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  1. Clear[[Beta],[Alpha]]
  2. f[z_]:=[Pi]^2 Cot[([Pi]  z [Alpha])^(1/2)]
  3. g[z_]:=f[z+(-([Pi]/2)+2 k [Pi])^2/([Pi] [Alpha])]
  4. res[k_]=Residue[FullSimplify[TrigToExp@g[z],k[Element]Integers],{z,0}]
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