# Calculate the sum of a series of lorentzain function?

GROUPS:
 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?
 You can apply FullSimplify[] and the result will be more appropriate with your expectations: In[10]:= Sum[1/((x + n)^2 + a^2), {n, -Infinity, Infinity}] Out[10]= (\[Pi] Sinh[2 a \[Pi]])/(a (-Cos[2 \[Pi] x] + Cosh[2 a \[Pi]])) In[11]:= Sum[a/((x + n)^2 + a^2), {n, -Infinity, Infinity}] Out[11]= 1/2 (\[Pi] Coth[a \[Pi] - I \[Pi] x] + \[Pi] Coth[a \[Pi] + I \[Pi] x]) In[12]:= FullSimplify[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])] Out[12]= 1/2 \[Pi] (Coth[\[Pi] (a - I x)] + Coth[\[Pi] (a + I x)])