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 Cot

 http://functions.wolfram.com/01.09.21.0251.01

 Input Form

 Integrate[(Cot[a t] - 1/(a t)) (z - t)^n, {t, 0, w}] == (z^n/a) (-2 I Pi Floor[3/4 - Arg[a w]/(2 Pi)] - Log[-2 I] - Log[a w] - (I (-1)^n a w^(1 + n))/(z^n (1 + n)) + Sum[(Binomial[n, k] k! PolyLog[1 + k, 1])/(2 I a z)^k, {k, 1, n}] - Sum[(Binomial[n, k]/(k + 1)) (-(w/z))^k (I a w + (k w - n w)/(z + k z)), {k, 0, n - 1}] - Sum[(-(w/z))^k Binomial[n, k] Sum[(Binomial[k, j] j! PolyLog[1 + j, E^(2 I a w)])/(-2 I a w)^j, {j, 0, k}], {k, 0, n}]) /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 0 w ( cot ( a t ) - 1 a t ) ( z - t ) n t z n a ( - ( - 1 ) n a w n + 1 z - n n + 1 - 2 π 3 4 - arg ( a w ) 2 π - log ( - 2 ) - log ( a w ) - k = 0 n - 1 1 k + 1 ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( - w z ) k ( a w + k w - n w k z + z ) + k = 1 n ( 2 a z ) - k ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] k ! Li PolyLog k + 1 ( 1 ) - k = 0 n ( - w z ) k ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] j = 0 k ( - 2 a w ) - j ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["j", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] j ! Li PolyLog j + 1 ( 2 a w ) ) /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] Condition t 0 w a t -1 1 a t -1 z -1 t n z n a -1 -1 -1 n a w n 1 z -1 n n 1 -1 -1 2 3 4 -1 a w 2 -1 -1 -2 -1 a w -1 k 0 n -1 1 k 1 -1 Binomial n k -1 w z -1 k a w k w -1 n w k z z -1 k 1 n 2 a z -1 k Binomial n k k PolyLog k 1 1 -1 k 0 n -1 w z -1 k Binomial n k j 0 k -2 a w -1 j Binomial k j j PolyLog j 1 2 a w n [/itex]

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2007-05-02