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/03.20.06.0051.01

 Input Form

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

 Standard Form

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"z"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[FractionBox["1", "4"], "+", "\[Nu]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[SqrtBox["2"], " ", "z"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["z", "2"]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]], "]"]]]], ")"]]]], "+", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", SqrtBox["2"], " ", "z"]]], " ", "z"]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[FractionBox["1", "4"], "+", "\[Nu]"]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]], "]"]]]], ")"]]]]]], ")"]]]]]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 ker ν ( z ) - π ν 2 - ( 1 + ) z 2 8 2 π - - 1 4 z ( ( - 1 ) 3 / 4 z ) 3 / 2 ( 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] ( 2 k ) ! ( 4 z 2 ) k ( π 2 ( ( - 1 ) k + 3 4 2 ( 4 ( - 1 ) ν - 3 ( 1 + ) 2 z ) ( - - 1 4 z ) 3 / 2 + 3 ( - 1 ) k + ν ( 1 - ) z 2 - - 1 4 z + ( - 1 ) 3 / 4 z ( 2 2 z ( - ) z - ( 1 + ) ( - 1 ) ν 2 z ( 2 2 ( - 1 + ) z + - z 2 ) ) ) + 4 ( - 1 ) 3 / 4 z ( 2 z z - ( - 1 ) ν + 3 4 2 z - z 2 ) ( log ( - - 1 4 z ) - log ( z ) ) + 4 ( - 1 ) k - - 1 4 z ( ( 1 + ) 2 z z + ( - 1 ) ν + 1 4 z 2 ) ( log ( ( - 1 ) 3 / 4 z ) - log ( z ) ) ) - ( - 1 ) 3 / 4 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] ( 2 k + 1 ) ! ( 4 z 2 ) k ( ( 1 + ) π 2 ( ( - 1 ) k + 3 4 ( 4 ( - 1 ) ν + 3 ( 1 + ) 2 z ) ( - 1 + ) ( - - 1 4 z ) 3 / 2 + 3 ( - 1 ) k + ν + 3 4 ( 1 - ) z 2 - - 1 4 z + ( - 1 ) 3 / 4 z ( 2 z ( - 1 + ) z + ( - 1 ) ν 2 z ( 4 ( 1 + ) z - 2 - z 2 ) ) ) + 4 ( - 1 ) 3 / 4 z ( ( - 1 ) ν + 1 4 2 z - z 2 - 2 z z ) ( log ( - - 1 4 z ) - log ( z ) ) + 4 ( - 1 ) k - - 1 4 z ( ( 1 + ) 2 z z - ( - 1 ) ν + 1 4 z 2 ) ( log ( ( - 1 ) 3 / 4 z ) - log ( z ) ) ) + ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) ν TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] n Condition Proportional KelvinKer ν z -1 ν 2 -1 -1 1 z 2 1 2 -1 8 2 1 2 -1 -1 1 4 z 1 2 -1 3 4 z 3 2 -1 k 0 n 2 -1 Pochhammer 1 2 -1 ν 2 k Pochhammer ν 1 2 2 k 2 k -1 4 z 2 -1 k 2 1 2 -1 -1 k 3 4 2 1 2 4 -1 ν -1 3 1 2 1 2 z -1 -1 1 4 z 3 2 3 -1 k ν 1 -1 z 2 1 2 -1 -1 1 4 z 1 2 -1 3 4 z 1 2 2 1 2 2 1 2 z -1 z -1 1 -1 ν 2 1 2 z 2 2 1 2 -1 z -1 z 2 1 2 4 -1 3 4 z 1 2 2 1 2 z z -1 -1 ν 3 4 2 1 2 z -1 z 2 1 2 -1 -1 1 4 z -1 z 4 -1 k -1 -1 1 4 z 1 2 1 2 1 2 z z -1 ν 1 4 z 2 1 2 -1 3 4 z -1 z -1 -1 3 4 2 z -1 k 0 n -1 2 -1 Pochhammer 1 2 -1 ν 2 k 1 Pochhammer ν 1 2 2 k 1 2 k 1 -1 4 z 2 -1 k 1 2 -1 -1 k 3 4 4 -1 ν 3 1 2 1 2 z -1 -1 -1 1 4 z 3 2 3 -1 k ν 3 4 1 -1 z 2 1 2 -1 -1 1 4 z 1 2 -1 3 4 z 1 2 2 1 2 z -1 z -1 ν 2 1 2 z 4 1 z -1 2 1 2 -1 z 2 1 2 4 -1 3 4 z 1 2 -1 ν 1 4 2 1 2 z -1 z 2 1 2 -1 2 1 2 z z -1 -1 1 4 z -1 z 4 -1 k -1 -1 1 4 z 1 2 1 2 1 2 z z -1 -1 ν 1 4 z 2 1 2 -1 3 4 z -1 z Rule z ν n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", "z"]], SqrtBox["2"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Nu]"]], "2"]]]], " ", 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"]]]], "]"]]]], ")"]], " ", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02