$Assumptions = m ∈ Integers
$Assumptions = n ∈ Integers
Integrate[Sin[m x] Sin[n x], {x, 0, π} ]

But no luck

Do you have a space (multiplication) between m, n and x ? — b.gates.you.know.what, Dec 05 '19 at 11:01
I didn't.... Now i am getting this ((-1)^n n Sin[m π])/(m^2 - n^2) — alicia roberts, Dec 05 '19 at 11:04
Note that the assumption m != n is not needed to get a result of 0. See https://mathematica.stackexchange.com/questions/174011/how-to-get-a-universal-answer-using-integrate and https://mathematica.stackexchange.com/questions/67080/how-to-code-around-known-mma-special-case-failures — Michael E2, Dec 05 '19 at 11:21

score 10 · Accepted Answer · edited Feb 11 '20 at 06:21

10

You overwrite $Assumptions several times, so it does not contain all the information that you meant to provide. Also Simplify can help where Integrate gave up the simplification. Try this:

$Assumptions = m != n && m ∈ Integers && n ∈ Integers
Simplify@Integrate[Sin[m x]*Sin[n x], {x, 0, π}]

edited Feb 11 '20 at 06:21

none

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answered Dec 05 '19 at 11:05

Henrik Schumacher

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Great, thanks!! – alicia roberts Dec 05 '19 at 11:07
You're welcome! – Henrik Schumacher Dec 05 '19 at 11:07
4

Also $Assumptions = m \[Element] Integers && n \[Element] Integers gives the same result 0 which is obviously wrong for the case n==m . Still (v12) a Mathematica problem? – Ulrich Neumann Dec 05 '19 at 11:17
Uh, yes. Must be the thing about Simplify assuming some generacity of the symbols... =/ – Henrik Schumacher Dec 05 '19 at 11:18
@UlrichNeumann Funny, I was making the same comment at the same time under the OP. :) – Michael E2 Dec 05 '19 at 11:23
@MichaelE2 Really funny, but you made it with more helpful details... – Ulrich Neumann Dec 05 '19 at 11:25

Orthogonality Sin(nx)Sin(mx)

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