from digital library of mathematical functions
First attempt:
Limit[BesselJ[v, z] Gamma[v + 1]/(1/2 z)^v, z -> 0,
GenerateConditions -> True]
Output : 1
That is, regardless the domain of v, the asymptotic equivalence holds.
I checked the following:
Limit[BesselJ[-3, z] Gamma[-3 + 1]/(1/2 z)^-3, z -> 0]
Output : ComplexInfinity
Next i tried the assumption:
Assuming[v \[Element] NegativeIntegers,
Limit[BesselJ[v, z] Gamma[v + 1]/(1/2 z)^v, z -> 0]]
Output : Indeterminate
Finally, i tried the correct :
Assuming[v \[NotElement] NegativeIntegers,
Limit[BesselJ[v, z] Gamma[v + 1]/(1/2 z)^v, z -> 0]]
Output : 1
Why is this happening?