Hw2 Problem I.2.  Bessel function result for the final integral after the branch cut manipulation.

Integrate[u E^(-u)/Sqrt[u^2 - b^2], {u, b, Infinity}]// HoldForm//TraditionalForm
i = Integrate[u E^(-u)/Sqrt[u^2 - b^2], {u, b, Infinity}];
Assuming[b > 0, i // Simplify]

(*c = Cell[BoxData[{RowBox[{RowBox[{"i"," ","="," ",RowBox[{"Integrate","[",RowBox[{RowBox[{"u"," ",RowBox[{RowBox[{"E","^",RowBox[{"(",RowBox[{"-","u"}],")"}]}],"/",RowBox[{"Sqrt","[",RowBox[{RowBox[{"u","^","2"}]," ","-"," ",RowBox[{"b","^","2"}]}],"]"}]}]}],","," ",RowBox[{"{",RowBox[{"u",","," ","b",","," ","Infinity"}],"}"}]}],"]"}]}],";"}],"
",RowBox[{"Assuming","[",RowBox[{RowBox[{"b"," ",">"," ","0"}],","," ",RowBox[{"i"," ","//"," ","Simplify"}]}],"]"}]}],"Input",CellChangeTimes->{{3.749921190349818*^9,3.749921288621295*^9},3.749921783227393*^9,{3.7499219085108633`*^9,3.7499219092769117`*^9},{3.7499221930087233`*^9,3.74992232223874*^9}}];
c2  = Cell[BoxData[RowBox[{"b"," ",RowBox[{"BesselK","[",RowBox[{"1",",","b"}],"]"}]}]],"Output",CellChangeTimes->{{3.749921237882144*^9,3.749921259026997*^9},3.749921290251644*^9,3.749921788326624*^9,3.749921910168478*^9,3.7499221970998087`*^9,{3.7499223095734043`*^9,3.749922323095406*^9}}];*)

Import["https://raw.githubusercontent.com/jkuczm/MathematicaCellsToTeX/master/NoInstall.m"]

(*CellToTeX[c]*)

(*CellToTeX[c2]*)

(*Series[a BesselK[1, a]/Exp[a], {a, 0, 10}]*)