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DLMF:5.9.E6 - MaRDI portal

DLMF:5.9.E6 (Q2671)

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DLMF:5.9.E6
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    Statements

    DLMF defining formula
    Γ ⁡ ( z ) ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ z ) = ∫ 0 ∞ t z - 1 ⁢ cos ⁡ t ⁢ d t , Euler-Gamma 𝑧 1 2 𝜋 𝑧 superscript subscript 0 superscript 𝑡 𝑧 1 𝑡 𝑡 {\displaystyle{\displaystyle\Gamma\left(z\right)\cos\left(\tfrac{1}{2}\pi z% \right)=\int_{0}^{\infty}t^{z-1}\cos t\mathrm{d}t,}}
    0 references
    constraint
    0 < ℜ ⁡ z < 1 0 𝑧 1 {\displaystyle{\displaystyle 0<\Re z<1}}
    0 references
    Symbols used
    gamma function
    DLMF defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
    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.m2aadec
    0 references
    cosine function
    DLMF defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
    0 references
    real part
    DLMF defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1abdec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C5.S1.XMD7.m1edec
    0 references

    Identifiers

     
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