DLMF:7.5.E8 (Q2354)

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DLMF:7.5.E8
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    Statements

    Defining formula
    C ⁡ ( z ) ± i ⁢ S ⁡ ( z ) = 1 2 ⁢ ( 1 ± i ) ⁢ erf ⁡ ζ . plus-or-minus Fresnel-cosine-integral 𝑧 𝑖 Fresnel-sine-integral 𝑧 1 2 plus-or-minus 1 𝑖 error-function 𝜁 {\displaystyle{\displaystyle C\left(z\right)\pm iS\left(z\right)=\tfrac{1}{2}(% 1\pm i)\operatorname{erf}\zeta.}}
    0 references
    DLMF id
    DLMF:7.5.E8
    0 references
    Symbols used
    Fresnel integral
    Defining formula
    C ⁡ ( z ) Fresnel-cosine-integral 𝑧 {\displaystyle{\displaystyle C\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E7.m2acdec
    0 references
    Fresnel integral
    Defining formula
    S ⁡ ( z ) Fresnel-sine-integral 𝑧 {\displaystyle{\displaystyle S\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E8.m2acdec
    0 references
    error function
    Defining formula
    erf ⁡ z error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erf}\NVar{z}}}
    xml-id
    C7.S2.E1.m2aadec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
    0 references
    Q11312
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C7.S1.XMD2.m1gdec
    0 references
    change of variable (locally)
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C7.S5.XMD1.m1adec
    0 references
     
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