DLMF:13.6.E14 (Q4401)

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DLMF:13.6.E14
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    Statements

    Defining formula
    M ⁡ ( 1 2 ⁢ a + 1 4 , 1 2 , 1 2 ⁢ z 2 ) = 2 1 2 ⁢ a - 3 4 ⁢ Γ ⁡ ( 1 2 ⁢ a + 3 4 ) ⁢ e 1 4 ⁢ z 2 π ⁢ ( U ⁡ ( a , z ) + U ⁡ ( a , - z ) ) , Kummer-confluent-hypergeometric-M 1 2 𝑎 1 4 1 2 1 2 superscript 𝑧 2 superscript 2 1 2 𝑎 3 4 Euler-Gamma 1 2 𝑎 3 4 superscript 𝑒 1 4 superscript 𝑧 2 𝜋 parabolic-U 𝑎 𝑧 parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle M\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},% \tfrac{1}{2}z^{2}\right)=\frac{2^{\frac{1}{2}a-\frac{3}{4}}\Gamma\left(\tfrac{% 1}{2}a+\tfrac{3}{4}\right)e^{\frac{1}{4}z^{2}}}{\sqrt{\pi}}\*\left(U\left(a,z% \right)+U\left(a,-z\right)\right),}}
    0 references
    DLMF id
    DLMF:13.6.E14
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    $$={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$$ 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.m2afdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aidec
    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.m1abdec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1mdec
    0 references
     
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