html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 KelvinKer

 http://functions.wolfram.com/03.20.06.0041.01

 Input Form

 KelvinKer[\[Nu], z] \[Proportional] (Sqrt[Pi]/(Sqrt[2] Sqrt[-z])) (Sum[((Pochhammer[1/2 - \[Nu], 2 k] Pochhammer[1/2 + \[Nu], 2 k] (1/(4 z^2))^k)/(2 k)!) ((I Sin[(1/8) (Pi (-3 + 4 \[Nu]) + 4 Sqrt[2] z) + (k Pi)/2])/ E^(z/Sqrt[2]) + 2 Cos[Pi \[Nu]] E^(z/Sqrt[2]) Cos[(1/8) (Pi + 4 Pi \[Nu] - 4 Sqrt[2] z) + (k Pi)/2]), {k, 0, Floor[n/2]}] + (1/(2 z)) Sum[((Pochhammer[1/2 - \[Nu], 1 + 2 k] Pochhammer[1/2 + \[Nu], 1 + 2 k] (1/(4 z^2))^k)/(1 + 2 k)!) (((-I) Sin[(1/8) (Pi (-1 + 4 \[Nu]) + 4 Sqrt[2] z) + (k Pi)/2])/ E^(z/Sqrt[2]) + 2 Cos[Pi \[Nu]] E^(z/Sqrt[2]) Cos[(1/8) (Pi (3 + 4 \[Nu]) - 4 Sqrt[2] z) + (k Pi)/2]), {k, 0, Floor[(n - 1)/2]}] + \[Ellipsis]) /; (z -> -Infinity) && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[" ", SqrtBox["\[Pi]"], " "]], RowBox[List[SqrtBox["2"], SqrtBox[RowBox[List["-", "z"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["4", SuperscriptBox["z", "2"]]]], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "+", RowBox[List["4", " ", "\[Pi]", " ", "\[Nu]"]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]]]], ")"]]]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", "z"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["4", SuperscriptBox["z", "2"]]]], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]]]], ")"]]]]]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "\[Infinity]"]]]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 ker ν ( z ) π 2 - z ( k = 0 n 2 ( 1 2 - ν ) 2 k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], RowBox[List["2", " ", "k"]]], Pochhammer] ( ν + 1 2 ) 2 k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["2", " ", "k"]]], Pochhammer] ( 1 4 z 2 ) k ( 2 k ) ! ( 2 z 2 cos ( π k 2 + 1 8 ( - 4 2 z + 4 π ν + π ) ) cos ( π ν ) + - z 2 sin ( π k 2 + 1 8 ( 4 2 z + π ( 4 ν - 3 ) ) ) ) + 1 2 z k = 0 n - 1 2 ( 1 2 - ν ) 2 k + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], Pochhammer] ( ν + 1 2 ) 2 k + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], Pochhammer] ( 1 4 z 2 ) k ( 2 k + 1 ) ! ( 2 cos ( π ν ) z 2 cos ( π k 2 + 1 8 ( π ( 4 ν + 3 ) - 4 2 z ) ) - - z 2 sin ( π k 2 + 1 8 ( 4 2 z + π ( 4 ν - 1 ) ) ) ) + ) /; ( z "\[Rule]" - ) n Condition Proportional KelvinKer ν z 1 2 2 1 2 -1 z 1 2 -1 k 0 n 2 -1 Pochhammer 1 2 -1 ν 2 k Pochhammer ν 1 2 2 k 1 4 z 2 -1 k 2 k -1 2 z 2 1 2 -1 k 2 -1 1 8 -4 2 1 2 z 4 ν ν -1 z 2 1 2 -1 k 2 -1 1 8 4 2 1 2 z 4 ν -3 1 2 z -1 k 0 n -1 2 -1 Pochhammer 1 2 -1 ν 2 k 1 Pochhammer ν 1 2 2 k 1 1 4 z 2 -1 k 2 k 1 -1 2 ν z 2 1 2 -1 k 2 -1 1 8 4 ν 3 -1 4 2 1 2 z -1 -1 z 2 1 2 -1 k 2 -1 1 8 4 2 1 2 z 4 ν -1 Rule z -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], ")"]], "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "+", RowBox[List["4", " ", "\[Pi]", " ", "\[Nu]"]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]]]], "+", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], ")"]], "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]]]]], RowBox[List["2", " ", "z"]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["-", "z"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "\[Infinity]"]]]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02