DLMF:18.19.E8 (Q5838)

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DLMF:18.19.E8
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    Statements

    Defining formula
    w ⁢ ( x ) = w ( λ ) ⁢ ( x ; ϕ ) = | Γ ⁡ ( λ + i ⁢ x ) | 2 ⁢ e ( 2 ⁢ ϕ - π ) ⁢ x , 𝑤 𝑥 superscript 𝑤 𝜆 𝑥 italic-ϕ superscript Euler-Gamma 𝜆 imaginary-unit 𝑥 2 superscript 𝑒 2 italic-ϕ 𝜋 𝑥 {\displaystyle{\displaystyle w(x)=w^{(\lambda)}(x;\phi)=\left|\Gamma\left(% \lambda+\mathrm{i}x\right)\right|^{2}e^{(2\phi-\pi)x},}}
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    DLMF id
    DLMF:18.19.E8
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    constraint
    0 < ϕ < π 0 italic-ϕ 𝜋 {\displaystyle{\displaystyle 0<\phi<\pi}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
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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.m2abdec
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