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 KelvinKer

 http://functions.wolfram.com/03.20.06.0055.01

 Input Form

 KelvinKer[\[Nu], z] == (1/4) Sum[Residue[((Gamma[s - \[Nu]/4] Gamma[s + (\[Nu] + 2)/4] Gamma[s + (2 - \[Nu])/4])/((z/4)^(4 s) (Gamma[(1 + \[Nu])/2 + s] Gamma[(1 - \[Nu])/2 - s]))) Gamma[s + \[Nu]/4], {s, -j - \[Nu]/4}], {j, 0, Infinity}] + (1/4) Sum[Residue[((Gamma[s + \[Nu]/4] Gamma[s + (\[Nu] + 2)/4] Gamma[s + (2 - \[Nu])/4])/((z/4)^(4 s) (Gamma[(1 + \[Nu])/2 + s] Gamma[(1 - \[Nu])/2 - s]))) Gamma[s - \[Nu]/4], {s, -j + \[Nu]/4}], {j, 0, Infinity}] + (1/4) Sum[Residue[((Gamma[s + \[Nu]/4] Gamma[s - \[Nu]/4] Gamma[s + (2 - \[Nu])/4])/((z/4)^(4 s) (Gamma[(1 + \[Nu])/2 + s] Gamma[(1 - \[Nu])/2 - s]))) Gamma[s + (\[Nu] + 2)/4], {s, -j - (\[Nu] + 2)/4}], {j, 0, Infinity}] + (1/4) Sum[Residue[((Gamma[s + \[Nu]/4] Gamma[s - \[Nu]/4] Gamma[s + (2 + \[Nu])/4])/((z/4)^(4 s) (Gamma[(1 + \[Nu])/2 + s] Gamma[(1 - \[Nu])/2 - s]))) Gamma[s + (2 - \[Nu])/4], {s, -j - (2 - \[Nu])/4}], {j, 0, Infinity}] /; !Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "4"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "4"]], "s"]]], RowBox[List["Gamma", "[", RowBox[List["s", "-", FractionBox["\[Nu]", "4"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "+", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "-", "s"]], "]"]]]]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "4"]]], "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "j"]], "-", FractionBox["\[Nu]", "4"]]]]], "}"]]]], "]"]]]]]], "+", RowBox[List[FractionBox["1", "4"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "4"]], "s"]]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "4"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "+", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "-", "s"]], "]"]]]]], RowBox[List["Gamma", "[", RowBox[List["s", "-", FractionBox["\[Nu]", "4"]]], "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "j"]], "+", FractionBox["\[Nu]", "4"]]]]], "}"]]]], "]"]]]]]], "+", RowBox[List[FractionBox["1", "4"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "4"]], "s"]]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "4"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["s", "-", FractionBox["\[Nu]", "4"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "+", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "-", "s"]], "]"]]]]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"]]], "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "j"]], "-", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"]]]]], "}"]]]], "]"]]]]]], "+", RowBox[List[FractionBox["1", "4"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "4"]], "s"]]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "4"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["s", "-", FractionBox["\[Nu]", "4"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "+", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "-", "s"]], "]"]]]]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"]]], "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "j"]], "-", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"]]]]], "}"]]]], "]"]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]

 MathML Form

 ker ν ( z ) 1 4 j = 0 res s ( ( z 4 ) - 4 s Γ ( s + ν 4 ) Γ ( s - ν 4 ) Γ ( s + 2 - ν 4 ) Γ ( s + ν + 1 2 ) Γ ( 1 - ν 2 - s ) Γ ( ν + 2 4 + s ) ) ( - j - ν + 2 4 ) + 1 4 j = 0 res s ( ( z 4 ) - 4 s Γ ( s + ν 4 ) Γ ( s - ν 4 ) Γ ( s + ν + 2 4 ) Γ ( s + ν + 1 2 ) Γ ( 1 - ν 2 - s ) Γ ( s + 2 - ν 4 ) ) ( - j - 2 - ν 4 ) + 1 4 j = 0 res s ( ( z 4 ) - 4 s Γ ( s - ν 4 ) Γ ( ν + 2 4 + s ) Γ ( s + 2 - ν 4 ) Γ ( s + ν + 1 2 ) Γ ( 1 - ν 2 - s ) Γ ( s + ν 4 ) ) ( - j - ν 4 ) + 1 4 j = 0 res s ( ( z 4 ) - 4 s Γ ( s + ν 4 ) Γ ( ν + 2 4 + s ) Γ ( s + 2 - ν 4 ) Γ ( s + ν + 1 2 ) Γ ( 1 - ν 2 - s ) Γ ( s - ν 4 ) ) ( - j + ν 4 ) /; ν TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition KelvinKer ν z 1 4 j 0 Subscript res s z 4 -1 -4 s Gamma s ν 4 -1 Gamma s -1 ν 4 -1 Gamma s 2 -1 ν 4 -1 Gamma s ν 1 2 -1 Gamma 1 -1 ν 2 -1 -1 s -1 Gamma ν 2 4 -1 s -1 j -1 ν 2 4 -1 1 4 j 0 Subscript res s z 4 -1 -4 s Gamma s ν 4 -1 Gamma s -1 ν 4 -1 Gamma s ν 2 4 -1 Gamma s ν 1 2 -1 Gamma 1 -1 ν 2 -1 -1 s -1 Gamma s 2 -1 ν 4 -1 -1 j -1 2 -1 ν 4 -1 1 4 j 0 Subscript res s z 4 -1 -4 s Gamma s -1 ν 4 -1 Gamma ν 2 4 -1 s Gamma s 2 -1 ν 4 -1 Gamma s ν 1 2 -1 Gamma 1 -1 ν 2 -1 -1 s -1 Gamma s ν 4 -1 -1 j -1 ν 4 -1 1 4 j 0 Subscript res s z 4 -1 -4 s Gamma s ν 4 -1 Gamma ν 2 4 -1 s Gamma s 2 -1 ν 4 -1 Gamma s ν 1 2 -1 Gamma 1 -1 ν 2 -1 -1 s -1 Gamma s -1 ν 4 -1 -1 j ν 4 -1 ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "4"]], " ", "s"]]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "-", FractionBox["\[Nu]", "4"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"]]], "]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "4"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "+", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "-", "s"]], "]"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "j"]], "-", FractionBox["\[Nu]", "4"]]]]], "}"]]]], "]"]]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "4"]], " ", "s"]]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "4"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"]]], "]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "-", FractionBox["\[Nu]", "4"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "+", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "-", "s"]], "]"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "j"]], "+", FractionBox["\[Nu]", "4"]]]]], "}"]]]], "]"]]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "4"]], " ", "s"]]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "4"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "-", FractionBox["\[Nu]", "4"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"]]], "]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "+", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "-", "s"]], "]"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "j"]], "-", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"]]]]], "}"]]]], "]"]]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "4"]], " ", "s"]]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "4"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "-", FractionBox["\[Nu]", "4"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"]]], "]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "+", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "-", "s"]], "]"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "j"]], "-", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"]]]]], "}"]]]], "]"]]]]]]]], "/;", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]

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

 2007-05-02