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 Cot

 http://functions.wolfram.com/01.09.21.0252.01

 Input Form

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

 Standard Form

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 MathML Form

 Integrate [ ( z - t ) n a t , { t , 0 , p } , GenerateConditions "\[Rule]" False ] + Integrate [ ( z - t ) n ( cot ( a t ) - 1 a t ) , { t , 0 , p } , GenerateConditions "\[Rule]" False ] z n a ( - a p n + 1 ( - p z ) n - 2 π 3 4 - arg ( a p ) 2 π - log ( - 2 ) + log ( p ) - log ( a p ) - a p 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]] ( - p z ) k + 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 ( - p 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 p ) - 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 p ) ) /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] Condition Rule GenerateConditions t 0 p z -1 t n a t -1 Rule GenerateConditions t 0 p z -1 t n a t -1 1 a t -1 z n a -1 -1 a p n 1 -1 -1 p z -1 n -1 2 3 4 -1 a p 2 -1 -1 -2 p -1 a p -1 a p k 0 n -1 1 k 1 -1 Binomial n k -1 p z -1 k k 1 n 2 a z -1 k Binomial n k k PolyLog k 1 1 -1 k 0 n -1 p z -1 k Binomial n k j 0 k -2 a p -1 j Binomial k j j PolyLog j 1 2 a p n [/itex]

 Rule Form

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

 2007-05-02