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 BesselK

 http://functions.wolfram.com/03.04.06.0032.01

 Input Form

 BesselK[\[Nu], z] == Subscript[F, Infinity][z, \[Nu]] /; Subscript[F, m][z, \[Nu]] == (1/2) ((Gamma[\[Nu]] Sum[(z/2)^(2 k)/(Pochhammer[1 - \[Nu], k] k!), {k, 0, m}])/(z/2)^\[Nu] + Gamma[-\[Nu]] (z/2)^\[Nu] Sum[(z/2)^(2 k)/(Pochhammer[1 + \[Nu], k] k!), {k, 0, m}]) == BesselK[\[Nu], z] + (Pi/(Sin[\[Nu] Pi] (m + 1)!)) (((2^(-\[Nu] - 3 - 2 m) z^(2 m + \[Nu] + 2))/Gamma[2 + m + \[Nu]]) HypergeometricPFQ[{1}, {2 + m, 2 + m + \[Nu]}, z^2/4] - ((2^(\[Nu] - 3 - 2 m) z^(2 m - \[Nu] + 2))/Gamma[2 + m - \[Nu]]) HypergeometricPFQ[{1}, {2 + m, 2 + m - \[Nu]}, z^2/4]) && Element[m, Integers] && m >= 0 /; !Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "\[Nu]", "]"]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["-", "\[Nu]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "\[Nu]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]]]], ")"]]]], "\[Equal]", RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "+", RowBox[List[FractionBox[RowBox[List["\[Pi]", " "]], RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "3", "-", RowBox[List["2", "m"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", "m"]], "+", "\[Nu]", "+", "2"]]], " "]], RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "m", "+", "\[Nu]"]], "]"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "+", "\[Nu]"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], "-", " ", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "-", "3", "-", RowBox[List["2", "m"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", "m"]], "-", "\[Nu]", "+", "2"]]], " "]], RowBox[List["Gamma", "[", RowBox[List["2", "+", "m", "-", "\[Nu]"]], "]"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "-", "\[Nu]"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]]]], ")"]]]]]]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]], ")"]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]

 MathML Form

 K ν ( z ) F ( z , ν ) /; ( ( F m ( z , ν ) 1 2 ( Γ ( ν ) ( z 2 ) - ν k = 0 m ( z 2 ) 2 k ( 1 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] k ! + Γ ( - ν ) ( z 2 ) ν k = 0 m ( z 2 ) 2 k ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] k ! ) K ν ( z ) + π sin ( ν π ) ( m + 1 ) ! ( 2 - 2 m - ν - 3 z 2 m + ν + 2 Γ ( m + ν + 2 ) 1 F 2 ( 1 ; m + 2 , m + ν + 2 ; z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["m", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "\[Nu]", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "4"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - 2 - 2 m + ν - 3 z 2 m - ν + 2 Γ ( m - ν + 2 ) 1 F 2 ( 1 ; m + 2 , m - ν + 2 ; z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["m", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "-", "\[Nu]", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "4"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) ) m ) /; ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Condition BesselK ν z Subscript F z ν Subscript F m z ν 1 2 Gamma ν z 2 -1 -1 ν k 0 m z 2 -1 2 k Pochhammer 1 -1 ν k k -1 Gamma -1 ν z 2 -1 ν k 0 m z 2 -1 2 k Pochhammer ν 1 k k -1 BesselK ν z ν m 1 -1 2 -2 m -1 ν -3 z 2 m ν 2 Gamma m ν 2 -1 HypergeometricPFQ 1 m 2 m ν 2 z 2 4 -1 -1 2 -2 m ν -3 z 2 m -1 ν 2 Gamma m -1 ν 2 -1 HypergeometricPFQ 1 m 2 m -1 ν 2 z 2 4 -1 m ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "\[Nu]", "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]]]], ")"]]]], "\[Equal]", RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "+", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "3", "-", RowBox[List["2", " ", "m"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "m"]], "+", "\[Nu]", "+", "2"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "+", "\[Nu]"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "+", "m", "+", "\[Nu]"]], "]"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "-", "3", "-", RowBox[List["2", " ", "m"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "m"]], "-", "\[Nu]", "+", "2"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "-", "\[Nu]"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "+", "m", "-", "\[Nu]"]], "]"]]]]], ")"]]]], RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]]]]]]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]], "/;", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]

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

 2007-05-02