FunctionDomain ouput conflict with Limit output

Question

I was just trying to find out the available domain for a function using the following :

FunctionDomain[-((I*a*(-1 + E^(2*I*a*Pi)))/(-1 + a^2)), a, Complexes]

& this unexpectedly returned : -1 + a^2 != 0 whereas I had previously checked :

Limit[-((I*a*(-1 + E^(2*I*a*Pi)))/(-1 + a^2)), a -> 1]

which returned as expected a Pi.

Can someone please explain why FunctionDomain failed? How can one trust it's output when it didn't calculate the limits here? I suspect that I am doing something incorrectly.

Furthermore, I also didn't understand the output of the following :

FunctionDomain[-((I*a*(-1 + E^(2*I*a*Pi)))/(-1 + a^2)), a, Reals] (* False *)

Also, how would someone extract the information about the domain for non-zero range (those values of a where my expression is not zero) in this case?

Answer 1

Clear["Global`*"]

expr = -((I*a*(-1 + E^(2*I*a*Pi)))/(-1 + a^2));

Limit[expr, a -> #] & /@ {-1, 1}

(* {π, π} *)

FunctionDomain[expr, a, Complexes]

(* -1 + a^2 != 0 *)

This indicates that the function is undefined for a = 1 or a = -1. The fact that the limits exist does not indicate that the function is defined for these values.

This is similar to Sin[x]/x which is undefined for x = 0 but the limit as x -> 0 is 1. In this case, Sinc[x] is defined to include the limiting case in the function definition.

FunctionDomain[expr, a, Reals]

(* False *)

This indicates that there are no real values of a for which expr is real.