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 KelvinKer

 http://functions.wolfram.com/03.20.06.0046.01

 Input Form

 KelvinKer[\[Nu], z] \[Proportional] (-((Sqrt[Pi] Csc[Pi \[Nu]])/(4 Sqrt[2]))) (((((-1)^(3/4) z)^(-(1/2) + \[Nu]) (E^((-1)^(1/4) z)/(-1)^(3/4) + (-((I Sqrt[I z^2] Cos[Pi \[Nu]])/z) + Sin[Pi \[Nu]]/(-1)^(3/4))/ E^((-1)^(1/4) z)) (Sum[((Pochhammer[1/2 - \[Nu], 2 k] Pochhammer[1/2 + \[Nu], 2 k])/(2^(2 k) (2 k)!)) (-(I/z^2))^k, {k, 0, Floor[n/2]}] + O[1/z^(2 Floor[n/2] + 2)]))/ E^((5 I Pi \[Nu])/4) + E^((I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^ (-(1/2) + \[Nu]) (E^((-1)^(3/4) z)/(-1)^(3/4) - ((Sqrt[(-I) z^2] Cos[Pi \[Nu]])/z - Sin[Pi \[Nu]]/(-1)^(3/4))/ E^((-1)^(3/4) z)) (Sum[((Pochhammer[1/2 - \[Nu], 2 k] Pochhammer[1/2 + \[Nu], 2 k])/(2^(2 k) (2 k)!)) (I/z^2)^k, {k, 0, Floor[n/2]}] + O[1/z^(2 Floor[n/2] + 2)]) + (1/(2 z)) ((((-1)^(3/4) z)^(-(1/2) + \[Nu]) (-E^((-1)^(1/4) z) + (((-1)^(1/4) Sqrt[I z^2] Cos[Pi \[Nu]])/z + Sin[Pi \[Nu]])/ E^((-1)^(1/4) z)) (Sum[((Pochhammer[1/2 - \[Nu], 1 + 2 k] Pochhammer[1/2 + \[Nu], 1 + 2 k])/(2^(2 k) (1 + 2 k)!)) (-(I/z^2))^k, {k, 0, Floor[(n - 1)/2]}] + O[1/z^(2 Floor[(n - 1)/2] + 2)]))/E^((5 I Pi \[Nu])/4) + E^((I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^(-(1/2) + \[Nu]) (I E^((-1)^(3/4) z) - (((-1)^(1/4) Sqrt[(-I) z^2] Cos[Pi \[Nu]])/z + I Sin[Pi \[Nu]])/E^((-1)^(3/4) z)) (Sum[((Pochhammer[1/2 - \[Nu], 1 + 2 k] Pochhammer[1/2 + \[Nu], 1 + 2 k])/(2^(2 k) (1 + 2 k)!)) (I/z^2)^k, {k, 0, Floor[(n - 1)/2]}] + O[1/z^(2 Floor[(n - 1)/2] + 2)])))/ z^\[Nu] + z^\[Nu] (E^((-(-1)^(1/4)) z + (I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) - \[Nu]) (I (Sqrt[I z^2]/z) Cos[Pi \[Nu]] + (-1)^(1/4) (E^(2 (-1)^(1/4) z) - Sin[Pi \[Nu]])) (Sum[((Pochhammer[1/2 - \[Nu], 2 k] Pochhammer[1/2 + \[Nu], 2 k])/ (2^(2 k) (2 k)!)) (-(I/z^2))^k, {k, 0, Floor[n/2]}] + O[1/z^(2 Floor[n/2] + 2)]) - E^((-(-1)^(3/4)) z + (3 I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^(-(1/2) - \[Nu]) ((-(Sqrt[(-I) z^2]/z)) Cos[Pi \[Nu]] + (-1)^(1/4) (-E^(2 (-1)^(3/4) z) + Sin[Pi \[Nu]])) (Sum[((Pochhammer[1/2 - \[Nu], 2 k] Pochhammer[1/2 + \[Nu], 2 k])/ (2^(2 k) (2 k)!)) (I/z^2)^k, {k, 0, Floor[n/2]}] + O[1/z^(2 Floor[n/2] + 2)]) - (1/(2 z)) (E^((-(-1)^(1/4)) z + (I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) - \[Nu]) ((-1)^(1/4) (Sqrt[I z^2]/z) Cos[Pi \[Nu]] - E^(2 (-1)^(1/4) z) - Sin[Pi \[Nu]]) (Sum[((Pochhammer[1/2 - \[Nu], 1 + 2 k] Pochhammer[1/2 + \[Nu], 1 + 2 k])/(2^(2 k) (1 + 2 k)!)) (-(I/z^2))^k, {k, 0, Floor[(n - 1)/2]}] + O[1/z^(2 Floor[(n - 1)/2] + 2)]) + I E^((-(-1)^(3/4)) z + (3 I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^ (-(1/2) - \[Nu]) ((-1)^(3/4) (Sqrt[(-I) z^2]/z) Cos[Pi \[Nu]] + E^(2 (-1)^(3/4) z) + Sin[Pi \[Nu]]) (Sum[((Pochhammer[1/2 - \[Nu], 1 + 2 k] Pochhammer[1/2 + \[Nu], 1 + 2 k])/(2^(2 k) (1 + 2 k)!)) (I/z^2)^k, {k, 0, Floor[(n - 1)/2]}] + O[1/z^(2 Floor[(n - 1)/2] + 2)])))) /; (Abs[z] -> Infinity) && !Element[\[Nu], Integers] && Element[n, Integers] && n >= 0

 Standard Form

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RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]]], "+", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "k"]]], " ", 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"]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]]], SuperscriptBox[RowBox[List["(", FractionBox["\[ImaginaryI]", SuperscriptBox["z", "2"]], ")"]], "k"]]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List[RowBox[List["2", RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]]], "+", "2"]]]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 ker ν ( z ) - π csc ( π ν ) 4 2 ( z - ν ( π ν 4 ( - - 1 4 z ) ν - 1 2 ( ( - 1 ) - 3 / 4 ( - 1 ) 3 / 4 z - - ( - 1 ) 3 / 4 z ( - z 2 cos ( π ν ) z - ( - 1 ) - 3 / 4 sin ( π ν ) ) ) ( k = 0 n 2 2 - 2 k ( 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 ) ! ( z 2 ) k + O ( 1 z 2 n 2 + 2 ) ) + - 5 π ν 4 ( ( - 1 ) 3 / 4 z ) ν - 1 2 ( - - 1 4 z ( ( - 1 ) - 3 / 4 sin ( π ν ) - z 2 cos ( π ν ) z ) + ( - 1 ) - 3 / 4 - 1 4 z ) ( k = 0 n 2 2 - 2 k ( 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 ) ! ( - z 2 ) k + O ( 1 z 2 n 2 + 2 ) ) + 1 2 z ( π ν 4 ( - - 1 4 z ) ν - 1 2 ( ( - 1 ) 3 / 4 z - - ( - 1 ) 3 / 4 z ( - 1 4 - z 2 cos ( π ν ) z + sin ( π ν ) ) ) ( k = 0 n - 1 2 2 - 2 k ( 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 ) ! ( z 2 ) k + O ( 1 z 2 n - 1 2 + 2 ) ) + - 5 π ν 4 ( ( - 1 ) 3 / 4 z ) ν - 1 2 ( - - 1 4 z ( - 1 4 z 2 cos ( π ν ) z + sin ( π ν ) ) - - 1 4 z ) ( k = 0 n - 1 2 2 - 2 k ( 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 ) ! ( - z 2 ) k + O ( 1 z 2 n - 1 2 + 2 ) ) ) ) + z ν ( - 3 π ν 4 - ( - 1 ) 3 / 4 z ( - - 1 4 z ) - ν - 1 2 ( - 1 4 ( sin ( π ν ) - 2 ( - 1 ) 3 / 4 z ) - - z 2 cos ( π ν ) z ) ( k = 0 n 2 2 - 2 k ( 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 ) ! ( z 2 ) k + O ( 1 z 2 n 2 + 2 ) ) + π ν 4 - - 1 4 z ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ( z 2 cos ( π ν ) z + - 1 4 ( 2 - 1 4 z - sin ( π ν ) ) ) ( k = 0 n 2 2 - 2 k ( 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 ) ! ( - z 2 ) k + O ( 1 z 2 n 2 + 2 ) ) - 1 2 z ( 3 π ν 4 - ( - 1 ) 3 / 4 z ( - - 1 4 z ) - ν - 1 2 ( ( - 1 ) 3 / 4 - z 2 cos ( π ν ) z + 2 ( - 1 ) 3 / 4 z + sin ( π ν ) ) ( k = 0 n - 1 2 2 - 2 k ( 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 ) ! ( z 2 ) k + O ( 1 z 2 n - 1 2 + 2 ) ) + π ν 4 - - 1 4 z ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ( - 1 4 z 2 cos ( π ν ) z - 2 - 1 4 z - sin ( π ν ) ) ( k = 0 n - 1 2 2 - 2 k ( 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 ) ! ( - z 2 ) k + O ( 1 z 2 n - 1 2 + 2 ) ) ) ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] n Condition Proportional KelvinKer ν z -1 1 2 ν 4 2 1 2 -1 z -1 ν ν 4 -1 -1 -1 1 4 z ν -1 1 2 -1 -3 4 -1 3 4 z -1 -1 -1 3 4 z -1 z 2 1 2 ν z -1 -1 -1 -3 4 ν k 0 n 2 -1 2 -2 k Pochhammer 1 2 -1 ν 2 k Pochhammer ν 1 2 2 k 2 k -1 z 2 -1 k O 1 z 2 n 2 -1 2 -1 -1 5 ν 4 -1 -1 3 4 z ν -1 1 2 -1 -1 1 4 z -1 -3 4 ν -1 z 2 1 2 ν z -1 -1 -3 4 -1 1 4 z k 0 n 2 -1 2 -2 k Pochhammer 1 2 -1 ν 2 k Pochhammer ν 1 2 2 k 2 k -1 -1 z 2 -1 k O 1 z 2 n 2 -1 2 -1 1 2 z -1 ν 4 -1 -1 -1 1 4 z ν -1 1 2 -1 3 4 z -1 -1 -1 3 4 z -1 1 4 -1 z 2 1 2 ν z -1 ν k 0 n -1 2 -1 2 -2 k Pochhammer 1 2 -1 ν 2 k 1 Pochhammer ν 1 2 2 k 1 2 k 1 -1 z 2 -1 k O 1 z 2 n -1 2 -1 2 -1 -1 5 ν 4 -1 -1 3 4 z ν -1 1 2 -1 -1 1 4 z -1 1 4 z 2 1 2 ν z -1 ν -1 -1 1 4 z k 0 n -1 2 -1 2 -2 k Pochhammer 1 2 -1 ν 2 k 1 Pochhammer ν 1 2 2 k 1 2 k 1 -1 -1 z 2 -1 k O 1 z 2 n -1 2 -1 2 -1 z ν -1 3 ν 4 -1 -1 -1 3 4 z -1 -1 1 4 z -1 ν -1 1 2 -1 1 4 ν -1 2 -1 3 4 z -1 -1 z 2 1 2 ν z -1 k 0 n 2 -1 2 -2 k Pochhammer 1 2 -1 ν 2 k Pochhammer ν 1 2 2 k 2 k -1 z 2 -1 k O 1 z 2 n 2 -1 2 -1 ν 4 -1 -1 -1 1 4 z -1 3 4 z -1 ν -1 1 2 z 2 1 2 ν z -1 -1 1 4 2 -1 1 4 z -1 ν k 0 n 2 -1 2 -2 k Pochhammer 1 2 -1 ν 2 k Pochhammer ν 1 2 2 k 2 k -1 -1 z 2 -1 k O 1 z 2 n 2 -1 2 -1 -1 1 2 z -1 3 ν 4 -1 -1 -1 3 4 z -1 -1 1 4 z -1 ν -1 1 2 -1 3 4 -1 z 2 1 2 ν z -1 2 -1 3 4 z ν k 0 n -1 2 -1 2 -2 k Pochhammer 1 2 -1 ν 2 k 1 Pochhammer ν 1 2 2 k 1 2 k 1 -1 z 2 -1 k O 1 z 2 n -1 2 -1 2 -1 ν 4 -1 -1 -1 1 4 z -1 3 4 z -1 ν -1 1 2 -1 1 4 z 2 1 2 ν z -1 -1 2 -1 1 4 z -1 ν k 0 n -1 2 -1 2 -2 k Pochhammer 1 2 -1 ν 2 k 1 Pochhammer ν 1 2 2 k 1 2 k 1 -1 -1 z 2 -1 k O 1 z 2 n -1 2 -1 2 -1 Rule z ν n [/itex]

 Rule Form

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

 2007-05-02