DLMF:13.8.E13 (Q4438): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Bessel function of the first kind / rank
 
Normal rank
Property / Symbols used: Bessel function of the first kind / qualifier
 
Defining formula:

J ν ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}

\BesselJ{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Bessel function of the first kind / qualifier
 
xml-id: C10.S2.E2.m2abdec

Revision as of 15:58, 2 January 2020

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DLMF:13.8.E13
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    Statements

    𝐌 ( - a , b , z ) ( z / a ) ( 1 - b ) / 2 e z / 2 Γ ( 1 + a ) Γ ( a + b ) ( J b - 1 ( 2 a z ) s = 0 p s ( z ) ( - a ) s - z / a J b ( 2 a z ) s = 0 q s ( z ) ( - a ) s ) , asymptotic-expansion Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 superscript 𝑧 𝑎 1 𝑏 2 superscript 𝑒 𝑧 2 Euler-Gamma 1 𝑎 Euler-Gamma 𝑎 𝑏 Bessel-J 𝑏 1 2 𝑎 𝑧 superscript subscript 𝑠 0 subscript 𝑝 𝑠 𝑧 superscript 𝑎 𝑠 𝑧 𝑎 Bessel-J 𝑏 2 𝑎 𝑧 superscript subscript 𝑠 0 subscript 𝑞 𝑠 𝑧 superscript 𝑎 𝑠 {\displaystyle{\displaystyle{\mathbf{M}}\left(-a,b,z\right)\sim\left(z/a\right% )^{(1-b)/2}\frac{e^{z/2}\Gamma\left(1+a\right)}{\Gamma\left(a+b\right)}\*\left% (J_{b-1}\left(2\sqrt{az}\right)\sum_{s=0}^{\infty}\frac{p_{s}(z)}{(-a)^{s}}-% \sqrt{z/a}J_{b}\left(2\sqrt{az}\right)\sum_{s=0}^{\infty}\frac{q_{s}(z)}{(-a)^% {s}}\right),}}
    0 references
    DLMF:13.8.E13
    0 references
    J ν ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    C10.S2.E2.m2abdec
    0 references