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 KelvinKei

 http://functions.wolfram.com/03.19.06.0036.01

 Input Form

 KelvinKei[\[Nu], z] \[Proportional] ((Sqrt[Pi] Csc[Pi \[Nu]])/(4 Sqrt[2])) (((-z^\[Nu]) ((-1)^(3/4) E^((I z)/Sqrt[2] + (3 I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^(-(1/2) - \[Nu]) + E^(-((I z)/Sqrt[2]) + (I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) - \[Nu]) ((Sqrt[I z^2] Cos[Pi \[Nu]])/z + (-1)^(3/4) Sin[Pi \[Nu]])) + ((-1)^(3/4) E^((I z)/Sqrt[2] + (I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^ (-(1/2) + \[Nu]) + E^(-((I z)/Sqrt[2]) - (5 I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) + \[Nu]) ((Sqrt[I z^2] Cos[Pi \[Nu]])/z - (-1)^(3/4) Sin[Pi \[Nu]]))/z^\[Nu])/E^(z/Sqrt[2]) + E^(z/Sqrt[2]) (z^\[Nu] ((-1)^(3/4) E^((I z)/Sqrt[2] + (I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) - \[Nu]) + E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^ (-(1/2) - \[Nu]) (-((I Sqrt[(-I) z^2] Cos[Pi \[Nu]])/z) + (-1)^(3/4) Sin[Pi \[Nu]])) - ((-1)^(3/4) E^((I z)/Sqrt[2] - (5 I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) + \[Nu]) - E^(-((I z)/Sqrt[2]) + (I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^ (-(1/2) + \[Nu]) ((I Sqrt[(-I) z^2] Cos[Pi \[Nu]])/z + (-1)^(3/4) Sin[Pi \[Nu]]))/z^\[Nu]) + ((-1 + 4 \[Nu]^2)/(8 z)) ((z^\[Nu] (E^((I z)/Sqrt[2] + (3 I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^ (-(1/2) - \[Nu]) + E^(-((I z)/Sqrt[2]) + (I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) - \[Nu]) (((-1)^(3/4) Sqrt[I z^2] Cos[Pi \[Nu]])/z - I Sin[Pi \[Nu]])) - (E^((I z)/Sqrt[2] + (I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^ (-(1/2) + \[Nu]) + E^(-((I z)/Sqrt[2]) - (5 I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) + \[Nu]) (((-1)^(3/4) Sqrt[I z^2] Cos[Pi \[Nu]])/z + I Sin[Pi \[Nu]]))/z^\[Nu])/E^(z/Sqrt[2]) + E^(z/Sqrt[2]) (z^\[Nu] ((-I) E^((I z)/Sqrt[2] + (I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) - \[Nu]) + E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^ (-(1/2) - \[Nu]) (((-1)^(3/4) Sqrt[(-I) z^2] Cos[Pi \[Nu]])/z + Sin[Pi \[Nu]])) + (I E^((I z)/Sqrt[2] - (5 I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) + \[Nu]) - E^(-((I z)/Sqrt[2]) + (I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^ (-(1/2) + \[Nu]) (((-1)^(3/4) Sqrt[(-I) z^2] Cos[Pi \[Nu]])/z - Sin[Pi \[Nu]]))/z^\[Nu])) + ((9 - 40 \[Nu]^2 + 16 \[Nu]^4)/(128 z^2)) ((z^\[Nu] ((-1)^(1/4) E^((I z)/Sqrt[2] + (3 I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^(-(1/2) - \[Nu]) + E^(-((I z)/Sqrt[2]) + (I Pi \[Nu])/4) ((-1)^(3/4) z)^ (-(1/2) - \[Nu]) ((I Sqrt[I z^2] Cos[Pi \[Nu]])/z - (-1)^(1/4) Sin[Pi \[Nu]])) - ((-1)^(1/4) E^((I z)/Sqrt[2] + (I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^ (-(1/2) + \[Nu]) + E^(-((I z)/Sqrt[2]) - (5 I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) + \[Nu]) ((I Sqrt[I z^2] Cos[Pi \[Nu]])/z + (-1)^(1/4) Sin[Pi \[Nu]]))/z^\[Nu])/E^(z/Sqrt[2]) + E^(z/Sqrt[2]) (z^\[Nu] ((-1)^(1/4) E^((I z)/Sqrt[2] + (I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) - \[Nu]) + E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^ (-(1/2) - \[Nu]) ((Sqrt[(-I) z^2] Cos[Pi \[Nu]])/z - (-1)^(1/4) Sin[Pi \[Nu]])) - ((-1)^(1/4) E^((I z)/Sqrt[2] - (5 I Pi \[Nu])/4) ((-1)^(3/4) z)^ (-(1/2) + \[Nu]) + E^(-((I z)/Sqrt[2]) + (I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^(-(1/2) + \[Nu]) ((Sqrt[(-I) z^2] Cos[Pi \[Nu]])/ z + (-1)^(1/4) Sin[Pi \[Nu]]))/z^\[Nu])) + ((-225 + 1036 \[Nu]^2 - 560 \[Nu]^4 + 64 \[Nu]^6)/(3072 z^3)) ((z^\[Nu] (I E^((I z)/Sqrt[2] + (3 I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^ (-(1/2) - \[Nu]) + E^(-((I z)/Sqrt[2]) + (I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) - \[Nu]) (((-1)^(1/4) Sqrt[I z^2] Cos[Pi \[Nu]])/z - Sin[Pi \[Nu]])) - (I E^((I z)/Sqrt[2] + (I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^ (-(1/2) + \[Nu]) + E^(-((I z)/Sqrt[2]) - (5 I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) + \[Nu]) (((-1)^(1/4) Sqrt[I z^2] Cos[Pi \[Nu]])/z + Sin[Pi \[Nu]]))/z^\[Nu])/E^(z/Sqrt[2]) + E^(z/Sqrt[2]) ((-z^\[Nu]) (E^((I z)/Sqrt[2] + (I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) - \[Nu]) + E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^ (-(1/2) - \[Nu]) ((-1)^(1/4) ((Sqrt[(-I) z^2] Cos[Pi \[Nu]])/z) - I Sin[Pi \[Nu]])) + (E^((I z)/Sqrt[2] - (5 I Pi \[Nu])/4) ((-1)^(3/4) z)^(-(1/2) + \[Nu]) + E^(-((I z)/Sqrt[2]) + (I Pi \[Nu])/4) ((-(-1)^(1/4)) z)^ (-(1/2) + \[Nu]) (((-1)^(1/4) Sqrt[(-I) z^2] Cos[Pi \[Nu]])/z + I Sin[Pi \[Nu]]))/z^\[Nu])) + \[Ellipsis]) /; (Abs[z] -> Infinity) && !Element[\[Nu], Integers]

 Standard Form

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 MathML Form

 kei ν ( z ) π csc ( π ν ) 4 2 ( z 2 ( z ν ( 3 π ν 4 - z 2 ( ( - 1 ) 3 / 4 sin ( π ν ) - - z 2 cos ( π ν ) z ) ( - - 1 4 z ) - ν - 1 2 + ( - 1 ) 3 / 4 z 2 + π ν 4 ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ) - z - ν ( ( - 1 ) 3 / 4 z 2 - 5 π ν 4 ( ( - 1 ) 3 / 4 z ) ν - 1 2 - π ν 4 - z 2 ( - - 1 4 z ) ν - 1 2 ( - z 2 cos ( π ν ) z + ( - 1 ) 3 / 4 sin ( π ν ) ) ) ) + - z 2 ( z - ν ( ( - 1 ) 3 / 4 z 2 + π ν 4 ( - - 1 4 z ) ν - 1 2 + - z 2 - 5 π ν 4 ( ( - 1 ) 3 / 4 z ) ν - 1 2 ( z 2 cos ( π ν ) z - ( - 1 ) 3 / 4 sin ( π ν ) ) ) - z ν ( ( - 1 ) 3 / 4 z 2 + 3 π ν 4 ( - - 1 4 z ) - ν - 1 2 + π ν 4 - z 2 ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ( z 2 cos ( π ν ) z + ( - 1 ) 3 / 4 sin ( π ν ) ) ) ) + 4 ν 2 - 1 8 z ( z 2 ( ( z 2 - 5 π ν 4 ( ( - 1 ) 3 / 4 z ) ν - 1 2 - π ν 4 - z 2 ( - - 1 4 z ) ν - 1 2 ( ( - 1 ) 3 / 4 - z 2 cos ( π ν ) z - sin ( π ν ) ) ) z - ν + ( 3 π ν 4 - z 2 ( - - 1 4 z ) - ν - 1 2 ( ( - 1 ) 3 / 4 - z 2 cos ( π ν ) z + sin ( π ν ) ) - z 2 + π ν 4 ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ) z ν ) + - z 2 ( z ν ( z 2 + 3 π ν 4 ( - - 1 4 z ) - ν - 1 2 + π ν 4 - z 2 ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ( ( - 1 ) 3 / 4 z 2 cos ( π ν ) z - sin ( π ν ) ) ) - z - ν ( z 2 + π ν 4 ( - - 1 4 z ) ν - 1 2 + - z 2 - 5 π ν 4 ( ( - 1 ) 3 / 4 z ) ν - 1 2 ( ( - 1 ) 3 / 4 z 2 cos ( π ν ) z + sin ( π ν ) ) ) ) ) + 16 ν 4 - 40 ν 2 + 9 128 z 2 ( z 2 ( z ν ( 3 π ν 4 - z 2 ( - z 2 cos ( π ν ) z - - 1 4 sin ( π ν ) ) ( - - 1 4 z ) - ν - 1 2 + - 1 4 z 2 + π ν 4 ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ) - z - ν ( π ν 4 - z 2 ( - z 2 cos ( π ν ) z + - 1 4 sin ( π ν ) ) ( - - 1 4 z ) ν - 1 2 + - 1 4 z 2 - 5 π ν 4 ( ( - 1 ) 3 / 4 z ) ν - 1 2 ) ) + - z 2 ( z ν ( - 1 4 z 2 + 3 π ν 4 ( - - 1 4 z ) - ν - 1 2 + π ν 4 - z 2 ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ( z 2 cos ( π ν ) z - - 1 4 sin ( π ν ) ) ) - z - ν ( - 1 4 z 2 + π ν 4 ( - - 1 4 z ) ν - 1 2 + - z 2 - 5 π ν 4 ( ( - 1 ) 3 / 4 z ) ν - 1 2 ( z 2 cos ( π ν ) z + - 1 4 sin ( π ν ) ) ) ) ) + 64 ν 6 - 560 ν 4 + 1036 ν 2 - 225 3072 z 3 ( z 2 ( z - ν ( π ν 4 - z 2 ( - 1 4 - z 2 cos ( π ν ) z + sin ( π ν ) ) ( - - 1 4 z ) ν - 1 2 + z 2 - 5 π ν 4 ( ( - 1 ) 3 / 4 z ) ν - 1 2 ) - z ν ( 3 π ν 4 - z 2 ( - 1 4 ( - z 2 cos ( π ν ) ) z - sin ( π ν ) ) ( - - 1 4 z ) - ν - 1 2 + z 2 + π ν 4 ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ) ) + - z 2 ( z ν ( z 2 + 3 π ν 4 ( - - 1 4 z ) - ν - 1 2 + π ν 4 - z 2 ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ( - 1 4 z 2 cos ( π ν ) z - sin ( π ν ) ) ) - z - ν ( z 2 + π ν 4 ( - - 1 4 z ) ν - 1 2 + - z 2 - 5 π ν 4 ( ( - 1 ) 3 / 4 z ) ν - 1 2 ( - 1 4 z 2 cos ( π ν ) z + sin ( π ν ) ) ) ) ) + ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Proportional KelvinKei ν z 1 2 ν 4 2 1 2 -1 z 2 1 2 -1 z ν 3 ν 4 -1 -1 z 2 1 2 -1 -1 3 4 ν -1 -1 z 2 1 2 ν z -1 -1 -1 1 4 z -1 ν -1 1 2 -1 3 4 z 2 1 2 -1 ν 4 -1 -1 3 4 z -1 ν -1 1 2 -1 z -1 ν -1 3 4 z 2 1 2 -1 -1 5 ν 4 -1 -1 3 4 z ν -1 1 2 -1 ν 4 -1 -1 z 2 1 2 -1 -1 -1 1 4 z ν -1 1 2 -1 z 2 1 2 ν z -1 -1 3 4 ν -1 z 2 1 2 -1 z -1 ν -1 3 4 z 2 1 2 -1 ν 4 -1 -1 -1 1 4 z ν -1 1 2 -1 z 2 1 2 -1 -1 5 ν 4 -1 -1 3 4 z ν -1 1 2 z 2 1 2 ν z -1 -1 -1 3 4 ν -1 z ν -1 3 4 z 2 1 2 -1 3 ν 4 -1 -1 -1 1 4 z -1 ν -1 1 2 ν 4 -1 -1 z 2 1 2 -1 -1 3 4 z -1 ν -1 1 2 z 2 1 2 ν z -1 -1 3 4 ν 4 ν 2 -1 8 z -1 z 2 1 2 -1 z 2 1 2 -1 -1 5 ν 4 -1 -1 3 4 z ν -1 1 2 -1 ν 4 -1 -1 z 2 1 2 -1 -1 -1 1 4 z ν -1 1 2 -1 3 4 -1 z 2 1 2 ν z -1 -1 ν z -1 ν 3 ν 4 -1 -1 z 2 1 2 -1 -1 -1 1 4 z -1 ν -1 1 2 -1 3 4 -1 z 2 1 2 ν z -1 ν -1 z 2 1 2 -1 ν 4 -1 -1 3 4 z -1 ν -1 1 2 z ν -1 z 2 1 2 -1 z ν z 2 1 2 -1 3 ν 4 -1 -1 -1 1 4 z -1 ν -1 1 2 ν 4 -1 -1 z 2 1 2 -1 -1 3 4 z -1 ν -1 1 2 -1 3 4 z 2 1 2 ν z -1 -1 ν -1 z -1 ν z 2 1 2 -1 ν 4 -1 -1 -1 1 4 z ν -1 1 2 -1 z 2 1 2 -1 -1 5 ν 4 -1 -1 3 4 z ν -1 1 2 -1 3 4 z 2 1 2 ν z -1 ν 16 ν 4 -1 40 ν 2 9 128 z 2 -1 z 2 1 2 -1 z ν 3 ν 4 -1 -1 z 2 1 2 -1 -1 z 2 1 2 ν z -1 -1 -1 1 4 ν -1 -1 1 4 z -1 ν -1 1 2 -1 1 4 z 2 1 2 -1 ν 4 -1 -1 3 4 z -1 ν -1 1 2 -1 z -1 ν ν 4 -1 -1 z 2 1 2 -1 -1 z 2 1 2 ν z -1 -1 1 4 ν -1 -1 1 4 z ν -1 1 2 -1 1 4 z 2 1 2 -1 -1 5 ν 4 -1 -1 3 4 z ν -1 1 2 -1 z 2 1 2 -1 z ν -1 1 4 z 2 1 2 -1 3 ν 4 -1 -1 -1 1 4 z -1 ν -1 1 2 ν 4 -1 -1 z 2 1 2 -1 -1 3 4 z -1 ν -1 1 2 z 2 1 2 ν z -1 -1 -1 1 4 ν -1 z -1 ν -1 1 4 z 2 1 2 -1 ν 4 -1 -1 -1 1 4 z ν -1 1 2 -1 z 2 1 2 -1 -1 5 ν 4 -1 -1 3 4 z ν -1 1 2 z 2 1 2 ν z -1 -1 1 4 ν 64 ν 6 -1 560 ν 4 1036 ν 2 -225 3072 z 3 -1 z 2 1 2 -1 z -1 ν ν 4 -1 -1 z 2 1 2 -1 -1 1 4 -1 z 2 1 2 ν z -1 ν -1 -1 1 4 z ν -1 1 2 z 2 1 2 -1 -1 5 ν 4 -1 -1 3 4 z ν -1 1 2 -1 z ν 3 ν 4 -1 -1 z 2 1 2 -1 -1 1 4 -1 z 2 1 2 ν z -1 -1 ν -1 -1 1 4 z -1 ν -1 1 2 z 2 1 2 -1 ν 4 -1 -1 3 4 z -1 ν -1 1 2 -1 z 2 1 2 -1 z ν z 2 1 2 -1 3 ν 4 -1 -1 -1 1 4 z -1 ν -1 1 2 ν 4 -1 -1 z 2 1 2 -1 -1 3 4 z -1 ν -1 1 2 -1 1 4 z 2 1 2 ν z -1 -1 ν -1 z -1 ν z 2 1 2 -1 ν 4 -1 -1 -1 1 4 z ν -1 1 2 -1 z 2 1 2 -1 -1 5 ν 4 -1 -1 3 4 z ν -1 1 2 -1 1 4 z 2 1 2 ν z -1 ν Rule z ν [/itex]

 Rule Form

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

 2007-05-02