DLMF:18.12.E12 (Q5654)

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DLMF:18.12.E12
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    Statements

    Defining formula
    e x ⁢ z ⁢ J 0 ⁡ ( z ⁢ 1 - x 2 ) = ∑ n = 0 ∞ P n ⁡ ( x ) n ! ⁢ z n . superscript 𝑒 𝑥 𝑧 Bessel-J 0 𝑧 1 superscript 𝑥 2 superscript subscript 𝑛 0 Legendre-spherical-polynomial 𝑛 𝑥 𝑛 superscript 𝑧 𝑛 {\displaystyle{\displaystyle e^{xz}J_{0}\left(z\sqrt{1-x^{2}}\right)=\sum_{n=0% }^{\infty}\frac{P_{n}\left(x\right)}{n!}z^{n}.}}
    0 references
    DLMF id
    DLMF:18.12.E12
    0 references
    Symbols used
    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.m2abdec
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    Q10920
    Defining formula
    P n ⁡ ( x ) Legendre-spherical-polynomial 𝑛 𝑥 {\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r10.m2aadec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    Q11750
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1kdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1kdec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1jdec
    0 references
     
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