# What does this result of Limit mean?

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, AsymptoticsMultivariateLimitDumpff$$1612446 = AsymptoticsMultivariateLimitDumprealfpreproc[ AsymptoticsMultivariateLimitDumpff$$1612446, {a, n}, AsymptoticsMultivariateLimitDumpzero$$1612446]; AsymptoticsMultivariateLimitDumpRealMLimit[ AsymptoticsMultivariateLimitDumpff$$1612446, {a, n}, AsymptoticsMultivariateLimitDumpzero$$1612446, {Limit, True, False, Complexes, Automatic, Automatic, "Quality"}], AsymptoticsMultivariateLimitDumpComplexMLimit[ AsymptoticsMultivariateLimitDumpff$$1612446, {a, n}, AsymptoticsMultivariateLimitDumpzero$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]. • 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. Apr 16 at 6:19 • 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)\$. Apr 16 at 9:39