DLMF:2.4.E5 (Q758)

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DLMF:2.4.E5
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    Statements

    Defining formula
    q ⁢ ( t ) = 1 2 ⁢ π ⁢ i ⁢ ∫ σ - i ⁢ ∞ σ + i ⁢ ∞ e t ⁢ z ⁢ Q ⁢ ( z ) ⁢ d z , 𝑞 𝑡 1 2 𝜋 𝑖 superscript subscript 𝜎 𝑖 𝜎 𝑖 superscript 𝑒 𝑡 𝑧 𝑄 𝑧 𝑧 {\displaystyle{\displaystyle q(t)=\frac{1}{2\pi i}\int_{\sigma-i\infty}^{% \sigma+i\infty}e^{tz}Q(z)\mathrm{d}z,}}
    0 references
    DLMF id
    DLMF:2.4.E5
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    constraint
    0 < t < ∞ 0 𝑡 {\displaystyle{\displaystyle 0<t<\infty}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
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    Q10973
    Defining formula
    q ⁢ ( t ) 𝑞 𝑡 {\displaystyle{\displaystyle q(t)}}
    xml-id
    C2.S4.XMD16.m1cdec
    0 references
    Q10975
    Defining formula
    Q ⁢ ( z ) 𝑄 𝑧 {\displaystyle{\displaystyle Q(z)}}
    xml-id
    C2.S4.XMD6.m1cdec
    0 references
    Q10976
    Defining formula
    σ 𝜎 {\displaystyle{\displaystyle\sigma}}
    xml-id
    C2.S4.XMD7.m1adec
    0 references
     
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