DLMF:7.2.E3 (Q2323)

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DLMF:7.2.E3
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    Statements

    Defining formula
    w ⁡ ( z ) = e - z 2 ⁢ ( 1 + 2 ⁢ i π ⁢ ∫ 0 z e t 2 ⁢ d t ) = e - z 2 ⁢ erfc ⁡ ( - i ⁢ z ) . error-function-w 𝑧 superscript 𝑒 superscript 𝑧 2 1 2 𝑖 𝜋 superscript subscript 0 𝑧 superscript 𝑒 superscript 𝑡 2 𝑡 superscript 𝑒 superscript 𝑧 2 complementary-error-function 𝑖 𝑧 {\displaystyle{\displaystyle{}w\left(z\right)=e^{-z^{2}}\left(1+\frac{2i}{% \sqrt{\pi}}\int_{0}^{z}e^{t^{2}}\mathrm{d}t\right)=e^{-z^{2}}\operatorname{% erfc}\left(-iz\right).}}
    0 references
    DLMF id
    DLMF:7.2.E3
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
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    complementary error function
    Defining formula
    erfc ⁡ z complementary-error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erfc}\NVar{z}}}
    xml-id
    C7.S2.E2.m2aadec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
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    Q11312
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C7.S1.XMD2.m1bdec
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