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Can someone help decipher what is the meaning of this computation that Mathematica has done?

Assuming[Re[x] >= 0 && a ∈ Reals,
  Reduce [1 / ((x Tanh[π x]) / (x^2 +  a^2)) == 0, x, Complexes]]

  (C[1] ∈ Integers && x != 0 && (x == (-((I π)/2) + 2 I π C[1])/π || 
     x == ((I π)/2 + 2 I π C[1])/π)) || (Sin[a π] != 0 && a != 0 && (x == -I a || x == I a))

My understanding is that the $\frac{x\;tanh (\pi\,x) } {x^2 + a^2 }$ blows up where $tanh(\pi\,x)= \infty $ and where $x^2 +a^2 = 0$. And these happen respectively at $x = i(\mathbb{Z} + 0.5)$ and $x = \pm i \vert a \vert$.

But why is Mathematica detecting a totally different set of imaginary zeroes for the reciprocal of the function than the first set of poles that I can see in the above?

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closed as off-topic by rasher, Michael E2, Sjoerd C. de Vries, bobthechemist, Yves Klett Mar 22 at 13:32

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Your code has the reciprocal of the function in the text - which one do you mean ? –  b.gatessucks Mar 22 at 8:24
@b.gatessucks Yes - I would have thought that "reduce" when asked for the zeroes of the reciprocal of the function would generate precisely the poles of the function that I enumerated below by hand. Why aren't the two answers matching? [...what are those "||" in Mathematica's output?..] –  user6818 Mar 22 at 8:34
|| is the logical Or. Try adding //TraditionalForm at the end of your command and see if the output is more readable. –  b.gatessucks Mar 22 at 8:46
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1 Answer 1

Adding FullSimplify might clarify things for you:

    Assuming[Re[x] >= 0 && a ∈ Reals, 
             Reduce[1/((x Tanh[π x])/(x^2 + a^2)) == 0, x, Complexes]] // FullSimplify

results in:

(C[1] ∈ Integers && x != 0 && (x == -(I/2) + 2 I C[1] || x == I/2 + 2 I C[1])) 
|| (Sin[a π] != 0 && (x == -I a || x == I a))

This is telling you (first line) "for some constants (C[1]) of integer and x not zero, x is -I/2 + 2*I* or I/2 + 2*I*". This is your I(Z+.5).

The || denotes or, the && and.

The second line tells you that when Sin[a π] is not zero, x can be +/- I*a.

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The Mathematica's answers are $i( \pm 0.5 + 2\mathbb{Z})$ but my answer (and hopefully the correct answer!) is $i( 0.5 + \mathbb{Z})$ - how are these the same? –  user6818 Mar 22 at 8:55
@user6818: These are the same... try e.g. Table[.5 + x, {x, 1, 10}] and Table[{-.5 + 2 x, .5 + 2 x}, {x, 1, 5}] // Flatten –  rasher Mar 22 at 8:59
wonder why Mathematica writes in such a weird split way! –  user6818 Mar 24 at 4:05
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