DLMF:13.8.E12 (Q4437)

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DLMF:13.8.E12
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    Statements

    Defining formula
    𝐌 ⁑ ( a , b , z ) ∼ ( z / a ) ( 1 - b ) / 2 ⁒ e z / 2 ⁒ Ξ“ ⁑ ( 1 + a - b ) Ξ“ ⁑ ( a ) ⁒ ( I b - 1 ⁑ ( 2 ⁒ a ⁒ z ) ⁒ βˆ‘ s = 0 ∞ p s ⁒ ( z ) a s - z / a ⁒ I 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 𝑏 Euler-Gamma π‘Ž modified-Bessel-first-kind 𝑏 1 2 π‘Ž 𝑧 superscript subscript 𝑠 0 subscript 𝑝 𝑠 𝑧 superscript π‘Ž 𝑠 𝑧 π‘Ž modified-Bessel-first-kind 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-b\right)}{\Gamma\left(a\right)}\*\left(% I_{b-1}\left(2\sqrt{az}\right)\sum_{s=0}^{\infty}\frac{p_{s}(z)}{a^{s}}-\sqrt{% z/a}I_{b}\left(2\sqrt{az}\right)\sum_{s=0}^{\infty}\frac{q_{s}(z)}{a^{s}}% \right),}}
    0 references
    DLMF id
    DLMF:13.8.E12
    0 references
    Symbols used
    gamma function
    Defining formula
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2agdec
    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.m2adec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
    0 references
    modified Bessel function of the first kind
    Defining formula
    I Ξ½ ⁑ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2adec
    0 references
    Q11558
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C13.S1.XMD3.m1ddec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1fdec
    0 references
     
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