DLMF:7.5.E10 (Q2356)

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DLMF:7.5.E10
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    Statements

    test
    g ⁡ ( z ) ± i ⁢ f ⁡ ( z ) = 1 2 ⁢ ( 1 ± i ) ⁢ e ζ 2 ⁢ erfc ⁡ ζ . plus-or-minus Fresnel-auxilliary-function-g 𝑧 𝑖 Fresnel-auxilliary-function-f 𝑧 1 2 plus-or-minus 1 𝑖 superscript 𝑒 superscript 𝜁 2 complementary-error-function 𝜁 {\displaystyle{\displaystyle\mathrm{g}\left(z\right)\pm i\mathrm{f}\left(z% \right)=\tfrac{1}{2}(1\pm i)e^{\zeta^{2}}\operatorname{erfc}\zeta.}}
    0 references
    DLMF id
    DLMF:7.5.E10
    0 references
    Symbols used
    auxiliary function for Fresnel integrals
    test
    f ⁡ ( z ) Fresnel-auxilliary-function-f 𝑧 {\displaystyle{\displaystyle\mathrm{f}\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E10.m2addec
    0 references
    auxiliary function for Fresnel integrals
    test
    g ⁡ ( z ) Fresnel-auxilliary-function-g 𝑧 {\displaystyle{\displaystyle\mathrm{g}\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E11.m2addec
    0 references
    complementary error function
    test
    erfc ⁡ z complementary-error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erfc}\NVar{z}}}
    xml-id
    C7.S2.E2.m2adec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2agdec
    0 references
    Q11312
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C7.S1.XMD2.m1idec
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    change of variable (locally)
    test
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C7.S5.XMD1.m1cdec
    0 references
     
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