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I don't know if this has already been discussed.

Integrate[BesselJ[2 m + 1, x], {x, 0, ∞}, Assumptions -> m ϵ Integers]

ConditionalExpression[1, Re[m] > -1]

Integrate[BesselJ[2 m, x], {x, 0, ∞}, Assumptions -> m ϵ Integers]

ConditionalExpression[1, Re[m] > -(1/2)]

Assuming[m ∈ Integers, Integrate[BesselJ[2 m + 1, x], {x, 0, ∞}]]

ConditionalExpression[1, m > -1]

Assuming[m ∈ Integers, Integrate[BesselJ[2 m, x], {x, 0, ∞}]]

ConditionalExpression[1, m > -(1/2)]

Why doesn't Mathematica consider the evenness or oddness of the 1st argument (i.e., 2 m or 2 m + 1)?

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  • $\begingroup$ @Karsten 7: Thanks for the edit. $\endgroup$
    – Dimitris
    Commented Oct 16, 2015 at 8:53
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    $\begingroup$ Related (possibly duplicates): (19833), (42114) -- See this answer to (19833) especially. $\endgroup$
    – Michael E2
    Commented Oct 16, 2015 at 10:12
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    $\begingroup$ Thanks for the links. However here the situation is different. Assuming[{k ∈ Integers}, Integrate[ Exp[ I k t], {t, -π, π}]] returns an incorrect result whereas Integrate[ Exp[ I k t], {t, -π, π},Assumptions->k ∈ Integers] returns the correct one. Here it seams that Integrate simply ignores (or cannot take into account) the Assumptions and proceed to its evaluation of the integrand without any Assumptions for both of the cases. $\endgroup$
    – Dimitris
    Commented Oct 16, 2015 at 12:10
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    $\begingroup$ I think Daniel's answer explains it: Try Simplify[Integrate[Exp[I k t], {t, -\[Pi], \[Pi]}, Assumptions -> k \[Element] Integers], k \[Element] Integers]. $\endgroup$
    – Michael E2
    Commented Oct 16, 2015 at 13:03
  • $\begingroup$ I agree with @MichaelE2. $\endgroup$
    – Daniel Lichtblau
    Commented Oct 16, 2015 at 14:48

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