DLMF:10.21.E42 (Q3349): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10919 / rank
 
Normal rank
Property / Symbols used: Q10919 / qualifier
 
Defining formula:

Ai ⁡ ( z ) Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}

\AiryAi@{\NVar{z}}
Property / Symbols used: Q10919 / qualifier
 
xml-id: C9.S2.SS1.m1adec
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.m2afdec
Property / Symbols used
 
Property / Symbols used: order not exceeding / rank
 
Normal rank
Property / Symbols used: order not exceeding / qualifier
 
Defining formula:

O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}

\bigO@{\NVar{x}}
Property / Symbols used: order not exceeding / qualifier
 
xml-id: C2.S1.E3.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11394 / rank
 
Normal rank
Property / Symbols used: Q11394 / qualifier
 
Defining formula:

a k zero-Airy-Ai 𝑘 {\displaystyle{\displaystyle a_{\NVar{k}}}}

\zAirya{\NVar{k}}
Property / Symbols used: Q11394 / qualifier
 
xml-id: C9.S9.SS1.p1.m5abdec
Property / Symbols used
 
Property / Symbols used: Q11456 / rank
 
Normal rank
Property / Symbols used: Q11456 / qualifier
 
Defining formula:

j ν , m zero-Bessel-J 𝜈 𝑚 {\displaystyle{\displaystyle j_{\NVar{\nu},\NVar{m}}}}

\zBesselj{\NVar{\nu}}{\NVar{m}}
Property / Symbols used: Q11456 / qualifier
 
xml-id: C10.S21.SS1.p2.m11andec
Property / Symbols used
 
Property / Symbols used: Q11438 / rank
 
Normal rank
Property / Symbols used: Q11438 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11438 / qualifier
 
xml-id: C10.S1.XMD1.m1qdec
Property / Symbols used
 
Property / Symbols used: Q11427 / rank
 
Normal rank
Property / Symbols used: Q11427 / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
 
xml-id: C10.S1.XMD7.m1ahdec
Property / Symbols used
 
Property / Symbols used: Q11464 / rank
 
Normal rank
Property / Symbols used: Q11464 / qualifier
 
Defining formula:

z ⁢ ( ζ ) 𝑧 𝜁 {\displaystyle{\displaystyle z(\zeta)}}

z(\zeta)
Property / Symbols used: Q11464 / qualifier
 
xml-id: C10.S21.XMD7.m1adec
Property / Symbols used
 
Property / Symbols used: function (locally) / rank
 
Normal rank
Property / Symbols used: function (locally) / qualifier
 
Defining formula:

h ⁢ ( ζ ) ℎ 𝜁 {\displaystyle{\displaystyle h(\zeta)}}

h(\zeta)
Property / Symbols used: function (locally) / qualifier
 
xml-id: C10.S21.XMD8.m1adec

Latest revision as of 15:46, 2 January 2020

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DLMF:10.21.E42
No description defined

    Statements

    Defining formula
    J ν ′ ⁡ ( j ν , m ) = - 2 ν 2 3 ⁢ Ai ′ ⁡ ( a m ) z ⁢ ( ζ ) ⁢ h ⁢ ( ζ ) ⁢ ( 1 + O ⁡ ( 1 ν 2 ) ) , diffop Bessel-J 𝜈 1 zero-Bessel-J 𝜈 𝑚 2 superscript 𝜈 2 3 diffop Airy-Ai 1 zero-Airy-Ai 𝑚 𝑧 𝜁 ℎ 𝜁 1 Big-O 1 superscript 𝜈 2 {\displaystyle{\displaystyle J_{\nu}'\left(j_{\nu,m}\right)=-\frac{2}{\nu^{% \frac{2}{3}}}\frac{\mathrm{Ai}'\left(a_{m}\right)}{z(\zeta)h(\zeta)}\left(1+O% \left(\frac{1}{\nu^{2}}\right)\right),}}
    0 references
    DLMF id
    DLMF:10.21.E42
    0 references
    constraint
    ζ = ν - 2 3 ⁢ a m 𝜁 superscript 𝜈 2 3 zero-Airy-Ai 𝑚 {\displaystyle{\displaystyle\zeta=\nu^{-\frac{2}{3}}a_{m}}}
    0 references
    Symbols used
    Q10919
    Defining formula
    Ai ⁡ ( z ) Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m1adec
    0 references
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2afdec
    0 references
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2aadec
    0 references
    Q11394
    Defining formula
    a k zero-Airy-Ai 𝑘 {\displaystyle{\displaystyle a_{\NVar{k}}}}
    xml-id
    C9.S9.SS1.p1.m5abdec
    0 references
    Q11456
    Defining formula
    j ν , m zero-Bessel-J 𝜈 𝑚 {\displaystyle{\displaystyle j_{\NVar{\nu},\NVar{m}}}}
    xml-id
    C10.S21.SS1.p2.m11andec
    0 references
    Q11438
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C10.S1.XMD1.m1qdec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1ahdec
    0 references
    Q11464
    Defining formula
    z ⁢ ( ζ ) 𝑧 𝜁 {\displaystyle{\displaystyle z(\zeta)}}
    xml-id
    C10.S21.XMD7.m1adec
    0 references
    function (locally)
    Defining formula
    h ⁢ ( ζ ) ℎ 𝜁 {\displaystyle{\displaystyle h(\zeta)}}
    xml-id
    C10.S21.XMD8.m1adec
    0 references
     
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