DLMF:13.8.E5 (Q4430)

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DLMF:13.8.E5
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    Statements

    Defining formula
    U ⁡ ( a , b , z ) ∼ b - 1 2 ⁢ a ⁢ e 1 4 ⁢ ζ 2 ⁢ b ⁢ ( λ ⁢ ( λ - 1 ζ ) a - 1 ⁢ U ⁡ ( a - 1 2 , ζ ⁢ b ) - ( λ ⁢ ( λ - 1 ζ ) a - 1 - ( ζ λ - 1 ) a ) ⁢ U ⁡ ( a - 3 2 , ζ ⁢ b ) ζ ⁢ b ) asymptotically-equals Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 superscript 𝑏 1 2 𝑎 superscript 𝑒 1 4 superscript 𝜁 2 𝑏 𝜆 superscript 𝜆 1 𝜁 𝑎 1 parabolic-U 𝑎 1 2 𝜁 𝑏 𝜆 superscript 𝜆 1 𝜁 𝑎 1 superscript 𝜁 𝜆 1 𝑎 parabolic-U 𝑎 3 2 𝜁 𝑏 𝜁 𝑏 {\displaystyle{\displaystyle U\left(a,b,z\right)\sim b^{-\frac{1}{2}a}e^{\frac% {1}{4}\zeta^{2}b}\left(\lambda\left(\frac{\lambda-1}{\zeta}\right)^{a-1}U\left% (a-\tfrac{1}{2},\zeta\sqrt{b}\right)-\left(\lambda\left(\frac{\lambda-1}{\zeta% }\right)^{a-1}-\left(\frac{\zeta}{\lambda-1}\right)^{a}\right)\frac{U\left(a-% \tfrac{3}{2},\zeta\sqrt{b}\right)}{\zeta\sqrt{b}}\right)}}
    0 references
    DLMF id
    DLMF:13.8.E5
    0 references
    Symbols used
    Kummer confluent hypergeometric function
    Defining formula
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2adec
    0 references
    asymptotic equality
    Defining formula
    ∼ asymptotically-equals {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.E1.m2aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    Q11320
    Defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1aadec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1ddec
    0 references
     
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