Bessel function of the first kind: Series representations (formula 03.01.06.0075)

BesselJ

$Mathematica Notation$

Bessel-Type Functions

BesselJ[nu,z]

Series representations

Asymptotic series expansions

Expansions for any z in exponential form

Using exponential function with branch cut-free arguments

http://functions.wolfram.com/03.01.06.0075.01

Input Form

BesselJ[\[Nu], z] \[Proportional] Piecewise[{{((-1)^(1/4) I^\[Nu])/(E^(I z) (Sqrt[2 Pi] Sqrt[z])) - ((-1)^(3/4) I^\[Nu] E^(I z - I Pi \[Nu]))/(Sqrt[2 Pi] Sqrt[z]), Arg[z] <= 0}}, ((-1)^(1/4) I^\[Nu])/(E^(I z) (Sqrt[2 Pi] Sqrt[z])) + ((-1)^(3/4) I^\[Nu] E^(I z + I Pi \[Nu]))/(Sqrt[2 Pi] Sqrt[z])] /; (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