DLMF:13.8.E4 (Q4429): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11557 / rank
 
Normal rank
Property / Symbols used: Q11557 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11557 / qualifier
 
xml-id: C13.S1.XMD6.m1cdec

Latest revision as of 15:57, 2 January 2020

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DLMF:13.8.E4
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    Statements

    Defining formula
    M ⁡ ( 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-M 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 M\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.E4
    0 references
    Symbols used
    $$={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$$ 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.m2abdec
    0 references
    asymptotic equality
    Defining formula
    ∼ asymptotically-equals {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.E1.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    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.m1adec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1cdec
    0 references
     
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