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 KelvinKei

 http://functions.wolfram.com/03.19.06.0043.01

 Input Form

 KelvinKei[\[Nu], z] \[Proportional] (1/(8 Sqrt[2 Pi])) ((E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2) (1/Sqrt[(-1)^(3/4) z]) (-4 (-1)^(3/4) Pi - (3 Pi I Sqrt[I z^2])/z + 4 (-Log[z] + Log[(-1)^(3/4) z]) (Sqrt[I z^2]/z)) + E^((I z)/Sqrt[2] + (I Pi \[Nu])/2) (1/Sqrt[(-(-1)^(1/4)) z]) ((-1)^(1/4) Pi + 4 (-1)^(3/4) (-Log[z] + Log[(-(-1)^(1/4)) z])))/ E^(z/Sqrt[2]) + E^(z/Sqrt[2]) (E^((I z)/Sqrt[2] + (I Pi \[Nu])/2) (1/Sqrt[(-1)^(3/4) z]) ((3 Pi)/(-1)^(3/4) - 4 (-1)^(3/4) (-Log[z] + Log[(-1)^(3/4) z])) + E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2) (1/Sqrt[(-(-1)^(1/4)) z]) (4 (-1)^(3/4) Pi + Pi (Sqrt[(-I) z^2]/z) + (4 I Sqrt[(-I) z^2] (-Log[z] + Log[(-(-1)^(1/4)) z]))/z))) (1 + O[1/z^4]) + (E^(-(((1 + I) z)/Sqrt[2]) + (Pi I (\[Nu] + 1))/2)/ (8 Sqrt[2 Pi] Sqrt[(-(-1)^(1/4)) z] ((-1)^(3/4) z)^(3/2))) ((-(((-1)^(3/4) (1 - 4 \[Nu]^2))/(8 z))) (Sqrt[(-(-1)^(1/4)) z] (Pi (-4 (-1)^\[Nu] z - 3 I E^((1 + I) Sqrt[2] z) z + 3 (-1)^(3/4 + \[Nu]) Sqrt[I z^2]) + 4 (E^((1 + I) Sqrt[2] z) z - (-1)^(1/4 + \[Nu]) Sqrt[I z^2]) (-Log[z] + Log[(-1)^(3/4) z])) - Sqrt[(-1)^(3/4) z] ((1/2) (-2 E^(I Sqrt[2] z) Pi z + (1 + I) (-1)^\[Nu] E^(Sqrt[2] z) Pi ((4 + 4 I) z - I Sqrt[2] Sqrt[(-I) z^2])) + 4 ((-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]))) (1 + O[1/z^4]) - ((I (9 - 40 \[Nu]^2 + 16 \[Nu]^4))/(128 z^2)) (Sqrt[(-(-1)^(1/4)) z] (Pi (4 (-1)^\[Nu] z - 3 I E^((1 + I) Sqrt[2] z) z + ((3 - 3 I) (-1)^\[Nu] Sqrt[I z^2])/Sqrt[2]) + 4 (E^((1 + I) Sqrt[2] z) z + (-1)^(1/4 + \[Nu]) Sqrt[I z^2]) (-Log[z] + Log[(-1)^(3/4) z])) + Sqrt[(-1)^(3/4) z] ((-(Pi/Sqrt[2])) (I Sqrt[2] E^(I Sqrt[2] z) z + (1 + I) (-1)^\[Nu] E^(Sqrt[2] z) ((-2 + 2 I) Sqrt[2] z + Sqrt[(-I) z^2])) + 4 (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]))) (1 + O[1/z^4]) + (((-1)^(1/4) (-225 + 1036 \[Nu]^2 - 560 \[Nu]^4 + 64 \[Nu]^6))/(3072 z^3)) (Sqrt[(-(-1)^(1/4)) z] (Pi (-4 (-1)^\[Nu] z - 3 I E^((1 + I) Sqrt[2] z) z + 3 (-1)^(3/4 + \[Nu]) Sqrt[I z^2]) + 4 (E^((1 + I) Sqrt[2] z) z - (-1)^(1/4 + \[Nu]) Sqrt[I z^2]) (-Log[z] + Log[(-1)^(3/4) z])) + Sqrt[(-1)^(3/4) z] ((1/2) (-2 E^(I Sqrt[2] z) Pi z + (1 + I) (-1)^\[Nu] E^(Sqrt[2] z) Pi ((4 + 4 I) z - I Sqrt[2] Sqrt[(-I) z^2])) + 4 ((-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]))) (1 + O[1/z^4])) /; (Abs[z] -> Infinity) && Element[\[Nu], Integers]

 Standard Form

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 MathML Form

 kei ν ( z ) 1 8 2 π ( z 2 ( 3 π ν 2 - z 2 - - 1 4 z ( 4 - z 2 ( log ( - - 1 4 z ) - log ( z ) ) z + π - z 2 z + 4 ( - 1 ) 3 / 4 π ) + z 2 + π ν 2 ( - 1 ) 3 / 4 z ( 3 ( - 1 ) - 3 / 4 π - 4 ( - 1 ) 3 / 4 ( log ( ( - 1 ) 3 / 4 z ) - log ( z ) ) ) ) + - z 2 ( 3 π ν 2 - z 2 ( - 1 ) 3 / 4 z ( - 4 ( - 1 ) 3 / 4 π - 3 π z 2 z + 4 ( log ( ( - 1 ) 3 / 4 z ) - log ( z ) ) z 2 z ) + z 2 + π ν 2 - - 1 4 z ( 4 ( - 1 ) 3 / 4 ( log ( - - 1 4 z ) - log ( z ) ) + - 1 4 π ) ) ) ( 1 + O ( 1 z 4 ) ) + π ( ν + 1 ) 2 - ( 1 + ) z 2 8 2 π - - 1 4 z ( ( - 1 ) 3 / 4 z ) 3 / 2 ( - ( - 1 ) 3 / 4 ( 1 - 4 ν 2 ) 8 z ( b - ( - 1 ) 3 / 4 z ( 1 2 ( ( 1 + ) ( - 1 ) ν 2 z π ( ( 4 + 4 ) z - 2 - z 2 ) - 2 2 z π z ) + 4 ( ( - 1 ) ν + 1 4 2 z - z 2 - 2 z z ) ( log ( - - 1 4 z ) - log ( z ) ) ) + - - 1 4 z ( π ( - 3 ( 1 + ) 2 z z - 4 ( - 1 ) ν z + 3 ( - 1 ) ν + 3 4 z 2 ) + 4 ( ( 1 + ) 2 z z - ( - 1 ) ν + 1 4 z 2 ) ( log ( ( - 1 ) 3 / 4 z ) - log ( z ) ) ) ) ( 1 + O ( 1 z 4 ) ) TagBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "(", FractionBox["1", SuperscriptBox["z", "4"]], ")"]]]], ")"]], HypergeometricPFQ] - ( 16 ν 4 - 40 ν 2 + 9 ) 128 z 2 ( ( - 1 ) 3 / 4 z ( 4 ( 2 z z - ( - 1 ) ν + 3 4 2 z - z 2 ) ( log ( - - 1 4 z ) - log ( z ) ) - π 2 ( 2 2 z z + ( - 1 ) ν 2 z ( 1 + ) ( 2 ( - 2 + 2 ) z + - z 2 ) ) ) + - - 1 4 z ( π ( ( - 1 ) ν z 2 ( 3 - 3 ) 2 - 3 ( 1 + ) 2 z z + 4 ( - 1 ) ν z ) + 4 ( ( 1 + ) 2 z z + ( - 1 ) ν + 1 4 z 2 ) ( log ( ( - 1 ) 3 / 4 z ) - log ( z ) ) ) ) ( 1 + O ( 1 z 4 ) ) TagBox[TagBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "(", FractionBox["1", SuperscriptBox["z", "4"]], ")"]]]], ")"]], HypergeometricPFQ], HypergeometricPFQ] + - 1 4 ( 64 ν 6 - 560 ν 4 + 1036 ν 2 - 225 ) 3072 z 3 ( ( - 1 ) 3 / 4 z ( 1 2 ( ( 1 + ) ( - 1 ) ν 2 z π ( ( 4 + 4 ) z - 2 - z 2 ) - 2 2 z π z ) + 4 ( ( - 1 ) ν + 1 4 2 z - z 2 - 2 z z ) ( log ( - - 1 4 z ) - log ( z ) ) ) + - - 1 4 z ( π ( - 3 ( 1 + ) 2 z z - 4 ( - 1 ) ν z + 3 ( - 1 ) ν + 3 4 z 2 ) + 4 ( ( 1 + ) 2 z z - ( - 1 ) ν + 1 4 z 2 ) ( log ( ( - 1 ) 3 / 4 z ) - log ( z ) ) ) ) ( 1 + O ( 1 z 4 ) ) TagBox[TagBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "(", FractionBox["1", SuperscriptBox["z", "4"]], ")"]]]], ")"]], HypergeometricPFQ], HypergeometricPFQ] ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) ν TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition Proportional KelvinKei ν z 1 8 2 1 2 -1 z 2 1 2 -1 3 ν 2 -1 -1 z 2 1 2 -1 -1 -1 1 4 z 1 2 -1 4 -1 z 2 1 2 -1 -1 1 4 z -1 z z -1 -1 z 2 1 2 z -1 4 -1 3 4 z 2 1 2 -1 ν 2 -1 -1 3 4 z 1 2 -1 3 -1 -3 4 -1 4 -1 3 4 -1 3 4 z -1 z -1 z 2 1 2 -1 3 ν 2 -1 -1 z 2 1 2 -1 -1 3 4 z 1 2 -1 -4 -1 3 4 -1 3 z 2 1 2 z -1 4 -1 3 4 z -1 z z 2 1 2 z -1 z 2 1 2 -1 ν 2 -1 -1 -1 1 4 z 1 2 -1 4 -1 3 4 -1 -1 1 4 z -1 z -1 1 4 1 O 1 z 4 -1 ν 1 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 -1 -1 3 4 1 -1 4 ν 2 8 z -1 b -1 -1 3 4 z 1 2 1 2 1 -1 ν 2 1 2 z 4 4 z -1 2 1 2 -1 z 2 1 2 -1 2 2 1 2 z z 4 -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 -1 -1 1 4 z 1 2 -3 1 2 1 2 z z -1 4 -1 ν z 3 -1 ν 3 4 z 2 1 2 4 1 2 1 2 z z -1 -1 ν 1 4 z 2 1 2 -1 3 4 z -1 z HypergeometricPFQ 1 O 1 z 4 -1 -1 16 ν 4 -1 40 ν 2 9 128 z 2 -1 -1 3 4 z 1 2 4 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 -1 2 1 2 -1 2 1 2 2 1 2 z z -1 ν 2 1 2 z 1 2 1 2 -2 2 z -1 z 2 1 2 -1 -1 1 4 z 1 2 -1 ν z 2 1 2 3 -3 2 1 2 -1 -1 3 1 2 1 2 z z 4 -1 ν z 4 1 2 1 2 z z -1 ν 1 4 z 2 1 2 -1 3 4 z -1 z HypergeometricPFQ 1 O 1 z 4 -1 -1 1 4 64 ν 6 -1 560 ν 4 1036 ν 2 -225 3072 z 3 -1 -1 3 4 z 1 2 1 2 1 -1 ν 2 1 2 z 4 4 z -1 2 1 2 -1 z 2 1 2 -1 2 2 1 2 z z 4 -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 -1 -1 1 4 z 1 2 -3 1 2 1 2 z z -1 4 -1 ν z 3 -1 ν 3 4 z 2 1 2 4 1 2 1 2 z z -1 -1 ν 1 4 z 2 1 2 -1 3 4 z -1 z HypergeometricPFQ 1 O 1 z 4 -1 Rule z ν [/itex]

 Rule Form

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

 2007-05-02