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 Abs

 http://functions.wolfram.com/12.01.21.0008.01

 Input Form

 ContourIntegrate[z^(m - 1)/Abs[z - k]^(2 (n + 1)), {z, Ï Exp[I Ï†], {0, 2 Pi}}] == 2*Pi*I*(UnitStep[Ï - Abs[w]]*Sum[Binomial[n + k, k]* Binomial[n + m, n - k]*((w^m*Abs[w]^(2*k))/ (Ï^2 - Abs[w]^2)^(n + k + 1)), {k, 0, n}] + UnitStep[Abs[w] - Ï]*Sum[Binomial[n + k, k]* Binomial[n + m, n - k]*(Ï^(2*(m + k))/ (Conjugate[w]^m*(Abs[w]^2 - Ï^2)^(n + k + 1))), {k, 0, n}]) /; n âˆˆ Integers && n >= 0 && m âˆˆ Integers && m >= 0 && w != 0 && Abs[w] != Ï

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[Integral]", "\[ScriptCapitalC]"], RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List["m", "-", "1"]]], SuperscriptBox[RowBox[List["Abs", "[", RowBox[List["z", "-", "w"]], "]"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["\[Rho]", "-", RowBox[List["Abs", "[", "w", "]"]]]], "]"]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "k"]], ",", "k"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "m"]], ",", RowBox[List["n", "-", "k"]]]], "]"]], " ", FractionBox[RowBox[List[SuperscriptBox["w", "m"], " ", SuperscriptBox[RowBox[List["Abs", "[", "w", "]"]], RowBox[List["2", " ", "k"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["\[Rho]", "2"], "-", SuperscriptBox[RowBox[List["Abs", "[", "w", "]"]], "2"]]], ")"]], RowBox[List["n", "+", "k", "+", "1"]]]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "-", "\[Rho]"]], "]"]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "k"]], ",", "k"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "m"]], ",", RowBox[List["n", "-", "k"]]]], "]"]], " ", FractionBox[SuperscriptBox["\[Rho]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "+", "k"]], ")"]]]]], RowBox[List[SuperscriptBox[RowBox[List["Conjugate", "[", "w", "]"]], "m"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Abs", "[", "w", "]"]], "2"], "-", SuperscriptBox["\[Rho]", "2"]]], ")"]], RowBox[List["n", "+", "k", "+", "1"]]]]]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["w", "\[NotEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "\[NotEqual]", "\[Rho]"]]]]]]]]

 MathML Form

 𝒞 z m - 1 "\[LeftBracketingBar]" z - w "\[RightBracketingBar]" 2 ( n + 1 ) z 2 π ( θ UnitStep ( "\[LeftBracketingBar]" w "\[RightBracketingBar]" - ρ ) ( k = 0 n ( k + n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "+", "n"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( m + n n - k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["m", "+", "n"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["n", "-", "k"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ρ 2 ( k + m ) w _ m ( "\[LeftBracketingBar]" w "\[RightBracketingBar]" 2 - ρ 2 ) k + n + 1 ) + θ UnitStep ( ρ - "\[LeftBracketingBar]" w "\[RightBracketingBar]" ) ( k = 0 n ( k + n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "+", "n"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( m + n n - k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["m", "+", "n"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["n", "-", "k"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] w m "\[LeftBracketingBar]" w "\[RightBracketingBar]" 2 k ( ρ 2 - "\[LeftBracketingBar]" w "\[RightBracketingBar]" 2 ) k + n + 1 ) ) /; n m w 0 "\[LeftBracketingBar]" w "\[RightBracketingBar]" ρ FormBox RowBox RowBox RowBox SubscriptBox ErrorBox 𝒞 RowBox FractionBox SuperscriptBox z RowBox m - 1 SuperscriptBox RowBox RowBox z - w RowBox 2 RowBox ( RowBox n + 1 ) RowBox z RowBox 2 π RowBox ( RowBox RowBox RowBox InterpretationBox θ UnitStep Rule Editable Rule Selectable ( RowBox RowBox w - ρ ) RowBox ( RowBox UnderoverscriptBox RowBox k = 0 n RowBox TagBox RowBox ( GridBox TagBox RowBox k + n Rule Editable TagBox k Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable TagBox RowBox ( GridBox TagBox RowBox m + n Rule Editable TagBox RowBox n - k Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable FractionBox SuperscriptBox ρ RowBox 2 RowBox ( RowBox k + m ) RowBox SuperscriptBox OverscriptBox w _ m SuperscriptBox RowBox ( RowBox SuperscriptBox RowBox w 2 - SuperscriptBox ρ 2 ) RowBox k + n + 1 ) + RowBox RowBox InterpretationBox θ UnitStep Rule Editable Rule Selectable ( RowBox ρ - RowBox w ) RowBox ( RowBox UnderoverscriptBox RowBox k = 0 n RowBox TagBox RowBox ( GridBox TagBox RowBox k + n Rule Editable TagBox k Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable TagBox RowBox ( GridBox TagBox RowBox m + n Rule Editable TagBox RowBox n - k Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable FractionBox RowBox SuperscriptBox w m SuperscriptBox RowBox w RowBox 2 k SuperscriptBox RowBox ( RowBox SuperscriptBox ρ 2 - SuperscriptBox RowBox w 2 ) RowBox k + n + 1 ) ) /; RowBox RowBox n RowBox m RowBox w 0 RowBox RowBox w ρ TraditionalForm [/itex]

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

 2002-12-18