DLMF:7.11.E4 (Q2406)

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DLMF:7.11.E4
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    Statements

    Defining formula
    erf ⁡ z = 2 ⁢ z π ⁢ M ⁡ ( 1 2 , 3 2 , - z 2 ) = 2 ⁢ z π ⁢ e - z 2 ⁢ M ⁡ ( 1 , 3 2 , z 2 ) , error-function 𝑧 2 𝑧 𝜋 Kummer-confluent-hypergeometric-M 1 2 3 2 superscript 𝑧 2 2 𝑧 𝜋 superscript 𝑒 superscript 𝑧 2 Kummer-confluent-hypergeometric-M 1 3 2 superscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{erf}z=\frac{2z}{\sqrt{\pi}}M\left(% \tfrac{1}{2},\tfrac{3}{2},-z^{2}\right)=\frac{2z}{\sqrt{\pi}}e^{-z^{2}}M\left(% 1,\tfrac{3}{2},z^{2}\right),}}
    0 references
    DLMF id
    DLMF:7.11.E4
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    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.m2adec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
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    error function
    Defining formula
    erf ⁡ z error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erf}\NVar{z}}}
    xml-id
    C7.S2.E1.m2aadec
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