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I feel confident this problem has been posed before but alas I cannot find it.

Clear[k, m];
Assuming[Element[{k, m}, NonNegativeIntegers], 
 Integrate[E^(2 Pi I (k - m) t), {t, 0, 1}]]

yields 0, which is only valid when $k \neq m$. The answer should instead be a KroneckerDelta with indices $k$ and $m$.

How do I ensure the case $k=m$ is also included in the result?

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9
  • $\begingroup$ Mathematica gives result which is generic. You could always add extra assumption. Here is screen shot !Mathematica graphics which gives zero only when k!=m and gives 1 when k=m but I do not know if there is way to make Integrate do that on its own. $\endgroup$
    – Nasser
    Commented Nov 22, 2023 at 19:48
  • $\begingroup$ @Nasser: Thanks. The result $0$ isn't exactly "generic," but instead the typical case. I realize I could split the cases into $k=m$ and $k \neq m$, but I would have thought that Mathematica could do this explicitly. $\endgroup$
    – David G. Stork
    Commented Nov 22, 2023 at 19:50
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    $\begingroup$ Should be a Kronecker delta, not a Dirac delta. $\endgroup$
    – Roman
    Commented Nov 22, 2023 at 19:58
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    $\begingroup$ Using FullSimplify on that result with the same assumptions reduces it to 0 though. $\endgroup$
    – eyorble
    Commented Nov 22, 2023 at 23:23
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    $\begingroup$ Here is one older version among several others. $\endgroup$
    – Daniel Lichtblau
    Commented Nov 24, 2023 at 2:28

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