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 PolyGamma

 http://functions.wolfram.com/06.15.16.0023.01

 Input Form

 PolyGamma[-n, z] == (-1)^n PolyGamma[-n, -z] + Sum[(2 UnitStep[Re[z]] - 1) Sum[(1/k!) ((-1)^((n + k) UnitStep[Re[z]]) (z - (2 UnitStep[Re[z]] - 1) p)^k + (-1)^((n + k) UnitStep[-Re[z]]) (z - (2 UnitStep[Re[z]] - 1) p + 2 UnitStep[Re[z]] - 1)^k) Sum[((-1)^j/j!) PolyGamma[j + k - n, 1], {j, 0, n - k - 2}], {k, 0, n - 2}] + (1/(n - 1)!) ((z - (2 UnitStep[Re[z]] - 1) p)^(-1 + n) (-EulerGamma + Log[-p + (2 UnitStep[Re[z]] - 1) z] - PolyGamma[n]) + (z - (2 UnitStep[Re[z]] - 1) p + (2 UnitStep[Re[z]] - 1))^(-1 + n) (EulerGamma - Log[-1 + p - (2 UnitStep[Re[z]] - 1) z] + PolyGamma[n])), {p, 1, Floor[Abs[Re[z]]]}] + ((z - (2 UnitStep[Re[z]] - 1) Floor[Abs[Re[z]]])^(-1 + n)/(n - 1)!) (EulerGamma + (I Pi ((2 UnitStep[Re[z]] - 1) z - Floor[Abs[Re[z]]]))/n + 2 I Pi Floor[3/4 - Arg[(2 UnitStep[Re[z]] - 1) z - Floor[Abs[Re[z]]]]/ (2 Pi)] + PolyGamma[n] + Log[-2 Pi I] - Log[Floor[Abs[Re[z]]] - (2 UnitStep[Re[z]] - 1) z] - Sum[(Binomial[-1 + n, k] k! PolyLog[1 + k, 1])/ (2 Pi I ((2 UnitStep[Re[z]] - 1) z - Floor[Abs[Re[z]]]))^k, {k, 1, -1 + n}] + Sum[(-1)^k Binomial[-1 + n, k] Sum[(Binomial[k, j] j! PolyLog[1 + j, E^(2 I Pi (2 UnitStep[Re[z]] - 1) z)])/(2 Pi I (Floor[Abs[Re[z]]] - (2 UnitStep[Re[z]] - 1) z))^j, {j, 0, k}], {k, 0, -1 + n}]) /; Element[n, Integers] && n > 0

 Standard Form

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 MathML Form

 ψ TagBox["\[Psi]", PolyGamma] ( - n ) ( z ) ( - 1 ) n ψ TagBox["\[Psi]", PolyGamma] ( - n ) ( - z ) + p = 1 "\[LeftBracketingBar]" Re ( z ) "\[RightBracketingBar]" ( 1 ( n - 1 ) ! ( ( - log ( p - ( 2 θ UnitStep ( Re ( z ) ) - 1 ) z - 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( n ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ( - p ( 2 θ UnitStep ( Re ( z ) ) - 1 ) + ( 2 θ UnitStep ( Re ( z ) ) - 1 ) + z ) n - 1 + ( z - ( 2 θ UnitStep ( Re ( z ) ) - 1 ) p ) n - 1 ( log ( ( 2 θ UnitStep ( Re ( z ) ) - 1 ) z - p ) - ψ TagBox["\[Psi]", PolyGamma] ( n ) - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ) + ( 2 θ UnitStep ( Re ( z ) ) - 1 ) k = 0 n - 2 1 k ! ( ( - 1 ) ( k + n ) θ UnitStep ( - Re ( z ) ) ( - ( 2 θ UnitStep ( Re ( z ) ) - 1 ) p + z + 2 θ UnitStep ( Re ( z ) ) - 1 ) k + ( - 1 ) ( k + n ) θ UnitStep ( Re ( z ) ) ( z - ( 2 θ UnitStep ( Re ( z ) ) - 1 ) p ) k ) j = 0 - k + n - 2 ( - 1 ) j ψ TagBox["\[Psi]", PolyGamma] ( j + k - n ) ( 1 ) j ! ) + ( z - ( 2 θ UnitStep ( Re ( z ) ) - 1 ) "\[LeftBracketingBar]" Re ( z ) "\[RightBracketingBar]" ) n - 1 ( n - 1 ) ! ( 2 π 3 4 - arg ( ( 2 θ UnitStep ( Re ( z ) ) - 1 ) z - "\[LeftBracketingBar]" Re ( z ) "\[RightBracketingBar]" ) 2 π + π ( ( 2 θ UnitStep ( Re ( z ) ) - 1 ) z - "\[LeftBracketingBar]" Re ( z ) "\[RightBracketingBar]" ) n + log ( - 2 π ) - log ( "\[LeftBracketingBar]" Re ( z ) "\[RightBracketingBar]" - ( 2 θ UnitStep ( Re ( z ) ) - 1 ) z ) + ψ TagBox["\[Psi]", PolyGamma] ( n ) - k = 1 n - 1 ( 2 π ( ( 2 θ UnitStep ( Re ( z ) ) - 1 ) z - "\[LeftBracketingBar]" Re ( z ) "\[RightBracketingBar]" ) ) - k ( n - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] k ! Li PolyLog k + 1 ( 1 ) + k = 0 n - 1 ( - 1 ) k ( n - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] j = 0 k ( 2 π ( "\[LeftBracketingBar]" Re ( z ) "\[RightBracketingBar]" - ( 2 θ UnitStep ( Re ( z ) ) - 1 ) z ) ) - j ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] j ! Li PolyLog j + 1 ( 2 π ( 2 θ UnitStep ( Re ( z ) ) - 1 ) z ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) /; n + Condition PolyGamma -1 n z -1 n PolyGamma -1 n -1 z p 1 z 1 n -1 -1 -1 p -1 2 UnitStep z -1 z -1 PolyGamma n -1 p 2 UnitStep z -1 2 UnitStep z -1 z n -1 z -1 2 UnitStep z -1 p n -1 2 UnitStep z -1 z -1 p -1 PolyGamma n -1 2 UnitStep z -1 k 0 n -2 1 k -1 -1 k n UnitStep -1 z -1 2 UnitStep z -1 p z 2 UnitStep z -1 k -1 k n UnitStep z z -1 2 UnitStep z -1 p k j 0 -1 k n -2 -1 j PolyGamma j k -1 n 1 j -1 z -1 2 UnitStep z -1 z n -1 n -1 -1 2 3 4 -1 2 UnitStep z -1 z -1 z 2 -1 2 UnitStep z -1 z -1 z n -1 -2 -1 z -1 2 UnitStep z -1 z PolyGamma n -1 k 1 n -1 2 2 UnitStep z -1 z -1 z -1 k Binomial n -1 k k PolyLog k 1 1 k 0 n -1 -1 k Binomial n -1 k j 0 k 2 z -1 2 UnitStep z -1 z -1 j Binomial k j j PolyLog j 1 2 2 UnitStep z -1 z n SuperPlus [/itex]

 Rule Form

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

 2007-05-02