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 KelvinKer

 http://functions.wolfram.com/03.20.06.0050.01

 Input Form

 KelvinKer[\[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)^(1/4) Pi - 4 I (-Log[z] + Log[(-1)^(3/4) z]) (Sqrt[I z^2]/z) - (3 Pi Sqrt[I z^2])/z) - E^((I z)/Sqrt[2] + (I Pi \[Nu])/2) (1/Sqrt[(-(-1)^(1/4)) z]) ((-(-1)^(3/4)) Pi + 4 (-1)^(1/4) (-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 (-1)^(3/4) Pi - 4 (-1)^(1/4) (-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)^(1/4) Pi + Pi I (Sqrt[(-I) z^2]/z) + (4 Sqrt[(-I) z^2] (Log[z] - Log[(-(-1)^(1/4)) z]))/z))) (1 + O[1/z^4]) - (E^(-(((1 + I) z)/Sqrt[2]) + (I Pi \[Nu])/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

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