Consider the following code:
IN: = Sum[1/((x + n)^2 + a^2), {n, -Infinity, Infinity}]
Out: = (\[Pi] Sinh[2 a \[Pi]])/(a (-Cos[2 \[Pi] x] + Cosh[2 a \[Pi]]))
But if I input
Sum[a/((x + n)^2 + a^2), {n, -Infinity, Infinity}]
just by multiplying a factor a, the result is totally different.
1/2 I (PolyGamma[0, 1 - I a - x] - PolyGamma[0, 1 + I a - x] +
PolyGamma[0, -I a + x] - PolyGamma[0, I a + x])
Apparently there is something wrong here... Could anybody help me?