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In version 13.0.1 on Windows 10:

Limit[E^(-2 a \[Pi])*Gamma[(I a)/2 - n] ((4^-n E^((3 a \[Pi])/2 + I n \[Pi]) \[Pi])/(
Gamma[-2 n] Gamma[1 + (I a)/2 + n]) + 
I (-1 + E^(2 a \[Pi])) *Hypergeometric2F1Regularized[1, 
1 + (I a)/2 + n, 1 + (I a)/2 - n, -1]), {a, n} -> {2*I, 4}, Direction -> Complexes]
If[Complexes == Reals, Asymptotics`MultivariateLimitDump`ff$1612446 = Asymptotics`MultivariateLimitDump`realfpreproc[ Asymptotics`MultivariateLimitDump`ff$1612446, {a, n}, Asymptotics`MultivariateLimitDump`zero$1612446]; Asymptotics`MultivariateLimitDump`RealMLimit[ Asymptotics`MultivariateLimitDump`ff$1612446, {a, n}, Asymptotics`MultivariateLimitDump`zero$1612446, {Limit, True, False, Complexes, Automatic, Automatic, "Quality"}], Asymptotics`MultivariateLimitDump`ComplexMLimit[ Asymptotics`MultivariateLimitDump`ff$1612446, {a, n}, Asymptotics`MultivariateLimitDump`zero$1612446, {Limit, True, False, Complexes, Automatic, Automatic, "Quality"}]]

This is not the returned input. The expression under the limit originates from Integrate[E^(-a*x)*(Cos[x]^2)^n, {x, 0, 2*Pi}, Assumptions -> n \[Element] PositiveIntegers, GenerateConditions -> True].

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2
  • $\begingroup$ The same issue with Limit[1/2 E^(-2 a \[Pi] - I n \[Pi]) (1 + E^( a \[Pi])) (1/(a + 2 I n) 4 (-1 + E^(a \[Pi])) Hypergeometric2F1[1, 1 + (I a)/2 + n, 1 + (I a)/2 - n, -1] (Cos[n \[Pi]] + I Sin[n \[Pi]]) + ( 4^-n E^((a \[Pi])/ 2) \[Pi] Gamma[(I a)/2 - n] (1 + I Tan[n \[Pi]]))/( Gamma[-2 n] Gamma[1 + (I a)/2 + n]), {a, n} -> {2*I, 4}, Direction -> Complexes] though the expression under the limit as the result of Integrate[E^(-a*x)*(Cos[x]^2)^n, {x, 0, 2*Pi}, Assumptions -> n >= 1, GenerateConditions -> True] should be continuous. $\endgroup$
    – user64494
    Apr 16 at 6:19
  • $\begingroup$ Moreover, the same issue with Limit[Evaluate[Integrate[E^(-a*x)*(Cos[x]^2)^n, {x, 0, 2*Pi}, Assumptions -> Re[n] >= 1, GenerateConditions -> True]], {n, a} -> {2, 2*I}, Direction -> Complexes] though the result should be analytical around $(2,2i)$. $\endgroup$
    – user64494
    Apr 16 at 9:39

1 Answer 1

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This looks like a piece of internal code with a rather obvious bug: "==" instead of "===" is used to compare domains, and hence we get an unevaluated If conditional instead of the result of running the "else" case. Will be fixed in the next version of Mathematica. Thanks for pointing it out.

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1
  • $\begingroup$ Thank you. I will be waiting for that version of Mathematica before making a decision. $\endgroup$
    – user64494
    Apr 16 at 5:44

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