DLMF:13.4.E3 (Q4366)

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DLMF:13.4.E3
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    Statements

    test
    𝐌 ⁡ ( a , b , - z ) = z 1 2 - 1 2 ⁢ b Γ ⁡ ( a ) ⁢ ∫ 0 ∞ e - t ⁢ t a - 1 2 ⁢ b - 1 2 ⁢ J b - 1 ⁡ ( 2 ⁢ z ⁢ t ) ⁢ d t , Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 superscript 𝑧 1 2 1 2 𝑏 Euler-Gamma 𝑎 superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑡 𝑎 1 2 𝑏 1 2 Bessel-J 𝑏 1 2 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,-z\right)=\frac{z^{\frac{1}{% 2}-\frac{1}{2}b}}{\Gamma\left(a\right)}\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}% b-\frac{1}{2}}J_{b-1}\left(2\sqrt{zt}\right)\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:13.4.E3
    0 references
    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2adec
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    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
     
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