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; }

 BesselK

 http://functions.wolfram.com/03.04.26.0082.01

 Input Form

 BesselK[\[Nu], z] Hypergeometric0F1[-1 - \[Nu], z^2/4] == 2^(-2 - \[Nu]) Gamma[-1 - \[Nu]] (-2 z^\[Nu] (1 + \[Nu]) + (1/Sqrt[2]) (Pi^(3/2) Csc[Pi (1/4 + \[Nu])] MeijerG[{{(2 + \[Nu])/2, (3 + \[Nu])/2}, {(1/4) (-1 - 2 \[Nu]), (1/4) (5 + 2 \[Nu])}}, {{-(\[Nu]/2), (4 + \[Nu])/2}, {(1/4) (-1 - 2 \[Nu]), \[Nu]/2, 2 + (3 \[Nu])/2, (1/4) (5 + 2 \[Nu])}}, z^2])) /; Inequality[-(Pi/2), Less, Arg[z], LessEqual, Pi/2]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]], "+", RowBox[List[FractionBox["1", SqrtBox["2"]], RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "4"], "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["4", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", FractionBox["\[Nu]", "2"], ",", RowBox[List["2", "+", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "}"]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]

 MathML Form

 K ν ( z ) 0 F 1 ( ; - ν - 1 ; z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "0"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]], Hypergeometric0F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "4"], Hypergeometric0F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric0F1] 2 - ν - 2 Γ ( - ν - 1 ) ( π 3 / 2 csc ( π ( ν + 1 4 ) ) 2 G 4 , 6 2 , 2 ( z 2 ν + 2 2 , ν + 3 2 , 1 4 ( - 2 ν - 1 ) , 1 4 ( 2 ν + 5 ) - ν 2 , ν + 4 2 , 1 4 ( - 2 ν - 1 ) , ν 2 , 3 ν 2 + 2 , 1 4 ( 2 ν + 5 ) ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "6"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[SuperscriptBox["z", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "3"]], "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], "-", "1"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "5"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "4"]], "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], "-", "1"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["\[Nu]", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"], "+", "2"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "5"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] - 2 z ν ( ν + 1 ) ) /; - π 2 < arg ( z ) π 2 Condition BesselK ν z Hypergeometric0F1 -1 ν -1 z 2 4 -1 2 -1 ν -2 Gamma -1 ν -1 3 2 ν 1 4 2 1 2 -1 MeijerG ν 2 2 -1 ν 3 2 -1 1 4 -2 ν -1 1 4 2 ν 5 -1 ν 2 -1 ν 4 2 -1 1 4 -2 ν -1 ν 2 -1 3 ν 2 -1 2 1 4 2 ν 5 z 2 -1 2 z ν ν 1 Inequality -1 2 -1 z 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]_"]], ",", FractionBox[SuperscriptBox["z_", "2"], "4"]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "4"], "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["4", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", FractionBox["\[Nu]", "2"], ",", RowBox[List["2", "+", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "}"]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], SqrtBox["2"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]]]

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

 2007-05-02