warning

```
Sum[(a^2 + (b + n)^2)^(-1), {n, -Infinity, Infinity}]
```

given

$$ frac { pi sinh (2 pi a)} {a ( cosh (2 pi a) – cos (2 pi b))}} $$

while the sum

```
Sum[(a^2 + (b + 2 [Pi] n) ^ 2) ^ (- 1), {n, -Infinity, Infinity}
```

given

$$ begin {array} {cc}

{And

begin {array} {cc}

frac { coth left ( frac {a} {2} + frac {ib} {2} right) + coth left ( frac {a} {2} – frac {ib} {2 } right)} {4 a} & arg (b + ia) geq 0 \

frac { coth left ( frac {1} {2} (a + ib) right)} {4a} + frac { coth left ( frac {1} {2} (ai b) right)} {4 a} – frac {1} {2 a} & text {True} \

end {array}

\

end {array} $$

listing two different cases. What is the source of this?

Also, ask a condition on $ a $ to be real and positive

```
Refine[Sum[(a^2 + (b + 2*[Pi] n) ^ 2) ^ (- 1), {n, -Infinity,
Infinite}], {Element[{a, b}, Reals], a> 0}]
```

do not get rid of "cases", as it should be (because then the imaginary part of $ (b + ia) $ always positive), the result remains exactly the same as above. Maybe I have not stated the assumptions on $ a $ correctly? I use Mathematica version 11.3.0.0.

Related to

What is the meaning of True in my result?