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DLMF:7.12.E7 - MaRDI portal

DLMF:7.12.E7 (Q3116)

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DLMF:7.12.E7
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    Statements

    DLMF defining formula
    R n ( g ) ⁢ ( z ) = ( - 1 ) n π ⁢ 2 ⁢ ∫ 0 ∞ e - π ⁢ z 2 ⁢ t / 2 ⁢ t 2 ⁢ n + ( 1 / 2 ) t 2 + 1 ⁢ d t . superscript subscript 𝑅 𝑛 Fresnel-auxilliary-function-g 𝑧 superscript 1 𝑛 𝜋 2 superscript subscript 0 superscript 𝑒 𝜋 superscript 𝑧 2 𝑡 2 superscript 𝑡 2 𝑛 1 2 superscript 𝑡 2 1 𝑡 {\displaystyle{\displaystyle R_{n}^{(\mathrm{g})}(z)=\frac{(-1)^{n}}{\pi\sqrt{% 2}}\int_{0}^{\infty}\frac{e^{-\pi z^{2}t/2}t^{2n+(1/2)}}{t^{2}+1}\mathrm{d}t.}}
    0 references
    Symbols used
    auxiliary function for Fresnel integrals
    DLMF defining formula
    g ⁡ ( z ) Fresnel-auxilliary-function-g 𝑧 {\displaystyle{\displaystyle\mathrm{g}\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E11.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C7.S1.XMD2.m1fdec
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    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C7.S1.XMD3.m1cdec
    0 references

    Identifiers

     
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