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DLMF:5.11.E9 - MaRDI portal

DLMF:5.11.E9 (Q2829)

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DLMF:5.11.E9
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    Statements

    DLMF defining formula
    | Γ ⁡ ( x + i ⁢ y ) | ∼ 2 ⁢ π ⁢ | y | x - ( 1 / 2 ) ⁢ e - π ⁢ | y | / 2 , asymptotically-equals Euler-Gamma 𝑥 𝑖 𝑦 2 𝜋 superscript 𝑦 𝑥 1 2 superscript 𝑒 𝜋 𝑦 2 {\displaystyle{\displaystyle|\Gamma\left(x+iy\right)|\sim\sqrt{2\pi}|y|^{x-(1/% 2)}e^{-\pi|y|/2},}}
    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.m2addec
    0 references
    asymptotic equality
    DLMF defining formula
    ∼ asymptotically-equals {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.E1.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.m2addec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C5.S1.XMD5.m1dec
    0 references
    real variable
    DLMF defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C5.S1.XMD6.m1dec
    0 references

    Identifiers

     
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