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I want to calculate the residue of an equation, such as:

equ = 1/((x - 2)Sqrt[x - 1])

So first I need to find the pole of the equation. My method is find the root of Denominator[equ] == 0:

pole = Solve[Denominator[equa] == 0, x]

and the solutions are

{{x -> 1}, {x -> 2}}

But obviously, the equation has no residue at x = 1 because its order is 1/2.

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How can I automatically find the pole of the equation where the residue exists? Or, alternatively, how to find the pole whose order is an integer?

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How about

r[x_] := 1/((x - 2) Sqrt[x - 1])

Select[x /. Solve[Denominator[r[x]] == 0, x],
       SeriesCoefficient[r[x], {x, #, -1}] != 0 &]
   {2}

Or a more direct solution (with thanks to xzczd):

Select[x /. Solve[Denominator[r[x]] == 0, x],
       FreeQ[Residue[r[x], {x, #}], _Residue] &]
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