DLMF:13.8.E13 (Q4438)

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DLMF:13.8.E13
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    Statements

    Defining formula
    𝐌 ⁑ ( - a , b , z ) ∼ ( z / a ) ( 1 - b ) / 2 ⁒ e z / 2 ⁒ Ξ“ ⁑ ( 1 + a ) Ξ“ ⁑ ( a + b ) ⁒ ( J b - 1 ⁑ ( 2 ⁒ a ⁒ z ) ⁒ βˆ‘ s = 0 ∞ p s ⁒ ( z ) ( - a ) s - z / a ⁒ J b ⁑ ( 2 ⁒ a ⁒ z ) ⁒ βˆ‘ s = 0 ∞ q s ⁒ ( z ) ( - a ) s ) , asymptotic-expansion Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 superscript 𝑧 π‘Ž 1 𝑏 2 superscript 𝑒 𝑧 2 Euler-Gamma 1 π‘Ž Euler-Gamma π‘Ž 𝑏 Bessel-J 𝑏 1 2 π‘Ž 𝑧 superscript subscript 𝑠 0 subscript 𝑝 𝑠 𝑧 superscript π‘Ž 𝑠 𝑧 π‘Ž Bessel-J 𝑏 2 π‘Ž 𝑧 superscript subscript 𝑠 0 subscript π‘ž 𝑠 𝑧 superscript π‘Ž 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 id
    DLMF:13.8.E13
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J Ξ½ ⁑ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2abdec
    0 references
    gamma function
    Defining formula
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ahdec
    0 references
    Olver’s confluent hypergeometric function
    Defining formula
    𝐌 ⁑ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}
    xml-id
    C13.S2.E3.m2aadec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ahdec
    0 references
    Q11558
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C13.S1.XMD3.m1edec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1gdec
    0 references
     
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