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DLMF:10.41.E14 - MaRDI portal

DLMF:10.41.E14 (Q4308)

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DLMF:10.41.E14
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    Statements

    DLMF defining formula
    J ν ⁡ ( ν ⁢ z ) = ( 4 ⁢ ζ 1 - z 2 ) 1 4 ⁢ ( Ai ⁡ ( ν 2 3 ⁢ ζ ) ν 1 3 ⁢ ( ∑ k = 0 ℓ A k ⁢ ( ζ ) ν 2 ⁢ k + O ⁡ ( 1 ζ 3 ⁢ ℓ + 3 ) ) + Ai ′ ⁡ ( ν 2 3 ⁢ ζ ) ν 5 3 ⁢ ( ∑ k = 0 ℓ - 1 B k ⁢ ( ζ ) ν 2 ⁢ k + O ⁡ ( 1 ζ 3 ⁢ ℓ + 1 ) ) ) , Bessel-J 𝜈 𝜈 𝑧 superscript 4 𝜁 1 superscript 𝑧 2 1 4 Airy-Ai superscript 𝜈 2 3 𝜁 superscript 𝜈 1 3 superscript subscript 𝑘 0 ℓ subscript 𝐴 𝑘 𝜁 superscript 𝜈 2 𝑘 Big-O 1 superscript 𝜁 3 ℓ 3 diffop Airy-Ai 1 superscript 𝜈 2 3 𝜁 superscript 𝜈 5 3 superscript subscript 𝑘 0 ℓ 1 subscript 𝐵 𝑘 𝜁 superscript 𝜈 2 𝑘 Big-O 1 superscript 𝜁 3 ℓ 1 {\displaystyle{\displaystyle J_{\nu}\left(\nu z\right)=\left(\frac{4\zeta}{1-z% ^{2}}\right)^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left(\nu^{\frac{2}{3}}\zeta% \right)}{\nu^{\frac{1}{3}}}\left(\sum_{k=0}^{\ell}\frac{A_{k}(\zeta)}{\nu^{2k}% }+O\left(\frac{1}{\zeta^{3\ell+3}}\right)\right)+\frac{\mathrm{Ai}'\left(\nu^{% \frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}}}\left(\sum_{k=0}^{\ell-1}\frac{B_{% k}(\zeta)}{\nu^{2k}}+O\left(\frac{1}{\zeta^{3\ell+1}}\right)\right)\right),}}
    0 references
    Symbols used
    Q10893
    DLMF 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
    DLMF defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2adec
    0 references
    order not exceeding
    DLMF defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2abdec
    0 references
    nonnegative integer
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1hdec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1kdec
    0 references
    complex parameter
    DLMF defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1hdec
    0 references
    coefficients (locally)
    DLMF defining formula
    A k ⁢ ( ζ ) subscript 𝐴 𝑘 𝜁 {\displaystyle{\displaystyle A_{k}(\zeta)}}
    xml-id
    C10.S20.XMD4.m1dec
    0 references
    coefficients (locally)
    DLMF defining formula
    B k ⁢ ( ζ ) subscript 𝐵 𝑘 𝜁 {\displaystyle{\displaystyle B_{k}(\zeta)}}
    xml-id
    C10.S20.XMD5.m1dec
    0 references

    Identifiers

     
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