DLMF:13.8.E9 (Q4434)

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DLMF:13.8.E9
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    Statements

    Defining formula
    M ⁡ ( a , b , x ) = Γ ⁡ ( b ) ⁢ e 1 2 ⁢ x ⁢ ( ( 1 2 ⁢ b - a ) ⁢ x ) 1 2 - 1 2 ⁢ b ⁢ ( J b - 1 ⁡ ( 2 ⁢ x ⁢ ( b - 2 ⁢ a ) ) + env ⁢ J b - 1 ⁡ ( 2 ⁢ x ⁢ ( b - 2 ⁢ a ) ) ⁢ O ⁡ ( | a | - 1 2 ) ) , Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑥 Euler-Gamma 𝑏 superscript 𝑒 1 2 𝑥 superscript 1 2 𝑏 𝑎 𝑥 1 2 1 2 𝑏 Bessel-J 𝑏 1 2 𝑥 𝑏 2 𝑎 envelope-Bessel-J 𝑏 1 2 𝑥 𝑏 2 𝑎 Big-O 𝑎 1 2 {\displaystyle{\displaystyle M\left(a,b,x\right)=\Gamma\left(b\right)e^{\frac{% 1}{2}x}\left((\tfrac{1}{2}b-a)x\right)^{\frac{1}{2}-\frac{1}{2}b}\*\left(J_{b-% 1}\left(\sqrt{2x(b-2a)}\right)+\mathrm{env}\mskip-2.0mu J_{b-1}\left(\sqrt{2x(% b-2a)}\right)O\left({\left|a\right|^{-\frac{1}{2}}}\right)\right),}}
    0 references
    DLMF id
    DLMF:13.8.E9
    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.m2adec
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    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2addec
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    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
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    $$={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$$ Kummer confluent hypergeometric function
    Defining formula
    M ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle M\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E2.m2addec
    0 references
    Q11090
    Defining formula
    env ⁢ J ν ⁡ ( x ) envelope-Bessel-J 𝜈 𝑥 {\displaystyle{\displaystyle\mathrm{env}\mskip-2.0mu J_{\NVar{\nu}}\left(\NVar% {x}\right)}}
    xml-id
    C2.S8.SS4.p5.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
     
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