DLMF:7.5.E6 (Q2352)

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DLMF:7.5.E6
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    Statements

    Defining formula
    e ± 1 2 ⁢ π ⁢ i ⁢ z 2 ⁢ ( g ⁡ ( z ) ± i ⁢ f ⁡ ( z ) ) = 1 2 ⁢ ( 1 ± i ) - ( C ⁡ ( z ) ± i ⁢ S ⁡ ( z ) ) . superscript 𝑒 plus-or-minus 1 2 𝜋 𝑖 superscript 𝑧 2 plus-or-minus Fresnel-auxilliary-function-g 𝑧 𝑖 Fresnel-auxilliary-function-f 𝑧 1 2 plus-or-minus 1 𝑖 plus-or-minus Fresnel-cosine-integral 𝑧 𝑖 Fresnel-sine-integral 𝑧 {\displaystyle{\displaystyle e^{\pm\frac{1}{2}\pi iz^{2}}(\mathrm{g}\left(z% \right)\pm i\mathrm{f}\left(z\right))=\tfrac{1}{2}(1\pm i)-(C\left(z\right)\pm iS% \left(z\right)).}}
    0 references
    DLMF id
    DLMF:7.5.E6
    0 references
    Symbols used
    auxiliary function for Fresnel integrals
    Defining formula
    f ⁡ ( z ) Fresnel-auxilliary-function-f 𝑧 {\displaystyle{\displaystyle\mathrm{f}\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E10.m2acdec
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    auxiliary function for Fresnel integrals
    Defining formula
    g ⁡ ( z ) Fresnel-auxilliary-function-g 𝑧 {\displaystyle{\displaystyle\mathrm{g}\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E11.m2acdec
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    Fresnel integral
    Defining formula
    C ⁡ ( z ) Fresnel-cosine-integral 𝑧 {\displaystyle{\displaystyle C\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E7.m2abdec
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    Fresnel integral
    Defining formula
    S ⁡ ( z ) Fresnel-sine-integral 𝑧 {\displaystyle{\displaystyle S\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E8.m2abdec
    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
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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.m2acdec
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    Q11312
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C7.S1.XMD2.m1edec
    0 references
     
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