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BesselJ






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselJ[nu,z] > Series representations > Asymptotic series expansions > Expansions for any z in trigonometric form > Using trigonometric functions with branch cut-free arguments





http://functions.wolfram.com/03.01.06.0078.01









  


  










Input Form





BesselJ[\[Nu], z] \[Proportional] (1/Sqrt[2 Pi]) ((1/Sqrt[z]) (1 - (I Sqrt[-z^2])/z) Cos[z - (Pi (2 \[Nu] + 1))/4] + (E^(I Pi \[Nu])/Sqrt[-z]) (1 + (I Sqrt[-z^2])/z) Cos[z + (Pi (2 \[Nu] + 1))/4]) (Sum[((Pochhammer[(1 - 2 \[Nu])/4, k] Pochhammer[(3 - 2 \[Nu])/4, k] Pochhammer[(1 + 2 \[Nu])/4, k] Pochhammer[(3 + 2 \[Nu])/4, k])/ (Pochhammer[1/2, k] k!)) (-(1/z^2))^k, {k, 0, n}] + O[1/z^(2 n + 2)]) + ((1 - 4 \[Nu]^2)/(8 z Sqrt[2 Pi])) ((1/Sqrt[z]) (1 - (I Sqrt[-z^2])/z) Sin[z - (Pi (2 \[Nu] + 1))/4] + (E^(I Pi \[Nu])/Sqrt[-z]) (1 + (I Sqrt[-z^2])/z) Sin[z + (Pi (2 \[Nu] + 1))/4]) (Sum[((Pochhammer[(3 - 2 \[Nu])/4, k] Pochhammer[(5 - 2 \[Nu])/4, k] Pochhammer[(3 + 2 \[Nu])/4, k] Pochhammer[(5 + 2 \[Nu])/4, k])/ (Pochhammer[3/2, k] k!)) (-(1/z^2))^k, {k, 0, n}] + O[1/z^(2 n + 2)]) /; (Abs[z] -> Infinity)










Standard Form





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MathML Form







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Rule Form





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





2007-05-02





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