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DLMF:7.14.E7 - MaRDI portal

DLMF:7.14.E7 (Q3137)

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DLMF:7.14.E7
No description defined

    Statements

    DLMF defining formula
    ∫ 0 ∞ e - a ⁢ t ⁢ C ⁡ ( 2 ⁢ t π ) ⁢ d t = ( a 2 + 1 + a ) 1 2 2 ⁢ a ⁢ a 2 + 1 , superscript subscript 0 superscript 𝑒 𝑎 𝑡 Fresnel-cosine-integral 2 𝑡 𝜋 𝑡 superscript superscript 𝑎 2 1 𝑎 1 2 2 𝑎 superscript 𝑎 2 1 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}C\left(\sqrt{\frac{2t}{\pi% }}\right)\mathrm{d}t=\frac{(\sqrt{a^{2}+1}+a)^{\frac{1}{2}}}{2a\sqrt{a^{2}+1}}% ,}}
    0 references
    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
    0 references
    Symbols used
    Fresnel integral
    DLMF defining formula
    C ⁡ ( z ) Fresnel-cosine-integral 𝑧 {\displaystyle{\displaystyle C\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E7.m2aadec
    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.m2aedec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
    0 references
    real part
    DLMF defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aedec
    0 references

    Identifiers

     
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