DLMF:7.7.E11 (Q2381)

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DLMF:7.7.E11
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    Statements

    Defining formula
    g ⁡ ( z ) = 1 π ⁢ 2 ⁢ ∫ 0 ∞ t ⁢ e - π ⁢ z 2 ⁢ t / 2 t 2 + 1 ⁢ d t , Fresnel-auxilliary-function-g 𝑧 1 𝜋 2 superscript subscript 0 𝑡 superscript 𝑒 𝜋 superscript 𝑧 2 𝑡 2 superscript 𝑡 2 1 𝑡 {\displaystyle{\displaystyle\mathrm{g}\left(z\right)=\frac{1}{\pi\sqrt{2}}\int% _{0}^{\infty}\frac{\sqrt{t}e^{-\pi z^{2}t/2}}{t^{2}+1}\mathrm{d}t,}}
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    DLMF id
    DLMF:7.7.E11
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    constraint
    | ph ⁡ z | ≤ 1 4 ⁢ π phase 𝑧 1 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\frac{1}{4}\pi}}
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    Symbols used
    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.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.m2ajdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ajdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ajdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ajdec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1addec
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